The Lerch zeta function III. Polylogarithms and special values

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $\Phi(s, z, c)$ is obtained from the Lerch zeta function $\zeta(s, a, c)$ by the change of variable $z=e^{2 \pi i a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions defining the multivaluedness. For positive integer values s=m and c=1 this function is closely related to the classical m-th order polylogarithm $Li_m(z)$ We study its behavior as a function of two variables $(z, c)$ for special values where s=m is an integer. For $m \ge 1$ it gives a one-parameter deformation of the polylogarithm, and satisfies a linear ODE with coefficients depending on c, of order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of $Li_m(z).$


Introduction
In this paper we study the Lerch  under the change of variable z := e 2πia . The Lerch transcendent Φ(s, z, c) is called by some authors the "Lerch zeta function" (e.g. Oberhettinger [61]), although ζ(s, a, c) is the function originally studied by Lerch [49] in 1887. One obtains by double specialization at z = 1 and c = 1 the Riemann zeta function ζ(s) = Φ(s, 1, 1) = ∞ n=0 1 (n + 1) s , (1.3) and this expansion is valid in the half-plane Re(s) > 1.
In his 1900 problem list Hilbert [33] raised a question related to the Lerch transcendent. This question appears just after the 18-th problem, perhaps intended as a prologue to several of the subsequent problems. Hilbert remarked that functions that satisfy algebraic partial differential equations form a class of "significant functions" , but that a number of important functions seem not to belong to this class. He wrote: The function of the two variables s and x defined by the infinite series ζ(s, x) = x + x 2 2 s + x 3 3 s + x 4 4 s + ... which stands in close relation with the function ζ(s), probably satisfies no algebraic partial differential equation. In the investigation of this question the functional equation x ∂ζ(s, x) ∂x = ζ(s − 1, x) will have to be used.
The function ζ(s, x) is sometimes called Jonquiére's function because it was studied in 1889 by de Jonquiére [35]. 1 It is obtained as ζ(s, x) = xΦ(s, x, 1), where Φ(s, x, 1) is from the specialization of the Lerch transcendent at value c = 1. In 1920 Ostrowski [63] justified Hilbert's assertion by proving that ζ(s, x) satisfies no algebraic differential equation. Further work done on this question is discussed in Garunkštis and Laurenčikas [25].
The Lerch transcendent Φ(s, z, c), which has an extra variable c, circumvents Hilbert's objection and belongs to Hilbert's class of "significant functions" . This comes about as follows. We introduce the two linear partial differential operators with polynomial coefficients so that it is a "significant function" in Hilbert's sense. We comment more on this linear PDE below. The Lerch transcendent yields classical polylogarithms under suitable specialization of its variables (up to an inessential factor). Taking s = m a positive integer, and further taking c = 1, yields a function closely related to the m-th order (Euler) polylogarithm z n+1 (n + c) s , (1.9) and call it the extended polylogarithm. This function interpolates all polylogarithms via the parameter s; simultaneously it gives a deformation of the polylogarithm with deformation parameter c. For nonpositive integers s = −m ≤ 0 it is known that the resulting function Li −m (z, c) is a rational function of the two variables (z, c); we term it the negative polylogarithm of order −m. The Lerch transcendent has a connection with mathematical physics. The first author will show in [44] that the Lerch zeta function has a fundamental association with the real Heisenberg group. The relation is visible in the commutation relations satisfied by the operators (1.5) and (1.6). We may reformulate the linear PDE that the Lerch transcendent satisfies, using the modified operator This linear partial differential operator formally has the xp-form suggested as a possible form of a Hilbert-Polya operator encoding the zeta zeros as eigenvalues (Berry and Keating [8], [9]); more details are given in Section 9. The work of this paper determines new basic analytic properties of this function, which may bring insight to its specializations such as the Riemann zeta function and the polylogarithms. We construct an analytic continuation of the Lerch transcendent in all three variables, revealing its fundamental character as a multivalued function. We give an exact determination of its multivaluedness, specified by monodromy functions, and determine the effect of this multivalued analytic continuation on the partial differential equations and difference equations above. An important feature is that this analytic continuation does not extend to certain sets over a base manifold which we term singular strata; these form the branch locus for the multivaluedness. In the (z, c)-variables these are points where either c is a nonpositive integer and/or where z = 1 or z = 0. In particular this three-variable analytic continuation omits the specialization to the Riemann zeta function given in (1.3), which occurs at the singular stratum point (z, c) = (1, 1). The Lerch transcendent does possess additional analytic continuations in fewer variables valid on some singular strata outside the analytic continuation in three variables; for example the Hurwitz zeta function ζ(s, c) = ∞ n=0 (n + c) −s arises on (one branch of) the singular stratum z = 1 with c variable. These additional analytic continuations typically include meromorphic continuations in the s-variable to all s ∈ C. However for many parameter ranges these functions in fewer variables are not continuous limits of the three-variable analytic continuations.
A particular goal of the paper is to understand the relation of this analytic continuation to the multivalued structure associated with the polylogarithm. The specialization to obtain the polylogarithm takes c = 1 (and also s = m ≥ 1, a positive integer), which lies on a singular stratum over the Lerch zeta manifold M in the (a, c)-variables described in Section 1.1. In part II however we showed that the three-variable analytic continuation has removable singularities at the points c = n for n ≥ 1, and hence extends to a larger manifold M ♯ . The polylogarithm case c = 1 is therefore covered in this extended analytic continuation when projected to the manifold N ♯ in the (z, c)-variables described below.
The Lerch transcendent has the new feature that its analytic continuation introduces a new singular stratum consisting of the z = 0 manifold, which is a branch locus around which it is multivalued, whose monodromy must be determined. This singular stratum is not directly visible in the (a, c) variables used in the Lerch zeta function. Upon specializing two variables to obtain the polylogarithm in the z-variable, we obtain a new determination of its monodromy structure, and we also obtain an interesting one-parameter deformation of the polylogarithm in the c-variable.
An additional reason for interest in this c-deformation, apart from containing the polylogarithms, concerns the behavior of functional equations satisfied by the Lerch transcendent. A major property of the Lerch zeta function ζ(s, a, c) is that it satisfies three-term and four-term functional equations relating certain linear combinations of functions at parameter values s to those at 1 − s. The functional equation of the Riemann zeta function ζ(s) and ζ(1 − s) can be derived for these functional equations, proceeding by a limiting process to a singular stratum yielding functional equations for the Hurwitz zeta function (when Re(s) > 0) and for the periodic zeta function (when Re(s) < 1), and from these recovering that of the Riemann zeta function, see Apostol [3,Chap. 12]. The Lerch transcendent inherits multivalued versions of these functional equations, well-defined for all non-integer values of the c-parameter, but they fail to extend to the parameter values corresponding to polylogarithms, as we explain in Section 1.3.
The following Sections 1.1-1.5 discuss the results of this paper in general terms; and the main results are stated in detail in Section 2.
1.1. Analytic continuation in three complex variables. We establish an analytic continuation of the Lerch transcendent in three complex variables (s, z, c) as a multivalued function of the variables (z, c), which are entire functions of s. The special choice s = n, for n a positive integer, gives a one-parameter deformation of the n-th order polylogarithm.
The existence of the analytic continuation follows using the results of Part II, which gave a multivalued analytic continuation of the Lerch zeta function ζ(s, a, c), to a covering manifold of the manifold That paper also gave an extended analytic continuation to a covering manifold of In this paper we lift this continuation to the Lerch transcendent Φ(s, z, c) using the multivalued inverse change of variable to the map z = e 2πia . We choose a branch 2 of the logarithm that agrees with the usual definition in the upper-half plane, and on its boundary takes log(1) = 0, log(−1) = πi.
Our choice of base point on M ♯ is (s 0 , a 0 , c 0 ) = ( 1 2 , 1 2 , 1 2 ) and with this choice of branch the point z 0 = −1 lifts to a 0 = 1 2 . The base manifold now becomes and, in the extended analytic continuation case, The formulas for the monodromy functions describing the multivaluedness become correspondingly more complicated than those in Part II, and give a representation ρ s for fixed s ∈ C of the fundamental group , acting on a direct sum vector space spanned by the monodromy functions 3 . This vector space is generally infinite dimensional, but it degenerates at values where s is an integer, see Theorem 2.4.
Two main features of the analytic continuation are: (1) The Lerch transcendent becomes single-valued on a certain covering manifold of N , which is a regular covering (i.e. Galois covering) with solvable covering group. (2) The Lerch transcendent continues to satisfy the two independent differentialdifference equations (1.5) and (1.6) on the covering manifold.
We distinguish between the regular stratum, which are all those parameter values where the analytic continuation exists, and singular strata which correspond to parameter values at which the analytic continuation gives a branch point. That is, the analytic continuation does not apply to certain (complex) codimension one singular strata of (z, c) parameter values, which include all values where the Lerch zeta function formally becomes the Hurwitz zeta function or Riemann zeta function. The singular strata values are all (z, c) such that either z = 0 or z = 1 or c is a nonpositive integer, or both. On certain singular strata, continuous limiting values may exist for restricted ranges of the s-parameter, as illustrated in some results in Part I, e.g. [45,Theorem 2.3].
In parallel to results given in Part II, the analytic continuation above has removable singularities at positive integer values of c, and extends to an analytic continuation over the larger manifold N ♯ above.

