A Kronecker limit formula for indefinite zeta functions

We prove an analogue of Kronecker’s second limit formula for a continuous family of “indefinite zeta functions”. Indefinite zeta functions were introduced in the author’s previous paper as Mellin transforms of indefinite theta functions, as defined by Zwegers. Our formula is valid in dimension g=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=2$$\end{document} at s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1$$\end{document} or 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}. For a choice of parameters obeying a certain symmetry, an indefinite zeta function is a differenced ray class zeta function of a real quadratic field, and its special value 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} was conjectured by Stark to be a logarithm of an algebraic unit. Our formula also permits practical high-precision computation of Stark ray class invariants.


Introduction
In a previous paper [5], we introduced indefinite zeta functions as Mellin transforms of certain indefinite theta functions associated to the intermediate Siegel half-space H (1) g , defined below.In this paper, we obtain a formula for the values of such an indefinite zeta function at s = 1 or s = 0, in the special case of dimension g = 2.Such formulas are traditionally called Kronecker limit formulas, after Kronecker's first and second limit formulas giving the constant term in the Laurent expansion at s = 1 of standard and twisted real analytic Eisenstein series.
When our parameters are specialised appropriately, our special value is a finite linear combination of Hecke L-values at s = 1.Our formula may be used to compute values of Hecke L-functions at s = 1 (resp.s = 0) relevant to the Stark conjectures, which we discuss in Section 1. 4.
For imaginary quadratic fields, Stark proved his conjectures using Kronecker's first and second limit formulas together with the theory of singular moduli [14].The Kronecker limit formulas give the constant Laurent series coefficient at s = 1 for families of Dirichlet series continuously interpolating the ray class zeta functions ζ(s, A)-namely, standard and twisted real analytic Eisenstein series (see [13] for details).We will discuss a generalisation of these Kronecker limit formulas, which correspond to the positive definite case, in [3].
Kronecker limit formulas applicable to real quadratic fields were developed by Hecke, Herglotz, and Zagier (in analogy with the first Kronecker limit formula), and by Shintani (in analogy with the second Kronecker limit formula).As in the imaginary quadratic case, these formulas are obtained by continuously interpolating between ray class zeta functions using a larger family of functions.Hecke's formula uses cycle integrals of real analytic Eisenstein series, whereas the formulas of Herglotz [2] and Zagier [21] (see also [1,8]) and the formulas of Shintani [10,11,12] use partial zeta functions defined by summing over a cone.Analogues of the Kronecker limit formulas in other settings have been found by Liu and Masri [6], Posingies [7], and Vlasenko and Zagier [20], among others.
The main theorem of this paper supplies a new real quadratic analogue of Kronecker's second limit formula based on a new interpolation between ray class zeta functions.The interpolation is by the indefinite zeta functions introduced in [5], which are Mellin transforms of nonholomorphic indefinite theta functions.Indefinite zeta functions have a nice functional equation, but they do not have a Dirichlet series representation for general parameters.
The main results on indefinite zeta functions-stated in Section 1.3-require a lot of notation, defined in Section 1.1 and Section 1.2.The proofs of the indefinite Kronecker limit formulas are provided in Section 2.
1.1.Notational conventions.We list some notational conventions used in the paper.
• e(z) := exp(2πiz) is the complex exponential, and this notation is used for z ∈ C not necessarily real.• H := {τ : Im τ > 0} is the complex upper half-plane.
• Non-transposed vectors v ∈ C g are always column vectors; the transpose v is a row vector.
, where f is any function taking values in an additive group.
). • We often express Ω = iM + N where M, N are real g × g symmetric matrices; N and M will always have real entries even when we do not say so explicitly.
We use complex logarithms throughout this paper.If f (τ ) is any nonvanishing holomorphic function on the upper half plane H, there is some holomorphic function (Log f )(τ ) such that exp ((Log f )(τ )) = f (τ ), because H is simply connected.Specifying a single value (or the limit as τ approaches some element of R ∪ {∞}) specifies Log f uniquely.It won't necessarily be true that (Log f )(τ ) = log(f (τ )).
Conventions for square roots, when not specified, follow [5].See Section 2.3 and Section 3.2 therein for details.
We recall the definition of the Siegel intermediate half-space, as defined in [5].
Definition 1.1.For 0 ≤ k ≤ g, we define the Siegel intermediate half-space of genus g and index k to be The H

