Indefinite zeta functions

We define generalised zeta functions associated with indefinite quadratic forms of signature (g-1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g-1,1)$$\end{document}—and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(g-1,1)$$\end{document}—and we investigate their properties. These indefinite zeta functions are defined as Mellin transforms of indefinite theta functions in the sense of Zwegers, which are in turn generalised to the Siegel modular setting. We prove an analytic continuation and functional equation for indefinite zeta functions. We also show that indefinite zeta functions in dimension 2 specialise to differences of ray class zeta functions of real quadratic fields, whose leading Taylor coefficients at s=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=0$$\end{document} are predicted to be logarithms of algebraic units by the Stark conjectures.


Introduction
The Dedekind zeta functions of imaginary quadratic fields are specialisations of real analytic Eisenstein series. For example, for the Gaussian field K = Q(i) and Re(s) > 1, where E(τ , s) is the real analytic Eisenstein series given for Im(τ ) > 0 and Re(s) > 1 by E(τ , s) := 1 2 (m,n)∈Z 2 gcd(m,n)=1 Im(τ ) s |mτ + n| 2s . (1.2) Placing the discrete set of Dedekind zeta functions into the continuous family of real analytic Eisenstein series allows us to prove many interesting properties of Dedekind zeta functions-for instance, the first Kronecker limit formula is seen to relate ζ K (0) to the value of the Dedekind eta function η(τ ) at a CM point.
In this paper, we find a new continuous family of functions, called indefinite zeta functions, in which ray class zeta functions of real quadratic fields sit as a discrete subset. Moreover, we construct indefinite zeta functions attached to quadratic forms of signature (g − 1, 1). In the case g = 2, norm forms of quadratic number fields give the specialisation of indefinite zeta functions to ray class zeta functions of real quadratic fields.
Indefinite zeta functions have analytic continuation with a functional equation about the line s = g 4 . This is in contrast to many zeta functions defined by a sum over a cone-such as multiple zeta functions and Shintani zeta functions-which have analytic continuation but no functional equation. Shintani zeta functions are used to give decompositions of ray class zeta functions attached to totally real number fields, which are then used to evaluate those ray class zeta functions (and the closely related Hecke L-functions) at non-positive integers [21] (see also Neukirch's treatment in [17], Chapter VII §9). Our specialisation of indefinite zeta functions to ray class zeta functions of real quadratic fields is an alternative to Shintani decomposition that gives a different interpolation between zeta functions attached to real quadratic number fields. Indefinite zeta functions also differ from archetypical zeta functions in that they are not (generally) expressed as Dirichlet series.
Indefinite zeta functions are Mellin transforms of indefinite theta functions. Indefinite theta functions were first described by Marie-France Vignéras, who constructed modular indefinite theta series with terms weighted by a weight function satisfying a particular differential equation [22,23]. Sander Zwegers rediscovered indefinite theta functions and used them to construct harmonic weak Maass forms whose holomorphic parts are essentially the mock theta functions of Ramanujan [27]. Zwegers's work triggered an explosion of interest in mock modular forms, with applications to partition identities [4], quantum modular forms and false theta functions [8], period integrals of the j-invariant [6], sporadic groups [7], and quantum black holes [5]. Readers looking for additional exposition on these topics may also be interested in the book [3] (especially section 8.2) and lecture notes [20,26].
The indefinite theta functions in this paper are a generalisation of those introduced by Zwegers to the Siegel modular setting. Our generalised indefinite zeta functions satisfy a modular transformation law with respect to the Siegel modular group Sp n (Z).