Specializations and Fuchsian ODE's.
We study consequences of this analytic continuation for functions of fewer variables obtained by specializing the variables. These specializations include the n-th order polylogarithms, corresponding to a "nonsingular" specialization at c = 1, and s = n ≥ 1 a positive integer. The classical specializations, giving rise to the Hurwitz zeta function or Riemann zeta function, approach singular strata where the analytic continuation breaks down. Here limiting values do exist for some ranges of the singular strata parameters, and a better understanding of the nature of these degenerations seems of particular interest.
First, we consider specialization to the point c = 1. This value is a "nonsingular" value for the extended analytic continuation to N ♯ . We deduce the complete multivalued analytic continuation of the extended polylogarithm. This covers the case of Hilbert's example function in the variables (s, x) above. As already noted, this specialization loses the algebraic PDE property.
Second, we consider the specialization of variables that treats s as a constant. This specialization retains the linear PDE property in the (z, c)-variables, but loses the differentialdifference equation property that depends on variation in s. In the case of s = −m a non-positive integer, the functions are rational functions of two variables (z, c), which are polynomial in the variable c, and which remain well-defined on certain singular strata in c and z. For positive integers, i.e. s = m ∈ Z ≥1 , this specialization gives a one-parameter 3 The base point can be moved to x ′ s = (s, −1, 1 2 ) since the manifold N has a product structure splitting off the s-coordinate, in which it is simply-connected. deformation, with deformation parameter c, of the classical m-th order polylogarithm Li m (z), which corresponds to taking c = 1.
Third, specializing to integer values s = m ≥ 1 and additionally specializing c ∈ C to be fixed, the specialized function Li m (z, c) satisfies a linear ordinary differential equation of order m + 1 in the z-variable. This ODE is of Fuchsian type with regular singular points at {0, 1, ∞} for all values of the c-parameter; in particular this differential equation is defined for singular stratum parameter values c ∈ Z ≤0 . We determine the monodromy representation of the fundamental group π 1 (P 1 {0, 1, ∞}, −1) for this equation as a function of the deformation parameter c. The monodromy is unipotent for c ∈ Z, is quasi-unipotent for c ∈ Q, and otherwise lies in a Borel subgroup of GL(m + 1, C) but is not quasi-unipotent. A second interesting feature is that this deformation of the monodromy varies continuously on the regular stratum , but has discontinuous behavior of the monodromy representation at the singular strata values c ∈ Z ≤0 .
It is known that in the case c = 1 a mixed Hodge structure can be attached to the collection of polylogarithms for all n ≥ 1, viewed as pro-unipotent connection over P 1 {0, 1, ∞} as described in Bloch [11]. We have not addressed the question whether a mixed Hodge structure can be associated to the singular strata cases c ∈ Z ≤0 , where the monodromy is unipotent.

Functional Equations.
It is well known that the Lerch zeta function ζ(s, a, c) satisfies a functional equation relating parameter values s and 1 − s, which can be given in an asymmetric three-term form given by Lerch [49] as hold for all (s, a, c) in a simply-connected domain, the fundamental polycylinder Written out, the first functional equation (1.13) has four terms, and states, These functional equations are non-local in the (a, c)-variables, and it is important that they leave the fundamental polycylinder Ω invariant. One can combine two four-term functional equations in such a way as to recover the three-term functional equation for the Lerch zeta function valid on the fundamental polycylinder. The four term functional equations (1.13), (1.14) project to functional equations for the Lerch transcendent, but become complicated to state since they involve exponential and logarithmic changes of variable. If we let z = e 2πia then we find the projected image of the fundamental polycylinder Ω to the manifold N is the simply-connected domain with all variables in D N and complete these two functions with appropriate archimedean Euler factors as above then we will obtain four-term functional equations with all four terms lying in D N . We then obtain under analytic continuation four-term functional equations in multivalued functions of shapê provided that correct choices are made of branches of all the multivalued functions on each side of the equation. In principle the results of this paper permit simultaneous determination of the multivaluedness of the four terms in the functional equation following paths in N starting from the fundamental polycylinder Ω, but this paper does not carry out such a determination. We conclude the topic of functional equations by pointing out two important issues, which remain to be resolved.
(1) The two four-term functional equations for the Lerch transcendent are well-defined on the manifold N but are not well-defined on the extended manifold N ♯ . The extended manifold N ♯ glues in the integer values c = n ≥ 1 in the Lerch transcendent (s, z, c) parameters, and these extra values include exactly the value c = 1 relevant to studying polylogarithms. This obstruction to extension occurs because always at least one of the four terms in the functional equation lies on a genuine singular stratum. A consequence is that four-term functional equations do not appear when studying the polylogarithm itself. Problems also occur with extending the three-term functional equations.
It is possible that further information can be extracted from these functional equations at the polylogarithm values, if one approaches these points along specific paths for restricted ranges of parameter values. In the c-deformation of the polylogarithm we study, the multivalued functional equations relating values at s and 1 − s "turn on" when c takes a non-integer value. Perhaps some modified functional equations in fewer variables survive in the limit as a value c = n ≥ 1 is approached for suitable ranges of the s-variable, because continuous limits to singular strata exist for some range of s, as shown in part I. One may also ask whether there is a "vanishing cycle" interpretation for some of this limit behavior.
(2) On the other hand, at the polylogarithmic points c = m ≥ 1, new functional equations appear. Polylogarithms Li m (z) are well known to satisfy functional equations of quite different shape, specific to each value of m, sometimes relating different values of m shifted by integers. These functional equations are relevant to geometry and physics, and relate these functions at different values of z. For the Euler dilogarithm there is a well known functional equation found by Spence [70] in 1809, often given in the form cf. Rogers [69], Zagier [80], [81]. These particular functional equations have an important relation to three dimensional geometry, specifically to Cheeger-Cherns-Simons invariants of hyperbolic 3-manifolds, cf. Dupont [19], Neumann [60]. The functional equations and related ones for higher polylogarithms seem specific to integer values s = m > 1 of the s-parameter, and are not known to survive deformation in s. Nonetheless one may ask what is the fate of the functional equations of the dilogarithm under the c-deformation presented in this paper, which in (1.17) involve the integer values s = 2 and s = 1.
We remark that there are geometric generalizations of the dilogarithm to higher dimensional cases, which aim to preserve functional equations having a geometric meaning, see Gel'fand and MacPherson [26], Hain and MacPherson [31]. These higher-dimensional generalizations have more variables but seem not directly relatable to higher polylogarithms Li n (x).