(k)
g are the open orbits of the action of Sp 2g (R) by fractional linear transformations on the space of complex symmetric matrices.In particular, H (0) g is the usual Siegel upper half-space.1.2.Indefinite theta and zeta functions.We review the relevant definitions from [5].Definition 1.2.For any complex number α, define the function where the integral runs along any contour from 0 to α.
Definition 1.3.Let Ω = iM + N be a complex symmetric matrix whose imaginary part has signature (g − 1, 1); that is, Ω ∈ H g .Define the (nonholomorphic) indefinite theta function where Nonholomorphic indefinite theta functions were first studied by Vignéras [18,19] and were rediscovered by Zwegers [22].Zwegers's theta function is defined for real c j when N is a scalar multiple of M .More precisely, if M is real symmetric matrix of signature (g − 1, 1), τ ∈ H, and c 1 , c 2 ∈ R g , then Θ c 1 ,c 2 (M z, τ M ) is equal up to an exponential factor to the function ϑ c 1 ,c 2 M (z, τ ) introduced by Zwegers on page 27 of [22].Our theta functions extend Zwegers's to the Siegel modular setting; a related generalisation has also been studied by Roehrig [9].
We define the indefinite zeta function using a Mellin transform of the indefinite theta function with characteristics.
g .The completed indefinite zeta function is where p, q ∈ R g , and c 1 , c 2 ∈ C g are parameters satisfying c 1 M c 1 < 0 and c 2 M c 2 < 0.
The completed indefinite zeta function has an analytic continuation and satisfies a functional equation, which is Theorem 1.1 of [5].p,q (Ω, s)).The function ζ c 1 ,c 2 p,q (Ω, s) may be analytically continued to an entire function on C. It satisfies the functional equation (1.6) 1.3.Kronecker limit formulas for indefinite zeta functions.The Kronecker limit formula for indefinite zeta functions involves the dilogarithm function and a rapidly convergent integral of a logarithm of an infinite product.We also require the following definition of the function κ c Ω (v), which is the square root of a rational function and will appear as a factor in the integrand. . (1.7) We now state the formula.
where τ + (c) is in the upper half-plane and τ − (c) is in the lower half-plane.Then, where (1.10) The function ϕ p 1 ,p 2 : H → C is defined by the a product expansion, and its logarithm (Log ϕ p 1 ,p 2 ) (ξ) is the unique continuous branch with the property Here log(1 − e(p 2 )) is the standard principal branch.
The following specialisation looks somewhat simpler and contains all of the cases of arithmetic zeta functions Z A (s) associated to real quadratic fields.Theorem 1.9 (Indefinite Kronecker limit formula at s = 1, pure imaginary case).Let M be a 2×2 real matrix of signature (1, 1), and let where Here, Log ϕ p 1 ,p 2 and κ c Ω are defined as in the statement of Theorem 1.8, and ξ = τ (c) is the unique root of the quadratic polynomial It is straightforward to use the functional equation for the indefinite zeta function to rephrase Theorem 1.8 and Theorem 1.9 as formulas for ζ c 1 ,c 2 0,q (Ω, 0).
where ω + (c) is in the upper half-plane and ω − (c) is in the lower half-plane.Then, where Here, Log ϕ and κ are defined as in the statement of Theorem 1.8.
Theorem 1.11 (Indefinite Kronecker limit formula at s = 0, pure imaginary case).Let M be a 2 × 2 real matrix of signature (1, 1), and let where Here, Log ϕ and κ are defined as in the statement of Theorem 1.8, and ξ = ω(c) is the unique root of the quadratic polynomial 1 in the upper half plane.
The Hecke L-value L K (1, χ) contains arithmetic information that is not well-understood in general.The abelian Stark conjectures predict that this value is an algebraic number times a regulator Reg χ , which is a determinant of a matrix of linear forms in logarithms of algebraic units in a particular abelian extension of the number field K [14,15,16,17].This conjecture is known when the base field K is equal to Q or an imaginary quadratic field, but not (for instance) when K is a real quadratic field.
The rank 1 abelian Stark conjectures give a partial answer to Hilbert's 12th Problem, which asked for explicit generators for the abelian extensions of a number field in terms of special values of transcendental functions.Computationally, the Stark conjectures are used to calculate class fields in the computer algebra systems Magma and PARI/GP.
The Stark conjectures are most precisely formulated in the rank 1 case-that is, when L K (s, χ) vanishes to order 1 at s = 0.The regulator Reg χ in that case is a determinant of a 1 × 1 matrix.The Stark conjectures are most succinctly written as a statement about the ray class zeta function (of a ray ideal class A) rather that as a statement about the Hecke L-function Just as definite zeta functions specialise to ray class zeta functions of imaginary quadratic fields, indefinite zeta functions specialise to differenced ray class zeta functions of real quadratic fields.The full details of this specialisation are given in Section 7 of [5].Definition 1.12 (Ray class zeta function).Let K be any number field and c an ideal of the maximal order O K .Let S be a subset of the real places of K (i.e., the embeddings K → R).
Let A be a ray ideal class modulo cS, that is, an element of the group The function Z A (s) extends to a holomorphic function on the whole complex plane.The rank 1 abelian Stark conjecture says that Z A (0) is the logarithm of an algebraic unit.
Conjecture 1.14 (Stark [16]).Let K be a real quadratic field and {ρ 1 , ρ 2 } the real embeddings of K.If R is not the identity of Cl c∞ 2 (O K ), then Z A (0) = log(ρ 1 (ε A )) for an algebraic unit ε A generating the ray class field L c∞ 2 corresponding to Cl c∞ 2 (O K ).The units are compatible with the Artin map: ε The specialisation of the indefinite zeta function to a differenced real quadratic zeta function is given by the following result, which is Theorem 1.3 of [5].
We give an example of such a computation in Section 3.