Main theorems
Given a positive definite quadratic form Q(x 1 , . . . x g ) with real coefficients, it is possible to associate a "definite zeta function" ζ Q (s), sometimes called the Epstein zeta function: ζ Q (s) := (n 1 ,...,n g )∈Z g \{0} 1 Q(n 1 , . . . , n g ) s . (1.3) However, if Q is instead an indefinite quadratic form, the series in Eq. (1.3) does not converge. One way to fix this issue it to restrict the sum to a closed subcone C of the double cone of positivity {v ∈ R g : Q(v) > 0}. This gives rise to a partial indefinite zeta function ζ C Q (s) := (n 1 ,...,n g )∈C∩Z g 1 Q(n 1 , . . . , n g ) s . (1.4) However, unlike the Epstein zeta function, this partial zeta function does not satisfy a functional equation.
Our family of completed indefinite zeta functions do satisfy a functional equation, although they are not (generally) Dirichlet series. The completed indefinite zeta function ζ c 1 ,c 2 p,q ( , s) is defined in terms of the following parameters: • a complex symmetric (not necessarily Hermitian) matrix = = iM + N , with M = Im( ) having signature (g − 1, 1); • "characteristics" p, q ∈ R g ; • "cone parameters" c 1 , c 2 ∈ C g satisfying the inequalities c j Mc j < 0; • a complex variable s ∈ C.
Due to invariance properties, ζ c 1 ,c 2 p,q ( , s) remains well defined with several of the parameters taken to be in quotient spaces: • the characteristics on a torus, p, q ∈ R g /Z g ; • the cone parameters in complex projective space, c 1 , c 2 ∈ P g−1 (C).
The functional equation for the completed indefinite zeta function is given by the following theorem. In the case of real cone parameters, the completed indefinite zeta function has a series expansion that may be viewed as an analogue of the Dirichlet series expansion of a definite zeta function. It decomposes (up to -factors) as a sum of an incomplete indefinite zeta function ζ c 1 ,c 2 p,q ( , s), which is a Dirichlet series, and correction terms ξ c j p,q ( , s) that depend only on the cone parameters c 1 and c 2 separately. Theorem 1.2 (Series decomposition) For real cone parameters c 1 , c 2 ∈ R g , and Re(s) > 1, the completed indefinite zeta function may be written as and ξ c p,q ( , s) = Here, for any complex symmetric matrix , Q (v) = v v denotes the associated quadratic form; also, 2 F 1 denotes a hypergeometric function (see Eq. (6.1)). The summand in Eq. (1.7) should always be interpreted as 0 when sgn(c 1 Mn) = sgn(c 2 Mn); whenever it is nonzero, Re(Q −i (n)) > 0, and the complex power is interpreted as Q −i (n) −s = exp (−s log (Q −i (n))) where log is the principal branch of the logarithm with a branch cut along the negative real axis.
The series defining the incomplete indefinite zeta function ζ c 1 ,c 2 p,q ( , s) is a variant of the partial indefinite zeta function 1.4, which may be seen by writing it as where C + = {v ∈ R g : sgn(c 2 Mv) ≤ 0 ≤ sgn(c 1 Mv)} and C − = {v ∈ R g : sgn(c 1 Mv) ≤ 0 ≤ sgn(c 2 Mv)} are subcones of the two components of the double cone of positivity of Q M (v), and the notation * means that points on the boundary of the cone are weighted by 1 2 , except for n = 0, which is excluded. The indefinite zeta function is defined as a Mellin transform of an indefinite theta function (literally, an indefinite theta null with real characteristics, see Definition 5.1 and the definitions in Sect. 3). Indefinite theta functions were introduced by Sander Zwegers in his PhD thesis [27]. The indefinite theta functions introduced in this paper generalise Zwegers's work to the Siegel modular setting.
Our definition of indefinite zeta functions is in part motivated by an application to the computation of Stark units over real quadratic fields, which will be covered more thoroughly in a companion paper [13]. In special cases, an important symmetry, which we call P-stability, causes the ξ c 1 and ξ c 2 terms in Eq. (1.6) to cancel, leaving a Dirichlet series ζ c 1 ,c 2 p,q ( , s). In the 2-dimensional case (g = 2), this Dirichlet series is a difference of two ray class zeta functions of an order in a real quadratic field. Theorem 1.3 (Specialisation to a ray class zeta function) Let K be a real quadratic number field, and let Cl c∞ 1 ∞ 2 denote the ray class group of O K modulo c∞ 1 ∞ 2 (see Eq. (7.1)). For each class A ∈ Cl c∞ 1 ∞ 2 and integral ideal b ∈ A −1 , there exists a real symmetric matrix M of signature (1,1), along with c 1 , c 2 , q ∈ C 2 , such that Here, Z A (s) is the differenced ray class zeta function associated with A (see Definition 7.2).
This paper is organised as follows. In Sect. 2, we review the theory of Riemann theta functions, which we extend to the indefinite case in Sect. 3, generalising Chapter 2 of Zwegers's PhD thesis [27]. In Sects. 4 and 5, we define definite and indefinite zeta functions, respectively, and prove their analytic continuations and functional equations; in particular, we prove Theorem 1.1. In Sect. 6, we prove a general series expansion for indefinite zeta functions, which is Theorem 1.2. In Sect. 7, we prove that indefinite zeta functions restrict to differences of ray class zeta functions of real quadratic fields, which is Theorem 1.3, and we work through an example computation of a Stark unit using indefinite zeta functions.