Prior work.
There is a long history of work on analytic continuation of the Lerch transcendent. After Lerch's 1887 work, in 1889 Jonquière [35] studied the two variable function ζ(s, x) := ∞ n=1 x n n s , obtaining various contour integral representations and a functional equation, with s and x allowed to take some complex values; this is the function considered by Hilbert [33]. In 1906 Barnes [4] studied the Lerch transcendent with some restrictions on its variables, and noted some aspects of its multivalued nature. In the period 1900-2000 there was much further work on these functions obtaining analytic continuations in two of the variables, omitting one of either a or x (resp. c for the Lerch zeta function) while pursuing other objectives, such as functional equations, which we pass over here.
Concerning analytic continuation in three variables, in 2000 Kanemitsu, Katsurada and Yoshimoto [37] obtained an analytic continuation of the Lerch transcendent in three variables to a single-valued function on various large domains in C 3 . These authors also obtained formulas for special values at negative integers, related to those given below in §5. They did not address the issue of further analytic continuation to a multivalued function. In 2008 Guillera and Sondow [29] also give a single-valued analytic continuation of Φ(s, z, c) for certain ranges of (s, z, c), mostly restricting c to be real-valued. Very recently Costin and Garoufalidis [17] obtained a multivalued analytic continuation for the function ζ(x, s), calling it the "fractional polylogarithm" and denoting it Li α (x) = ∞ n=1 x n n α in variables (α, x) on a cover of C × (P 1 (C) {0, 1, ∞}); such a continuation appears here as a special case of Theorem 3.6. Vepstas [72] also obtained results applicable to analytic continuation of the fractional polylogarithm.
The detailed multivalued nature of the Lerch zeta function ζ(s, a, c) itself in all variables appears to have been first worked out in Part II ( [46]). We note that an old approach of Barnes [4] might be further developed to effect an analytic continuation of the Lerch transcendent in three variables.
Polylogarithms have their own independent history, as generalizations of the logarithm, and trace back to work of Euler [21], cf. [43,Sect. 2.4]. Much classical work on them is presented in the book of Lewin [51] and in the volume [52]. The appearance of the dilogarithm in many contexts in mathematics and physics is described in Zagier [80], [81] and Oesterlé [62]. It appears in the computation of volumes of hyperbolic tetrahedra, and from there to define invariants of hyperbolic manifolds, related to its functional equations, see Neumann [60]. Polylogarithms appear in the theory of motives, in iterated integrals and mixed motives, see the discussion in Bloch [11] and Hain [30]. Generalized polylogarithms given by iterated integrals are considered in Minh et all [54] and Joyner [36]. They appear in Beilinson's conjectures on special values of L-functions, in defining regulators ( [5]), cf. Beilinson and Deligne [6], Huber and Wildeshaus [34]. Geometric versions of polylogarithms have been formulated (Goncharov [27], [28], Cartier [12]).
There has been much other work on the Lerch  1.5. Present work. From the viewpoint of earlier work, the main point of this paper is to make an explicit study of the multivaluedness of the analytic continuation of the Lerch transcendent, and to determine the effects of this analytic continuation on its other algebraic structures.
(i) We determine explicit formulas for the monodromy functions and their behavior under specialization. On the conceptual side, these formulas illuminate a new way in which the the non-positive integer values s = −n ≤ 0 are "special values" of the associated functions, namely they are distinguished points in the s-parameter space in the sense that these are the unique values where all monodromy functions vanish identically, see Theorem 2.4. (ii) It is well known that the special values ζ(−n) (n ≥ 0) at negative integers are rational numbers whose arithmetic properties allow p-adic interpolation 4 which leads to the construction of p-adic L-functions. In Section 6 we show that at these special values s = −n one can recover information from nearby nonsingular strata values zΦ(−n, z, c) (taking limits c → 0 + ) that is sufficient to interpolate p-adic L-functions; this is achieved using periodic zeta function values. (iii) From the viewpoint of polylogarithms and iterated integrals, we show that under specialization this Lerch transcendent provides a complete set of solutions to a one-parameter Fuchsian deformation of the polylogarithm differential equation in the parameter c, and we determine its monodromy representation. This deformation of the polylogarithm may in future shed interesting light on its behavior.
The results of this paper suggest that further study be made of the limiting structure of functional equations in a neighborhood of the polylogarithm point c = 1. As noted in Section 1.3 there are two sources of functional equations, which relate these functions for different values of the s-parameter. The three-term and four-term functional equations come from number theory, and represent a generalization of the functional equations for the Riemann zeta function, relating functions with values s and 1 − s. These functional equations break down at the integer parameter values c = m ≥ 1. But exactly at those cvalues these functions satisfy many additional functional equations which relate function values at various s-parameter values shifted by integers, some of which have geometric meaning. It seems of interest to determine how these additonal functional equations deform in the c-parameter.
The extra variables in the Lerch transcendent potentially make visible new connections between these number-theoretic and geometric viewpoints. The variable z in the Lerch transcendent, added to the Hurwitz zeta function variable c, gives it the property of satisfying a linear partial differential equation, together with raising and lowering operators, whose form connects to mathematical physics. On the number theory side, the Lerch transcendent may potentially yield new information about the Hurwitz zeta function and the Riemann zeta function, even though these functions live on singular strata. This potentially may occur by explicit limiting processes (for certain parameter ranges), using also regularization methods, and perhaps through analysis of the indirect influence of its monodromy. In section 9 we suggest a number of other directions for further work.

Summary of main results
We obtain the analytic continuation and monodromy functions for the Lerch transcendent as a multivalued function defined on the complex 3-fold The universal coverÑ of N can be identified with homotopy classes [γ] of paths γ in N , and we refer to a point [γ] ∈Ñ , where the curves start at the fixed base point . The universal covers for other manifolds containing N and for covers of the extended manifold N ♯ are defined similarly.
Our notation used here for paths γ generalizes the notation used in part II, which was restricted to be a loop having γ(0) = γ(1) = x 0 = ( 1 2 , 1 2 , 1 2 ), with associated homotopy class [γ] ∈ π 1 (M, x 0 ). In part II we wrote Z(s, a, c, [γ]) to denote the function element centered at the endpoint γ(1), with (s, a, c) denoting local coordinates in a neighborhood of the endpoint of the loop γ. Reaching the point (s, a, c) from γ(1) can be thought of as following an additional path γ ′ from γ ′ (0) = γ(1) to γ ′ (1) = (s, a, c) that remains in a simply connected region obtained by cutting the manifold {(s, a, c) ∈ C×(C Z)×(C Z)} along the lines {a = m + it : t ≤ 0} for m ∈ Z and similarly in the c-variable. In this paper γ denotes a path, to be thought of as the analogue of the composed path γ • γ ′ , paths being composed left to right, as in Hatcher [32, p. 26]. Thus γ need not be a closed path. (2) The functionZ(s, z, c, [γ]) becomes single-valued on a two-step solvable regular (i.e. Galois) covering manifoldÑ solv of N , which can be taken to be the manifold fixed by the second commutator subgroup of π 1 (N , This is established in Section 3, where Theorem 3.4 and Theorem 3.5 give more detailed statements, which imply the result above. In particular we show that all monodromy functions vanish identically on a certain normal subgroup Γ ′ of π 1 (N , x ′ 0 ) which contains the second derived subgroup (second commutator subgroup) (π 1 (N , Our next result shows that the singularities at c ∈ Z ≥1 are removable, giving an analytic continuation to a solvable covering of N # , as follows (cf. Theorem 3.6).
In Section 4 we observe that the Lerch transcendent and its analytic continuation satisfies two differential-difference equations and a linear partial differential equation, as follows (cf. Theorem 4.1). (2) The Lerch transcendent Φ(s, z, c) satisfies the linear partial differential equation The analytic continuationZ(s, z, c, [γ]) satisfies this equation on the universal coverÑ of N .
) of the Lerch transcendent satisfies onÑ the two differential-difference equations and the linear partial differential equation (2.5).
As explained in Section 3.1, the fundamental group π 1 (N , x ′ 0 ) is the product of π 1 (P 1 (C) {0, 1, ∞}, −1), a free group on two generators [Z 0 ] and [Z 1 ], and π 1 (C Z, 1 2 ), a free group on generators [Y n ] for n ∈ Z. We next determine the structure of the vector spaces W s , for fixed s ∈ C, spanned by all the branches of the multivalued analytic continuation of the Lerch transcendent, over a neighborhood of a given point (s, z, c) ∈ N , specifying a set of generators for these spaces. We define the space W s to be a (generally infinite) direct sum of one-dimensional vector spaces given by particular monodromy generators, see Section 4 . There is a generic basis for s ∈ Z and for s ∈ Z there are linear relations among the generators, effectively reducing their number. The following result is established as Theorem 4.3.

Theorem 2.4. ( Lerch Transcendent Monodromy Space)
The Lerch transcendent monodromy space W s at s depends on the parameter s ∈ C as follows.
(i) (Generic case) If s ∈ Z, then W s is an infinite-dimensional vector space, and has as a basis the set of functions , then W m is an infinite-dimensional vector space, and has as a basis the set of functions In Sections 5-8 we specialize the Lerch transcendent variables (s, z, c) to cases where s = m is an integer. These are exactly the cases where the monodromy functions satisfy "non-generic" linear relations. We show that as functions of the two complex variables (z, c) we obtain further analytic continuation into the singular strata of the three-variable analytic continuation given in Section 3. Here the singular strata correspond to z = 0, 1 and/or c ∈ Z ≤0 . For convenience we state results in terms of the extended polylogarithm Li m (c, z) = zΦ(m, c, z).
In Section 5 we treat the case where s = −m ≤ 0 is a non-positive integer. Here we may note that Li −m (c, z) satisfies the ordinary differential equation of order m + 1 in the which is independent of z. It implies that Li −m (c, z) is necessarily a polynomial in c of degree at most m, having coefficients which are functions of z. It is known that these coefficients are rational functions of z; this follows from the observation of Apostol [2] in 1951 that the function Li −m (z, c) extends to a rational function in the (z, c) variables.
Here we determine formulas for these rational functions using the differential equation, as follows (Theorem 5.1).
analytically continues to a rational function of z and c on P 1 (C) × P 1 (C).

10)
in which the q m (z) are rational functions of z given by q 0 (z) = 1 1−z and We also determine recursion relations for these rational functions, and show they have a reflection symmetry z m+1 r m ( 1 z ) = r m (z). (Theorem 5.3). The rational function Li −m (c, z) takes well-defined values on the Riemann sphere for all (z, c) ∈ P 1 (C) × P 1 (C), and thus extends to the singular strata regions given by the complex hyperplanes c ∈ Z ≤0 , resp. z = 0. On the singular stratum z = 0 these functions take finite values, but on the 'singular stratum z = 1, they have a nontrivial polar part, and take the constant value ∞ ∈ P 1 (C).
In Section 6 we consider the double specialization when s = −m ≤ 0 is a nonpositive integer and c = 0. We show the function then agrees with the analytically continued value of the periodic zeta function F (a, s) := ∞ n=1 e 2πina n −s , when z = e 2πia lies on the unit circle (Theorem 6.1).
where the limit is taken through values of c in 0 < ℜ(c) < 1.
analytically continues to an entire function of s. In particular, for s = −m ∈ Z ≤0 there holds (2.13) This equality (2.12) is non-trivial because it involves a limiting procedure, since the point c = 0 lies in a singular stratum of the analytic continuation of the Lerch zeta function given in part II. This equality permits one to construct p-adic L-functions by interpolation from values of the Lerch zeta function. In contrast, it appears that one cannot recover values of the Hurwitz zeta function at s = −m directly from the Lerch transcendent by such a limiting procedure, letting a → 0 + or a → 1 − ; these limits do not exist.
In Section 7 we treat the one-variable specialization where s = m ≥ 1 is a positive integer. We state results in terms of the function of two variables Li m (z, c), observing that it satisfies the (slightly different) linear PDE (2.14) We obtain the following result as Theorem 7.1. This result gives an analytic continuation of Li m (z, c) to negative integer c, which are points that fall in the singular strata outside of the analytic continuation given in Section 3 In Section 8 we consider the double specialization, in which s = m ≥ 1 is a positive integer and c is fixed. Here we consider the functions Li m (z, c) of one variable z and show they have monodromy functions which are a deformation of the monodromy of the polylogarithm Li m (z), viewing c as a deformation parameter, with c = 1 giving the polylogarithm-this value of c is inside the three-variable analytic continuation given in §3. Namely we observe that F (z) := Li m (z, c) satisfies an ordinary differential equation In Theorem 8.1 we obtain the following result.

Theorem 2.8. (c-Deformed Polylogarithm Ordinary Differential Equation)
Let m ∈ Z ≥0 and let c ∈ C be fixed.
(1) The function F (z) = Li m (z, c) satisfies the ordinary differential equation of order m + 1.
(2) The operator D c m+1 is a Fuchsian operator for all c ∈ C. For each c ∈ C its singular points on the Riemann sphere are all regular and are contained in the set {0, 1, ∞}. ( We next study the monodromy representation of the fundamental group on the multivalued solutions of this differential equation. The associated monodromy representation is finite-dimensional, of dimension m + 1 independent of c ∈ C. We show that the image of the monodromy representation lies in a Borel subgroup of GL(m + 1, C), and lies in a unipotent subgroup exactly when c ∈ Z. The result splits into two cases, one for the "non-singular" strata values of c and the other for the singular strata values c ∈ Z ≤0 . This first case is given in Theorem 8.4, as follows.