Notation and conventions
We list for reference the notational conventions used in this paper.
• Non-transposed vectors v ∈ C g are always column vectors; the transpose v is a row vector.
• If is a g × g matrix, then is its transpose, and (when is invertible) − is a shorthand for −1 .
where f is any function taking values in an additive group.
• Unless otherwise specified, the logarithm log(z) is the standard principal branch with log(1) = 0 and a branch cut along the negative real axis, and z a means exp(a log(z)). • Throughout the paper, will be used to denote a g × g complex symmetric matrix.
We will often need to express = iM + N where M, N are real symmetric matrices. Matrices denoted by M and N will always have real entries, even when we do not say so explicitly.

Applications and future work
A paper in progress will prove a Kronecker limit formula for indefinite zeta functions in dimension 2, which specialises to an analytic formula for Stark units [13]. This formula may be currently be found in the author's PhD thesis [12]. While one application of indefinite zeta functions (the new analytic formula for Stark units) is known, we are hopeful that others will be found. We formulate a few research questions to motivate future work.
• Can one formulate a full modular transformation law for indefinite theta functions c 1 ,c 2 [f ](z, ) for some general class of test functions f ?
• (How) can indefinite theta functions of arbitrary signature, as introduced by Alexandrov, Banerjee, Manschot, and Pioline [1], Nazaroglu [16], and Westerholt-Raum [25], be generalised to the Siegel modular setting? What do the Mellin transforms of indefinite theta functions of arbitrary signature look like? (Note: Since this paper was posted, a preprint of Roehrig [19] has appeared that answers the first question by providing a description of modular indefinite Siegel theta series by means of a system of differential equations, in the manner of Vignéras.) • The symmetry property we call P-stability is not the only way an indefinite theta function can degenerate to a holomorphic function of ; there is also the case when M = i , the quadratic form Q M factors as a product of two linear forms, and the cone parameters are sent to the boundary of the cone of positivity. How do the associated indefinite zeta functions degenerate in this case? • What can be learned by specialising indefinite zeta functions at integer values of s besides s = 0 and s = 1?

Definite theta functions
In this expository section, we discuss some classical results on (definite) theta functions to provide context for the new results on indefinite theta functions proved in Sect. 3. Most of the results in this section may be found in [14], [15], or [18]. The expert may skip most of this section but will need to refer to back Sect. 2.3 for conventions for square roots of determinants used in Sect. 3.2. Definite theta functions in arbitrary dimension were introduced by Riemann, building on Jacobi's earlier work in one dimension. The work of many mathematicians, including Hecke, Siegel, Schoenberg, and Mumford, further developed the theory of theta functions (including their relationship to zeta functions) and contributed ideas and perspectives used in this exposition.
The definite theta function-or Riemann theta function-of dimension (or genus) g is a function of an elliptic parameter z and a modular parameter . The elliptic parameter z lives in C g , but may (almost) be treated as an element of a complex torus C g / , which happens to be an abelian variety. The parameter is written as a complex g × g matrix and lives in the Siegel upper half-space H g , whose definition imposes a condition on M = Im( ).

Definitions and geometric context
An abelian variety over a field K is a connected projective algebraic group; it follows from this definition that the group law is abelian. (See [15] as a reference for all results mentioned in this discussion.) A principal polarisation on an abelian variety A is an isomorphism between A and the dual abelian variety A ∨ . Over K = C, every principally polarised abelian variety of dimension g is a complex torus of the form A( where is in the Siegel upper half-space (sometimes called the Siegel upper half-plane, although it is a complex manifold of dimension 2 ). The complex structure on A(C) determines the algebraic structure on A over C; indeed, the map A → A(C) defines an equivalence of categories from the category of abelian varieties over C to the category of polarisable tori (see Theorem 2.9 in [15]). Concretely, theta functions realise the algebraic structure from the analytic. The functions (z + t; ) for representatives t ∈ C g of 2-torsion points of A(C) may be used to define an explicit holomorphic embedding of A as an algebraic locus in complex projective space. These shifts t are called characteristics. More details may be found in Chapter VI of [14], in particular pages 104-108. The positive integer g is sometimes called the "genus" because the Jacobian Jac(C) of an algebraic curve of genus g is a principally polarised abelian variety of dimension g. Not all principally polarised abelian varieties are Jacobians of curves; the question of characterising the locus of Jacobians of curves inside the moduli space of all principally polarised abelian varieties is known as the Schottky problem.

The modular parameter and the symplectic group action
The Siegel upper half-space has a natural action of the real symplectic group. This group, and an important discrete subgroup, are defined as follows.

Definition 2.4
The real symplectic group is defined as the set of 2g × 2g real matrices preserving a standard symplectic form.
where I is the g × g identity matrix. The integer symplectic group is defined by Sp 2g (Z) := Sp 2g (R) ∩ GL 2g (Z).
The real symplectic group acts on the Siegel upper half-space by the fractional linear transformation action We will show in Proposition 3.3 (specifically, by the case k = 0) that H g is closed under this action.

A canonical square root
On the Siegel upper half-space H g , det(−i ) has a canonical square root.
Proof Equation (2.5) holds for diagonal and purely imaginary by reduction to the one-dimensional case ∞ −∞ e −π ax 2 dx = 1 √ a . Consequently, Eq. (2.5) holds for any purely imaginary by a change of basis, using spectral decomposition.
Consider the two sides of Eq. (2.5) as holomorphic functions in g(g+1) 2 complex variables (the entries of ); they agree whenever those variables are real. Because they are holomorphic, it follows by analytic continuation that they agree everywhere. Definition 2.6 Lemma 2.5 provides a canonical square root of det(−i ): Whenever we write " det(−i )" for ∈ H g , we will be referring to this square root.
We will later need to use this square root to evaluate a shifted version of the integral that defines it.

Corollary 2.7
Let ∈ H g and c ∈ C g . Then, Proof Fix . The left-hand side of Eq. (2.7) is constant for c ∈ R g , by Lemma 2.5. Because the left-hand side is holomorphic in c, it is in fact constant for all c ∈ C g .
Note that, if ∈ H g , then is invertible and − −1 ∈ H g . The latter is true because The behaviour of our canonical square root under the modular transformation → − −1 is given by the following proposition.
Proof This follows from Definition 2.6 by plugging in = iI, because the function is continuous and takes values in {±1}, and H g is connected.

Proposition 2.9
The definite theta function for z ∈ C g and ∈ H g satisfies the following transformation law with respect to the z variable, for a + b ∈ Z g + Z g : The proof is a straightforward calculation. It may be found (using slightly different notation) as Theorem 4 on pages 8-9 of [18].

Theorem 2.10
The definite theta function for z ∈ C g and ∈ H g satisfies the following transformation laws with respect to the variable, where A ∈ GL g (Z), B ∈ M g (Z), B = B : (2) (z; + 2B) = (z; ).
Proof The proof of (1) and (2) is a straightforward calculation. A more powerful version of this theorem, combining (1)-(3) into a single transformation law, appears as Theorem A on pages 86-87 of [18]. To prove (3), we apply the Poisson summation formula directly to the theta series. The Fourier transforms of the terms are given as follows.
In the last line, we used Lemma 2.5 and Definition 2.6. Now, by the Poisson summation formula, If is replaced by − −1 , we obtain (3).
As was mentioned, it is possible to combine all of the modular transformations into a single theorem describing the transformation of under the action of Sp 2g (Z), This rule is already fairly complicated in dimension g = 1, where the transformation law involves Dedekind sums. The general case is done in Chapter III of [18], with the main theorems stated on pages 86-90.

Definite theta functions with characteristics
There is another notation for theta functions, using "characteristics," and it will be necessary to state the transformation laws using this notation as well. We replace z with z = p + q for real variables p, q ∈ R g . The reader is cautioned that the literature on theta functions contains conflicting conventions, and some authors may use notation identical to this one to mean something slightly different.

Definition 2.11
Define the definite theta null with real characteristics p, q ∈ R g , for ∈ H g : The transformation laws for p,q ( ) follow directly from those for (z; ).