Theorem 2.9. (c-Deformed Polylogarithm Monodromy-Nonsingular Case)
For each integer s = m ≥ 1 and each c ∈ C Z ≤0 , the monodromy action for D c m of The special case c = 1 of this result corresponds to the polylogarithm case considered by Ramakrishnan [65], [66], [67]. (We remark at the end of §8 on the issue of reconciling our formulas with those of Ramakrishnan.) We also obtain the monodromy in the singular strata cases c ∈ Z ≤0 , outside the analytic continuation in §3. Here the monodromy representation exhibits a discontinuous jump from the "non-singular" strata values, that seems unresolvable by a change of basis of (2.23) Remark on notation for logarithms. We will need two different versions of the restriction of the logarithm to a single-valued function on a cut plane. The principal branch log u makes a cut on the negative real axis, and the semi-principal branch Log u makes a cut on the positive real axis. In both cases the cut line is connected to the upper half-plane, so that The two branches differ in the values assigned to the logarithm in the lower half-plane. However if one starts from a base point in the upper half-plane of u, the multivalued extensions of these two functions agree. We use ℜ(s) and ℑ(s) for the real and imaginary parts of a complex variable s. Finally, compositions of paths are read left to right, so g • h with g(1) = h(0) means follow path g then follow path h, see Hatcher [32, p. 26].

Analytic continuation of Lerch transcendent
In this section we analytically continue the Lerch transcendent Φ(s, z, c) in the three complex variables (s, z, c) to a multivalued function and compute its monodromy functions, using the results of part II.
3.1. Analytic continuation of Lerch zeta function. We recall some details of the analytic continuation of the Lerch zeta function given in part II [46]. This analytic continuation first extended the Lerch zeta function to a single-valued function of three complex variables on the extended fundamental polycylinder This region is simply connected, and is invariant under the Z/4Z-symmetry of the functional equation (s, a, c) → (1 − s, 1 − c, a).
To understand the multivalued structure, for fixed s ∈ C we introduce the "elementary functions" which has coefficients c n (s) given by The right side of (3.2) converges conditionally for 0 < ℜ(s) < 1 for real 0 < a, c < 1 with 0 < ℜ(s) < 1. More generally the coefficients c n (s) are meromorphic functions, with simple poles at s ∈ Z ≥1 . The multivaluedness of the Lerch zeta function is exactly that inherited from the individual multivalued members of the right sides of (3.1) and (3.2), as we now explain. To obtain the analytic continuation in three variables, we fix as base point The full analytic continuation outsideΩ is obtained by extending these functions to a simply connected domain covering the whole region M by making a series of cuts in the a-plane and c-plane separately.
We recall from part II generators for the homotopy group π 1 (M, x 0 ). In the a-variable, the generators for π 1 (C Z, a = 1 2 ) are {[X n ] : n ∈ Z} in which X n denotes a path from base point a = 1 2 that lies entirely in the upper half-plane to the point a = n+ǫi, followed by a small counterclockwise oriented loop of radius ǫ around the point a = n, followed by return along the path. The generators [X n ] are pictured in Figure 1.  The monodromy of the individual elementary functions is simple. In the case of the c = n singularity a counterclockwise loop traversed k times in the c-plane ( with homotopy type [Y n ] k ) sends φ s n (a, c) → e 2πiks φ s n (a, c), while for the a = n singularity a counterclockwise loop traversed k times in the a-plane (corresponding to [X n ] k ) sends ψ s n (a, c) → e 2πik(1−s) ψ s n (a, c). In part II ([46, Theorem 4.5]) we establish that, the multivaluedness comes directly from the individual terms in (3.1) and (3.2). We obtain onΩ that For the loops [X n ] we obtain formulas that 5 can be shown to be equivalent to Z(s, a, c, [X n ] k ) = ζ(s, a, c) + (e −2πiks − 1)c n (s)ψ s n (a, c).  The cases s = m ≥ 1 correspond to polylogarithm parameter values.
A picture of the covering manifold structure over M is given in Figure 3 in Section 3.2.

Homotopy generators for Lerch transcendent.
We regard the Lerch transcendent variables (s, z, c) as lying on the manifold   To obtain an analytic continuationZ of Φ(s, z, c) toÑ we use paths starting from the base point This agrees with Φ(s, z, c) in a small neighborhood of x ′ 0 , so effects its analytic continuation to the universal coverÑ ≡M. Theorem 4.2 of part II ( [46]) now shows thatZ is single-valued onM ab .
We now specify a set of generators G ′ of π 1 (N , x ′ 0 ). The group π 1 (N , x ′ 0 ) has π 1 (M, x 0 ) as a normal subgroup with quotient group Z. In particular it contains more closed loops than π 1 (M, x 0 ), e.g. a path with endpoints (s, a, c) and (s, a + 1, c) in M projects to a closed path on N . Since N is a product manifold, we have .  For the c-variable in C Z we retain the generators [Y n ] given in Section 3.1, and we obtain the full set of generators  (3) The image group H 1 := (π ′ ) * (π 1 (M, x 0 )) in π 1 (N , x ′ 0 ) is given by (3.13) (4) The image group H 1 contains the commutator subgroup of π 1 (N , (3.14) In particular, H 1 is a normal subgroup of π 1 (N , x ′ 0 ), and M is an abelian Galois covering of N .
Proof. (1). This is established by checking that images under π ′ of certain paths in the a-plane result in loops homotopic to [Z 0 ] (resp. [Z 1 ]) in the z-plane. For Z 0 we consider a path from a = 1 2 to the point a = 3 2 that consists of a line segment in a-plane, from a = 1 2 to a = 3 2 , except for a small clockwise-oriented half-circle in Im(a) > 0 centered at a = 1, made to detour around the point a = 1. When projected to the z-plane, the curve proceeds from z = −1 in a counterclockwise circle of radius 1 around z = 0, with an indentation near z = 1 to leave z = 1 outside the loop. This image is clearly homotopic to Z 0 . These are pictured in Figure 5. (2). We assert that the homotopy class of the projection of Z 1 to the z-plane equals that of X 0 , see Figure 6. To verify this, note that the path X 0 given in part II is a closed path in the a-plane that first moves vertically from the base point a = 1 2 to a = 1 2 + iǫ 2π , (for small enough ǫ) then moves horizontally to a = iǫ 2π , then moves in a counterclockwise loop of radius ǫ 2π around a = 0 back to a = iǫ 2π , and finally returns to a = 1 2 following the original path. One may verify that, when projected to the z-plane, the image of X 0 moves along the z-axis from z = −1 to z = −e −ǫ , then proceeds in a clockwise half-circle to z = e −ǫ . Next, the image of the counterclockwise loop in the a-plane around a = 0 is which is a nearly circular path that encircles z = 1 counterclockwise, reaching at θ = π the point z = e ǫ > 1 on the real axis, with the second half of its path from θ = π to θ = 2π being the reflection of the first half in the real axis. Then it returns to z = −1 along the outgoing path. This path is clearly homotopic to Z 1 , whence [Z 1 ] = [X 0 ]. This is pictured in Figure 6.  in which the homotopy class [X n ] is given by a path X n in the a-plane with basepoint a = 1 2 , holding s = 1 2 , c = 1 2 fixed throughout, that traverses a line segment in the upperhalf plane to a = n + ǫi, followed by a counterclockwise loop of radius ǫ around a = n, followed by return along the line segment; similarly [Y n ] is given by a path Y n in the c-plane with basepoint c = 1 2 , holding s = 1 2 and a = 1 2 fixed throughout, that traverses a line segment in the upper-half plane to c = n + ǫi, followed by a counterclockwise loop of radius ǫ around c = n, followed by return along the line segment. Extending the argument of (2) we find that since the projection is constant. It follows that the image group H 1 := (π ′ ) * (π 1 (M, x 0 )) has generating set (3.13).
(4). To verify the inclusion (3.14), note first that since both generators , and since all [Y n ] ∈ H 1 we see that all generators of (π 1 (N , x ′ 0 )) ′ are in H 1 and the inclusion (3.14) follows.
Finally, since all subgroups of a group that contain its commutator subgroup are normal, we conclude from (3.14) that H 1 is a normal subgroup of π 1 (N , We regard D as embedded in the universal coverÑ ≡M by lifting it toΩ in M followed by the embedding ofΩ inM.
To describe the multivalued nature of the analytic continuation of the Lerch transcen-dentZ, we recall some definitions from part II.
where γ is an arbitrary path with basepoint γ(0) = x ′ 0 and τ γ is the composed path (first follow τ and then γ, as in [32, p. 26]). (3.23) In the sequel we will need two different branches of the logarithm, defined as follows. We let log z denote the principal branch of the logarithm, cut along the negative real axis, with the negative real axis itself viewed as belonging to the upper half-plane, so log 1 = 0, log(−1) = πi, log(−i) = − πi 2 . The semi-principal branch Log z of the logarithm is defined on the complex plane cut along the positive real axis, whose value at z = −1 is πi, and with the positive real axis connected to the upper half-plane, so Log (1) = 0, Log (−1) = πi, Log (−i) = 3πi 2 . We now specify certain functions that will appear as monodromy functions of the Lerch transcendent. For each integer n, we now define the function on the simply-connected domain D. Here z −c = e −cLog z , and a = 1 2πi Log z, but in (3.24) we evaluate the term (a − n) s−1 := e (s−1) log(a−n) using the principal branch of the logarithm, noting that a − n (resp. n − a) always has positive real part when (s, a, c) ∈Ω. The semi-principal branch Log z in the formula above is needed to apply on the domain z ∈ C R ≥0 used in D. The function (3.24) then extends to a function f n ([γ]) onÑ by analytic continuation.

27)
and In addition We can now apply the formula . We can find a lifting of the path in the class of π(N , x ′ 0 ) acting on monodromy functions. We show the actions commute, and further analyze their actions to show that certain monodromy functions vanish identically.  (3.37) to be the identity, so that Then the following hold.
Proof. In this proof the argument is carried out for each homotopy class [γ] separately. We regard it as fixed, and so abbreviate We know that the generators [Z 0 ] and [Z 1 ] of π 1 (M, x ′ 0 ) each commute with all generators [Y n ], but this is not in itself sufficient to imply (3.38). The extra facts to be used are that the [Z j ]-generators annihilate all [Y n ]-monodromy functions, and the [Y n ]generators annihilate all [Z j ]-monodromy functions. More precisely, since z −n (c − n) −s is single-valued in the z-variable for fixed s and c (for n ≤ 0), it follows from Theorem 3.4 that for j ∈ {0, 1} and all k, n ∈ Z. Also since f p (s, z, c) is single-valued in the c-variable for fixed s and z, it follows from Theorem 3.4 that Applying the induction hypothesis to the terms on the right side gives Next, the relations (3.42) and (3.43) imply that which completes the induction step, yielding (3.38). (M, x 0 ). The map π ′ is injective when restricted to this subgroup, hence (3.39) is a direct consequence of the formula given in Theorem 4.6 of part II when restricted to the subgroup G Y .
(3) We observe that the homomorphism from π 1 (N , given by (3.37) to the sum of the exponents of [Z 0 ] occurring in (3.37) has kernel H 0 generated by That is, we have the exact sequence  Consequently all monodromy functions ofZ vanish on the image group Γ ′ := (π ′ ) * (Γ).
We next observe that H 0 is a normal subgroup of π 1 (N , x ′ 0 ) because we have the inclusion of the commutator subgroup It follows that Γ ′ = [H 0 , H 0 ] is a normal subgroup of π 1 (N , x ′ 0 ) because it is a characteristic subgroup of the normal subgroup H 0 of the group (cf. Robinson [68, 1.5

.6 (3)]). In addition the inclusion
It follows that π 1 (N , The formulas in Theorem 3.5 together with those of Theorem 3.4 suffice to evaluate any monodromy function where the last equality follows from (3.38)
Proof. The fact that the possible singularities at c = n are removable follows almost immediately from the corresponding result in Part II [46,Theorem 2.3]. It remains only to check that the additional monodromy functions in Theorems 3.4 and 3.5 remain holomorphic at points (s, a, c) with c = n ≥ 1, inÑ # . This is manifested from their form given in Theorem 3.4, i.e. the only locations where they are possibly not holomorphic is points (s, z, c) with z = 0, 1 or c ∈ Z ≤0 .
Remark 3.7. The k-th lower central series group γ (k) (G) of a group G is defined recursively by Each quotient group G/γ (k) (G) is a nilpotent group. It can be shown that the group π 1 (N , is the group given in Theorem 3.5. It follows that the lower central series group γ (k) (π 1 (N , x ′ 0 )) ⊆ Γ ′ for all n ≥ 1. Thus the quotient group π 1 (N , x ′ 0 )/Γ ′ falls outside the framework studied by Deligne [18].