Proposition 2.12
Let ∈ H g and p, q ∈ R g . The elliptic transformation law for the definite theta null with real characteristics is given by

Proposition 2.13
Let ∈ H g and p, q ∈ R g . The modular transformation laws for the definite theta null with real characteristics are given as follows, where A ∈ GL g (Z), B ∈ M g (Z), and B = B .

Indefinite theta functions
If we allow Im( ) to be indefinite, the series expansion in Eq. (2.2) no longer converges anywhere. We want to remedy this problem by inserting a variable coefficient into each term of the sum. In Chapter 2 of his PhD thesis [27], Sander Zwegers found-in the case when is purely imaginary-a choice of coefficients that preserves the transformation properties of the theta function.
The results of this section generalise Zwegers's work by replacing Zwegers's indefinite theta function ϑ c 1 ,c 2 M (z, τ ) by the indefinite theta function c 1 ,c 2 [f ](z; ). The function has been generalised in the following ways.
• Allowing a test function f (u), which must be specialised to f (u) = 1 for all the modular transformation laws to hold.
One motivation for introducing a test function f is to find transformation laws for a more general class of test functions (e.g. polynomials). We may investigate the behaviour of test functions under modular transformations in future work. However, for the purpose of this paper, only the cases u → |u| r will be relevant. We call a complex torus of the form

The Siegel intermediate half-space
Intermediate tori are usually not algebraic varieties. An example of intermediate tori in the literature are the intermediate Jacobians of Griffiths [9][10][11]. Intermediate Jacobians generalise Jacobians of curves, which are abelian varieties, but those defined by Griffiths are usually not algebraic. (In contrast, the intermediate Jacobians defined by Weil [24] are algebraic.) The symplectic group Sp 2g (R) acts on the set of g × g complex symmetric matrices by the fractional linear transformation action, Moreover, the H (k) g are the open orbits of the Sp 2g (R)-action on the set of g × g complex symmetric matrices.
Proof Trivial for Im( ) −1 , so Im(− −1 ) and Im( ) have the same signature. These three types of matrices generate Sp 2g (R). ( 2 ). For an appropriate choice of real symmetric B ∈ M g (R), we thus have A 1 A + B = 2 . That is, so 1 and 2 are in the same Sp 2g (R)-orbit.
Thus, the H (k) g are the open orbits of the Sp 2g (R)-action on the set of g × g symmetric complex matrices.

More canonical square roots
From now on, we will focus on the case of index k = 1, which is signature (g − 1, 1). The construction of modular theta series for k ≥ 2 utilises higher-order error functions arising in string theory [1]. More research is needed to develop the higher index theory in the Siegel modular setting.

Lemma 3.4
Let M be a real symmetric matrix of signature (g − 1, 1). On the region R M = {z ∈ C g : z Mz < 0}, there is a canonical choice of holomorphic function g(z) such that g(z) 2 = −z Mz.
Proof By Sylvester's law of inertia, there is some P ∈ GL + g (R) (i.e. with det(P) > 0) such that M = P JP, where The region S := {(z 2 , . . . , z g ) ∈ C g−1 : |z 2 | 2 + · · · |z g | 2 < 1} is simply connected (as it is a solid ball) and does not intersect there exists a unique continuous branch of the function 1 − z 2 2 − · · · − z 2 g on S sending (0, . . . , 0) → 1; this function is also holomorphic. For z ∈ R J , define This g J is holomorphic and satisfies g J (z) 2 = −z Jz, g J (αz) = αg J (z), and g J (e 1 ) = 1 where Conversely, if we have a continuous function g(z) satisfying g(z) 2 = −z Jz and g(e 1 ) = 1, it follows that g(αz) = αg(z), and thus g(z) = g J (z). Now, we'd like to define g M (z) := g J (Pz), so that we have g M (z) 2 = −z Mz. We need to check that this definition does not depend on the choice of P.
is a continuous square root of −z Jz sending e 1 to 1, so g J (Qz) = g J (z). Taking Q = P 2 P −1 1 and replacing z with P 1 z, we have g J (P 2 z) = g J (P 1 z), as desired.

Definition 3.5
If M is a real symmetric matrix of signature (g − 1, 1), we will write √ −z Mz for the function g M (z) in Lemma 3.4. We may also use similar notation, such We have now shown that A is positive definite, as it is positive definite on subspaces W and Cc, and these subspaces span C g and are perpendicular with respect to A.
Lemma 3.7 Let = N + iM be an invertible complex symmetric g × g matrix. Consider c ∈ C g such that c Mc < 0. The following identities hold: Proof Proof of (1): Proof of (2): Proof of (3): Note that det(I + A) = 1 + Tr(A) for any rank 1 matrix A. Thus, using (2) in the last step.
We can thus define det(−i ) as follows: where the square roots on the right-hand side are as defined in Definitions 2.6 and 3.5. This definition does not depend on the choice of c, because {c ∈ C g : c Mc < 0} is connected. where the integral may be taken along any contour from 0 to α. In particular, for the constant functions 1(u) = 1, set

Definition of indefinite theta functions
(3.33) When α is real, define E f (α) for an arbitrary continuous test function f : The function c 1 ,c 2 (z; ) = c 1 ,c 2 [1](z; ) is the function we are most interested in, because it will turn out to satisfy a symmetry in → − −1 . We will also show that the functions c 1 ,c 2 [u → |u| r ](z; ) are equal (up to a constant) for certain special values of the parameters.
Before we can prove the transformation laws of our theta functions, we must show that the series defining them converges.

Proposition 3.11
The indefinite theta series attached to f (Eq. (3.35)) converges absolutely and uniformly for z ∈ R g + iK , where K is a compact subset of R g (and for fixed , c 1 , c 2 , and f ).
Proof Let M = Im . We may multiply c 1 and c 2 by any complex scalar without changing the terms of the series Eq. where log p(n) = o n 2 . Thus, the terms of the series are o e − πε 2 ( n 2 + M −1 y ) , and so the series converges absolutely and uniformly for x ∈ R g and y ∈ K .

Transformation laws of indefinite theta functions
We will now prove the elliptic and modular transformation laws for indefinite theta functions. In all of these results, we assume that z ∈ C g , ∈ H (1) g , c j ∈ C g satisfying c j Im( )c j , and f is a function of one variable satisfying the conditions specified in Definition 3.10.

Proposition 3.12
The indefinite theta function attached to f satisfies the following transformation law with respect to the z variable, for a + b ∈ Z g + Z g : Proof By definition, (3.43) Because a ∈ Z g , Im(a) is zero and e(n a) = 1, so The identity is proved.   Proof The proof of (1) is a direct calculation.
by the change of basis m = An, so The proof of (2) is also a direct calculation.
where e n Bn = 1 because the n Bn are integers, and Im(B) = 0 because B is a real matrix. The proof of (3) is more complicated, and, like the proof of the analogous property for definite (Jacobi and Riemann) theta functions, uses Poisson summation. The argument that follows is a modification of the argument that appears in the proof of Lemma 2.8 of Zwegers's thesis [27].
We will find a formula for the Fourier transform of the terms of our theta series. Most of the work is done in the calculation of the integral that follows. In this calculation, M = Im , and z = x + iy for x, y ∈ C g . The differential operator x is a row vector with entries ∂ ∂x j , and similarly for n . Now compute, using Lemma 3.7, We have now shown that Define the following function on C g , suppressing the dependence of C(z) on and c. We have just shown that x C(z) = 0, so C(z + a) = C(z) for any a ∈ R g . By inspection, C(z + −1 b) = C(z) for any b ∈ R g . It follow from both of these properties that C(z) is constant. Moreover, by inspection, C(−z) = −C(z); therefore, C(z) = 0. In other words, (3.76) Now set g(z) := c 1 ,c 2 (z; ), which has Fourier coefficients By plugging in z −ν for z in Eq. (3.76) and multiplying both sides by e − 1 2 (z −ν) −1 (z − ν) , we obtain the following expression for the Fourier coefficients of g: It follows by Poisson summation that We obtain (3) by replacing with − −1 .

Indefinite theta functions with characteristics
Now we restate the transformation laws using "characteristics" notation, which will be used when we define indefinite zeta functions in Sect. 5.