Differential-difference operators and monodromy functions
As discussed in part II, the Lerch zeta function ζ(s, a, c) satisfies two differentialdifference equations. We introduce the operators Then we have and D + L ζ(s, a, c) = −sζ(s + 1, a, c). (4.5) The Lerch transcendent Φ(s, z, c) then satisfies suitable differential-difference equations and a linear partial differential equation inherited from these. The substitution z = e 2πia transforms the Lerch differential operator to the polyzeta operator This lifts to an operator on functions on the universal coverÑ ≡M which is equivariant with respect to the group G of diffeomorphisms ofÑ that preserve the projection from N to N ; the group G is isomorphic to π 1 (N , x ′ 0 ).
Thus in the (s, z, c)-variables, the corresponding differential-difference equations satisfied by the Lerch transcendent are and ∂ ∂c Φ(s, z, c) = −sΦ(s + 1, z, c), (4.11) and the corresponding differential equation lifts to that given in (4.9) by analytic continuation.
(3) The monodromy functions satisfy the same differential-difference equations and differential equation because the differential operators z ∂ ∂z and ∂ ∂c are equivariant with respect to the covering map fromÑ to N .
We now study the restricted Lerch transcendent monodromy functions M s [τ ′ ] (Z) obtained by holding the variable s fixed, in a fashion analogous to §7 of part II [46].
where π :Ñ → N is the universal covering map. HereZ s := Q [id] (Z) is the restriction ofZ toÑ s . This vector space is a direct sum of one dimensional spaces: consisting of all finite linear combinations of the given countable generating set of vectors.
These vector spaces have the following properties. (i) (Generic Case) If s ∈ Z, then W s is an infinite-dimensional vector space, and has as a basis the set of functions (4.14) (ii) (Positive Integer Case) If s = m ∈ Z >0 , then W m is an infinite-dimensional vector space, and has as a basis the set of functions Proof. We establish these cases in reverse order.
is a nonzero multiple of f n | s=m , where f n at (s, z, c) ∈ ∆ is given by (3.24). Thus Each finite subset of the functions f n (s, z, c)| s=m together withZ s , is easily checked to be linearly independent in a small neighborhood of the point (m, −1, 1 2 ), so (ii) follows. (i) The proof in the generic case parallels that of Theorem 7.1 of part II ( [46]). It also makes use of the independence formula (3.38) of Theorem 3.5.
Remark 4.4. Theorem 4.3 shows that values s ∈ Z are "special values" in the sense that the monodromy functions satisfy non-generic linear relations at these values. We note the coincidence that these same points s ∈ Z are "special values" in the sense of number theory, in that the values of these functions at these points encode important arithmetic information, discussed below. Non-positive integer s = −m ≤ 0 are especially interesting because all monodromy functions vanish identically: therefore the values at these points are well-defined on the base manifold N , without having to lift to any covering manifold. This observation strengthens the observation made in part II of the vanishing property of monodromy for the Lerch zeta function, because we have a larger set of monodromy functions. The "well-definedness" property of these values seems particularly significant in that these values contain arithmetic information: p-adic L-functions can be obtained by p-adic interpolation through values at these points. More precisely, results in §6 establish:  is a c-deformed polylogarithm of negative integer order. As mentioned above, it is known that the functions Li −m (z, c) meromorphically continue as rational functions of two variables (z, c) ∈ C × C. These rational functions automatically give a meromorphic continuation in the (z, c)-variables to all integer points c ∈ Z; thus including the singular strata points c = −n ≤ 0 outside the analytic continuation of part II ( [46]).
In this section we determine recursions for these rational functions and deduce various symmetry properties they exhibit (Theorem 5.3) . We begin with the following expression for c-deformed negative polylogarithms Li −m (z, c). Here Li 0 (z, c) = zq 0 (z) and in which the q m (z) are rational functions given by Proof. The case m = 0 with c = 0, ∞ follows immediately from (1.9), Since this function is independent of c, it extends trivially to all c ∈ P 1 (C) as Li 0 (z, c). We next use the identity which yields, taking s = −m, This holds for c = 0, ∞ when z is in a sufficiently small circular neighborhood of the origin (with radius depending on both c and m). The fact that the function Li −m (z, c) has the form (5.2) for all m ≥ 1 with q j (z) as described is proved by induction on m ≥ 1 using (5.5), taking the differential equation (5.4) as the definition of q m+1 (z). The differential equation (5.4) shows that each q m (z) is a rational function of z.
In the remainder of this section we study these "special values" with s = −m ∈ Z ≤0 in more detail. One easily shows by induction on m that the rational function q m (z) has the form m r m (z) 1 z 2 z 2 + z 3 z 3 + 4z 2 + z 4 z 4 + 11z 3 + 11z 2 + z 5 z 5 + 26z 4 + 66z 3 + 26z 2 + z Table 1. Values of r m (z).
To evaluate the rational functions q m (z) we use the following result, which is due to Apostol [2, (3.1)]. The right side of (5.8) gives a meromorphic continuation of G(z, c; u) to (z, c, u) ∈ C 3 .
We now establish various properties of the rational functions q m (z) appearing in these special values. (1−z) k around z = 1 is given by (1−z) k . Substituting this expression for q m (z) into the left hand side of (5.14) and interchanging the order of summation over m and k yields We express the right hand side of (5.14) also as an infinite series in powers of 1 1−z and u, valid for |1 − e −u | < |1 − z|: Comparing the coefficients of 1 (5.15) and (5.16) gives the expression for a m,k . (ii) The proof of this identity also uses (5.14). More precisely, replacing z by 1 z in (5.14), we obtain This gives the relation Since q m (z) = r m (z) (1 − z) m+1 , the above identity can be rewritten as which proves (ii).
(iii) Dividing both sides of (5.12) by (1 − z) m+1 , we convert (5.12) to an equivalent form When m = 1, the right hand side is z 1−z q 0 (z) = z (1−z) 2 = q 1 (z). We shall prove (5.17) by induction on m. Assume it holds for some m ≥ 1. Rewrite the identity in this case as Differentiating both sides of (5.18) and using the identity (5.4) that q n+1 (z) = z ∂ ∂z (q n (z)) for n ≥ 0, we arrive at In other words, as desired.

Double specialization: Periodic zeta function
We now consider the double specialization of the Lerch transcendent setting s = −m a non-positive integer and setting c = 0. The importance of these "special values" s = −m of the s-parameter is that values of zeta and L-functions at these points have arithmetic significance; they encode information about the arithmetic structure of number fields. We have the problem that the c-parameter value c = 0 lies on the singular stratum outside the analytic continuation of Φ(s, z, c) given in Section 3. Our object is to show that sufficiently many of these "special values" can be recovered by a limiting process from "regular stratum" values of the Lerch transcendent so as to carry out the numbertheoretic construction of p-adic L-functions. We shall take a limit as c → 0, and to do this we set z = e 2πia and will also suppose that 0 < ℜ(a) < 1, so that z ∈ C R >0 .
Concerning the number theoretic significance of the values s = −m (m ≥ 0) is that on a singular stratum of the Lerch transcendent giving the Riemann zeta function these values are the important rational numbers where the B k are Bernoulli numbers. We follow the convention on Bernoulli numbers that they are defined by More The arithmetic information in the "special values" L(−m, χ) include p-adic regularities captured by interpolating these values p-adically to obtain p-adic L-functions. Two different interpolation methods to construct p-adic L-functions are known. A original construction of Kubota and Leopoldt [42] in 1964 used interpolation of Hurwitz zeta function values; it is presented in Washington [73,Sect. 5.2,Theorem 5.11]. A second approach was given in 1977 by Morita [55]. which uses interpolation of periodic zeta function values, at the points (z, a) with a = j p k having 0 < j < p k and with z = e 2πij p k . More information on the periodic zeta function approach to p-adic L-functions is given in Amice and Fresnel [1] and Naito [57].
We use the periodic zeta function approach to the "special values", where limits exist, rather than the Hurwitz zeta function approach, where the limits do not exist, see Remark 6.2 below. The following result shows that it is possible to take a limit as c → 0 + of zΦ(s, z, c), with z = e 2πia to recover the values at s = −m of the (analytically continued) periodic zeta function n s .
That is, this limit exists, and these values F (z, s) are extractable from data in the nonsingular part of the analytic continuation of Φ(s, z, c) at z = −m, when approaching the singularity. We now suppose a is real and apply the results from part I ( [45]) on the limiting behavior of the Lerch zeta function ζ(s, a, c) approaching the boundary of • . Namely, for ℜ(s) < 0, Theorem 2.3 (iii) and equation (2.13) of [45] give for 0 < a < 1 that where the limits are taken over real 0 < c < 1. Finally, the validity of (6.2) extends from real a to 0 < ℜ(a) < 1 by uniqueness of analytic continuation in the a-variable. .  , c), as the parameter a = 1 2πi logz has a → 0 (resp. a → 1). Under a variable change this limit corresponds to taking z → 1, which is a singular stratum value. As indicated by Theorem 6.1 above we can indirectly access Hurwitz zeta function values by expressing them as a linear combination of periodic zeta function values, see Apostol [3,Theorem 12.3]. The periodic zeta function values are obtainable as limiting values using Theorem 6.1.

Specialization of Lerch transcendent: s a positive integer
In this section we treat specialization of variables related to polylogarithms, where we specialize the s-parameter to be a positive integer value s = m ≥ 1. We will state results in terms of the extended polylogarithm (1) The factor of z 2 at the front of the differential operator D c m+1 is included so that it will have polynomial coefficients, and belong to the Weyl algebra C[z, d dz ], as shown in Lemma 8.2. We check, for m ≥ 1, where k=0 c k (z) d k dz k , which can occur only at z = 0, 1. The condition for a singular point of first kind (which implies regular singular point) at z 0 ∈ C is that, for 0 ≤ j ≤ m + 1, the order of the pole at z 0 of the coefficient c k (z) is at most m + 1 − k. Now (8.12) shows that this condition holds at z = 0, 1. The necessary and sufficient condition for at most a regular singular point at z = ∞ is that, for 0 ≤ k ≤ m + 1, each c k (z) has a zero of order at least m + 1 − k at z = ∞. This clearly holds in (8.12) as well. Thus the singular points are always a subset of {0, 1, ∞} and each point is either nonsingular or else a regular singular point is regular, for all c ∈ C. Thus the differential operator is Fuchsian for all c ∈ C. (In fact z = 1 is not a singular point for m = 0 and all c ∈ C, but when m ≥ 1 all three points 0, 1, ∞ are genuine singularities for all values of c.) (3) For integer k ≥ 0 we have It follows that {z 1−c (log z) j : 0 ≤ j ≤ m − 1} are annihilated by z d dz + c − 1 m hence by D c m+1 . This shows that the m + 1 functions listed above in B m+1,c are all in the solution space of D c m+1 . It remains to check that they are linearly independent over C. The functions z 1−c (log z) k are well-defined solutions for all c ∈ C, they are linearly independent from powers of the logarithm, and none of them have a singularity at z = 1.
For c ∈ C Z ≤0 the function Li m,c (z) is a well-defined solution, and it does have a singularity there since it diverges approaching this point along the real axis z → 1 + . We conclude that these m + 1 functions are a basis of solutions of the differential equation (8.2).
(4) The values c = −k ∈ Z ≤0 are singular strata values of the three-variable analytic continuation. To see that D c m+1 Li * m,−k (z) = 0, observe that Combining these results gives which gives the result. Linear independence over C is verified as in (3). .
We now compute the monodromy representation of this differential equation. We view Li m,c (z) := zΦ(m, z, c) as a multivalued function of the variable z on the domain P 1 (C) {0, 1, ∞}. The fundamental group is a free group on two generators. It acts on the universal coverC 0,1,∞ by deck transformations. We treat the "non-singular" case that c ∈ C Z ≤0 and the "singular" case c ∈ Z ≤0 separately. Definition 8.3. Let W m+1,c denote the (m + 1)-dimensional complex vector space of functions on P 1 (C) {0, 1, ∞} spanned by the vectors in B m+1,c for c ∈ C Z ≤0 (resp. B * m+1,c if c ∈ Z ≤0 ), making cuts along the real axis (−∞, 0) and (1, ∞). The monodromy representation of Li m,c (z) is the induced action ρ m,c : F 2 → Aut(W m+1,c ) ≃ GL(m + 1, C) . (8.13) The matrix representation on GL(m + 1, C) is obtained by viewing the entries of B m+1,c as a 1 × (m + 1) column vector.
Now we treat the "singular" case c ∈ Z ≤0 , and determine that the monodromy representation jumps discontinuously between the "nonsingular" and "singular" cases. In these cases the image of ρ m,c falls in a unipotent subgroup of GL(m + 1, C).
Proof. Recall that for c = −k ∈ Z ≤0 , We compute the monodromy by taking a scaling limit approaching the singular point. Letting c = −k + ε for positive ε, we assert that To verify this, we use the expansion

Further directions
There are many directions for further investigation; we discuss a few of them.
(1) Can one better understand the nature of the singularities on the singular strata? In particular, the Riemann zeta function is (formally) obtained as a limit function on a doubly-specialized singular stratum. Part I showed that there are obstructions to this limiting process: for example, limiting values approaching the singular stratum a = 1 in the (a, c)-variables exist only for Re(s) > 1. It is an interesting problem to obtain "renormalized" limits on singular strata for other ranges of the s-variable. In part I [45, Sect. 6] the authors gave a way to do this for the Lerch zeta function for one fixed singular stratum, by removing a small number of divergent terms. In this paper in Section 6 we showed that one can extract data approaching the singular stratrum at c = 0 + and negative integer s sufficient to reconstruct p-adic L-functions.
(2) The Lerch zeta function possesses additional discrete symmetries. One can define an action of a commuting family of (two-variable) "Hecke operators" in the (a, c) variables on the Lerch zeta function (resp. (z, c)-variables for the Lerch transcendent) for which ζ(s, a, c) (resp. Φ(s, z, c)) is a simultaneous eigenfunction, acting on various function spaces. Part IV ( [47]) of this series considers such operators on a function space with real variables. These additional discrete symmetries together with the differential equation (1.12) suggest that there should be an automorphic interpretation of the Lerch zeta function, made in terms of the related functions L ± (s, a, c). The first author has found such an interpretation for the real variables form treated in parts I and IV of this series, showing that symmetrized Lerch functions with characters are Eisenstein series on the real Heisenberg group quotiented by the integer Heisenberg group ([?]). It is an open problem to find an automorphic interpretation for the complex-analytic version of the Lerch zeta function (reap. Lerch transcendent) treated in part II and this paper.
(3). One can ask if results of this paper might be interpretable in the framework of an algebraic D-module over the Weyl algebra C[c, z, ∂ ∂c , ∂ ∂z ], which is associated to a parametric family of linear PDE's ∆ Φ − sI with with s as eigenvalue parameter. Such a reformulation may involve a non-holonomic Dmodule with an infinite set of singularities.
(4) What are the properties of the extension of polylogarithms under deformation in the c-variable? As mentioned in Section 1.3, one can ask whether functional equations such as the five-term relation for the dilogarithm might survive in some fashion under the c-deformation of the polylogarithm studied in Section 8. Specifically, the integer points c = m ≥ 2 and s = n ≥ 1 have maximally unipotent monodromy with apparent discontinuity in the monodromy matrices. One can ask whether the functions at these special points satisfy interesting identities in parallel fashion to the polylogarithms.
The dilogarithm is known to have a single-valued variant, the Rogers dilogarithm, obtained by adding a correction term. One may wonder if there exists analogous singlevalued variant of the extended function in the c-variable, or at specific integer points c = m ≥ 2.
(5) One may investigate generalizations of the Lerch transcendent in the direction of an "elliptic Lerch zeta function", made in analogy with work of Beilinson [7] and Levin [50]) on the elliptic polylogarithm.
(6) The partial differential operator ∆ Φ in (1.12) in the introduction can be viewed as an unbounded operator acting on functions restricted to the domain acting on the Hilbert space L 2 [0, 1] 2 , dadc , which is treated in [?, Sect. 9.2]. One may search for natural skew-adjoint "boundary conditions" on the operator ∆ Φ or ∆ L which yield operators having spectra on the line ℜ(s − 1 2 ) = 0. One such set of boundary conditions will be presented in [44,Sect. 9]; the spectrum of the resulting operator is purely continuous. It is an open question whether one can formulate natural geometric boundary conditions on ∆ Φ that will yield a Hilbert-Polya operator for ζ(s).
Labs-Research and the second author consulted there; they thank AT&T for support. The first author received support from the Mathematics Research Center at Stanford University in 2009-2010. The second author received support from the National Center for Theoretical Sciences and National Tsing Hua University in Taiwan in 2009-2014. To these institutions the authors express their gratitude.