P-stable indefinite theta functions
We now introduce a special symmetry that may be enjoyed by the parameters (c 1 , c 2 , z, ), which we call P-stability. In this section, c 1 , c 2 will always be real vectors.
Remarkably, P-stable indefinite theta functions attached to f (u) = |u| r turn out to be independent of r (up to a constant factor).  Now we use the P-symmetry to show that these two series are, in fact, equal. Note that Im(P z) = Im(z) because P z ≡ z mod Z 2 , so Moreover, Thus, we may substitute Pn for n in the first series (involving c 2 ) to obtain the second (involving c 1 ). We've now shown the periodicity relation where p Re(r) (n) is a polynomial independent of Im(r). Hence, c 1 ,c 2 r (z, ) is bounded on the line Re(r) = σ by n∈Z g p σ (n)e −πε n+M −1 y 2 . It follows that it is bounded on any vertical strip. Along with periodicity, this implies that c 1 ,c 2 r (z, ) as a function of r is bounded and entire, thus constant.

Definite zeta functions and real analytic Eisenstein series
We will now consider definite zeta functions-the Mellin transforms of definite theta functions-in preparation for studying the Mellin transforms of indefinite theta functions in the next section. In dimension 2, definite zeta functions specialise to real analytic Eisenstein series for the congruence subgroup 1 (N ) (which specialise further to ray class zeta functions of imaginary quadratic ideal classes).

Definition and Dirichlet series expansion
We define the definite zeta function as a Mellin transform of the indefinite theta null with real characteristics.

Definition 4.1 Let ∈ H (0)
g and p, q ∈ R g . The definite zeta function is (4.1) By direct calculation, ζ p,q ( , s) has a Dirichlet series expansion.
where Q −i (n + q) −s is defined using the standard branch of the logarithm (with a branch cut on the negative real axis).

Specialisation to real analytic Eisenstein series
Now, suppose g = 2, = iM for some real symmetric, positive definite matrix M, p = 0 0 , and q / ∈ Z 2 . Then the definite zeta function may be written as follows. If q ∈ Q 2 and the gcd of the denominators of the entries of q is N , this is essentially an Eisenstein series of associated with 1 (N ). Choose k, ∈ Z such that q ≡ k/N /N (mod 1) and gcd(k, ) = 1. Then, we have The Eisenstein series associated with the cusp ∞ of 1 (N ) is Here, ∞ 1 (N ) is the stabiliser of ∞ under the fractional linear transformation action; that (4.14) Combining Eqs. (4.8) and (4.12), we see that

Indefinite zeta functions: definition, analytic continuation, and functional equation
We now turn our attention to the primary objects of interest, (completed) indefinite zeta functions-the Mellin transforms of indefinite theta functions. We will generally omit the word "completed" when discussing these functions. As usual, let ∈ H (1) g , p, q ∈ R g , c 1 , c 2 ∈ C g , c 1 Mc 1 < 0, c 2 Mc 2 < 0. We define the indefinite zeta function using a Mellin transform of the indefinite theta function with characteristics.

Definition 5.1 The (completed) indefinite zeta function is
The terminology "zeta function" here should not be taken to mean that ζ c 1 ,c 2 p,q ( , s) has a Dirichlet series-it (usually) doesn't (although it does have an analogous series expansion involving hypergeometric functions, as we'll see in Sect. 6). Rather, we think of it as a zeta function by analogy with the definite case, and (as we'll see) because is sometimes specialises to certain classical zeta functions.
By defining the zeta function as a Mellin transform, we've set things up so that a proof of the functional equation Theorem 1.1 is a natural first step. Analytic continuation and a functional equation will follow from Theorem 3.14 by standard techniques. Our analytic continuation also gives an expression that converges quickly everywhere and is therefore useful for numerical computation, unlike Eq. (5.1) or the series expansion in Sect. 6.

Theorem 1.1
The function ζ c 1 ,c 2 p,q ( , s) may be analytically continued to an entire function on C. It satisfies the functional equation Proof Fix r > 0, and split up the Mellin transform integral into two pieces, Replacing t by t −1 , and then using part (3) of Theorem 3.14, the second integral is (Recall that scaling the c j does not affect the value of c 1 ,c 2 p,q ( ).) Putting it all together, we have As we showed in the proof of Proposition 3.11, the -functions in both integrals decay exponentially as t → ∞, so the right-hand side converges for all s ∈ C. The right-hand side is obviously analytic for all s ∈ C, so we've analytically continued ζ c 1 ,c 2 p,q ( , s) to an entire function of s. Finally, we must prove the functional equation. If we plug in Eq. (5.8), factor out the coefficient of the second term, and switch the order of the two terms, we obtain The functional equation now follows from Eqs. (5.9) and (5.10).
The formula for the analytic continuation is useful in itself. In particular, we have used this formula for computer calculations, as it may be used to compute the indefinite zeta function to arbitrary precision in polynomial time.

Corollary 5.2
The following expression is valid on the entire s-plane.

Series expansion of indefinite zeta function
In this section, we give a series expansion for indefinite zeta functions, under the assumption that c 1 and c 2 are real. Specifically, we write ζ c 1 ,c 2 p,q ( , s) as a sum of three series, the first of which is a Dirichlet series and the others of which involve hypergeometric functions. This expansion is related to the decomposition of a weak harmonic Maass form into its holomorphic "mock" piece and a non-holomorphic piece obtained from a "shadow" form in another weight.
To proceed, we will need to introduce some special functions and review some of their properties.

Hypergeometric functions and modified beta functions
Let a, b, c be complex numbers, c not a negative integer or zero. If z ∈ C with |z| < 1, the power series converges. Here we are using the Pochhammer symbol (w) n := w(w + 1) · · · (w + n − 1).

Proposition 6.1 There is an identity
valid about z = 0 and using the principal branch for Proof This is part of Theorem 2.2.5 of [2].
Using this identity, we extend the domain of definition of 2 F 1 (a, b; c; x) from the unit disc {|z| < 1} to the union of the unit disc and a half- z)) with the logarithm having a branch cut along the negative real axis. At the boundary point z = 1, the hypergeometric series converges when Re(c) > Re(a + b), and its evaluation is a classical theorem of Gauss. .
Of particular interest to us will be a special hypergeometric function which is a modified version of the beta function.
Proof To prove (1), we use the substitution t = u 1−u .
To prove (2), expand G(x; a, b) as a power series in x (up to a non-integral power).

The series expansion
We are now ready to prove Theorem 1.2, which we first restate here for convenience.

Theorem 1.2
If c 1 , c 2 ∈ R g , and Re(s) > 1, then the indefinite zeta function may be written as Proof Take the Mellin transform of the theta series term-by-term, and apply Lemma 6.6. Note that the series for ξ c p,q ( , s) converges absolutely, so the series may be split up like this.
The function ζ c 1 ,c 2 p,q ( , s) here is a Dirichlet series summed over a double cone, with any lattice points on the boundary of the cone weighted by 1 2 . The coefficients of the terms are ±e p n , where the sign is determined by whether one is in the positive or negative part of the double cone.

Zeta functions of ray ideal classes in real quadratic fields
In this section, we will specialise indefinite zeta functions to obtain certain zeta functions to obtain certain zeta functions attached to real quadratic fields. We define two Dirichlet series, ζ (s, A) and Z A (s), attached to a ray ideal class A of the ring of integers of a number field. The function Z A (s) is holomorphic at s = 1. Now, specialise to the case where K = Q( √ D) be a real quadratic field of discriminant D. Let O K be the maximal order of K , and let c be an ideal of O K . Let A be a narrow ray ideal class modulo c, that is, an element of the group Cl c∞ 1 ∞ 2 (O K ). We show, as promised in the introduction, that the indefinite zeta function specialises to the L-series Z A (s) attached to a ray class of an order in a real quadratic field. Theorem 1.3 For each A ∈ Cl c∞ 1 ∞ 2 and integral ideal b ∈ A −1 , there exists a real symmetric 2 × 2 matrix M, vectors c 1 , c 2 ∈ R 2 , and q ∈ Q 2 such that (2πN (b)) −s (s)Z A (s) = ζ c 1 ,c 2 0,q (iM, s). Let ε 0 be the fundamental unit of O K , and let ε (= ε k 0 for some k) be the smallest totally positive unit of O K greater than 1 such that ε ≡ 1 (mod c).
Choose any c 1 ∈ R 2 such that Q M (c 1 ) < 0. Let P be the matrix describing the linear action of ε on b by multiplication, i.e. ε(β n) = β (Pn). Set c 2 = Pc 1 .
We have also computed Z I (0) a different way in PARI/GP, using its internal algorithms for computing Hecke L-values. We obtained the same numerical answer this way.