Proof of the Kronecker limit formulas
The method of proof is to compute the Fourier series in ξ for an indefinite theta function with respect to an action by a one-parameter unipotent subgroup {T ξ } of SL 2 (R), then take a Mellin transform and specialise variables.After taking the Mellin transform, we must allow ξ to be a complex parameter and perform a fairly delicate contour integration.Unlike in the definite case, the Fourier coefficients of the indefinite theta are not elementary functions, which ultimately leads to a more complicated Kronecker limit formula.
We fix the following notation for this section.Let c 1 , c 2 ∈ C 2 satisfying c j M c j < 0, and consider the indefinite theta Θ c 1 ,c 2 p,q with characteristics p, q ∈ R 2 , as defined in Definition 1.4.
We will also need the following inequality.
Proof.We have Q Ω ( z 1 ) = ω 11 z 2 + 2ω 12 z + ω 22 , and the expressions for the roots come from the quadratic formula.

Some integrals involving E(u).
We will now prove a few integral formulas that we will need.Lemma 2.3.Suppose that α, β ∈ C satisfy Re (α 2 − 2iβ) > 0.Then, using the standard branch of the square root function, Proof.By integration by parts, where the contour C is a ray from the origin through the point α 2 − 2iβ.If z ∈ C with x = Re(z) > 0, s ∈ C with σ = Re(s) > 0, and [z 1 , z 2 ] denotes the oriented line segment from z 1 to z 2 , then lim Thus, in particular, C e −u u 1/2 du u = Γ 1 2 = π 1/2 .Plugging this into eq.(2.16) gives eq.(2.11).
As usual, let M = Im(Ω).Define the following auxiliary function, which will appear as a factor in the integral in the indefinite Kronecker limit formula.
(2.21) Also, set (2.23) In the case s = 1, we will leave out s and set We have the following formula for κ c Λ (v) for a particular Λ that will come up in the calculation.
The following lemma will be needed to evaluate certain integrals.

2.3.
Fourier series of a unipotent transform of an indefinite theta function.Consider the function of ξ ∈ R (although ξ will be allowed to be complex later on) and t ∈ R ≥0 , (2.27) Write this function as a Fourier series, (2.28) We are ultimately interested in the Mellin transform of this function, where, as we will show, (2.31) where h j (ξ, t) is the sum over the terms with n 2 = j, and h(ξ, t) is the sum over all the terms where n 2 = 0. Also, assume that q 1 = q 2 = 0.
First, calculate h 0 (ξ, t): (2.33) The n 1 = 0 term of this sum vanishes.We write, for n 2 = 0, (2.37) The exponential sum , and to 0 otherwise.Thus, for all Our convention here is that a sum over n 2 |k ranges over both positive and negative n 2 , and over all integers when k = 0.
2.4.Shifting the contour vertically.Fix a positive real number λ to be specified later.Let C + (C − ) be the contour consisting of the horizontal line Im(z) = λ (Im(z) = −λ), oriented towards the right half-plane.For each The integrands in eq. ( 2.38) approach zero as Re(ξ) → ±∞, so we may rewrite this formula using contour integrals n 2 e(−kξ) dξ.

2.5.
Taking Mellin transforms term-by-term.To calculate the Mellin transform of h 0 (ξ, t), we need to check absolute convergence to justify reversing the order of summation/integration.Proposition 2.7.
Proof.We bound the integral as follows.
The sum converges for σ > 1 2 , and the integral converges for σ > 0 (because the integrand approaches a constant at t → 0 and decays exponentially as t → ∞).
Therefore, we can switch the sum and the integral.Using Lemma 2.3 and dropping the subscript on n 1 , Next, we're going to calculate the Mellin transform of h(ξ, t).We need an absolute convergence result to justify our calculation here, too.Proposition 2.8.Suppose σ = Re(s) > 1 2 .Then, (2.49) The proposition is proved.Now we may justify taking the Mellin transform of the Fourier series term-by-term.It follows from Proposition 2.8 that where (2.57) Proposition 2.8 also implies that we can switch the order of integration to compute ,n 2 e(n 2 (p 1 ξ + p 2 ) − kξ) κ c 1 ,c 2 Ω (ξ, s) dξ.
(2.60) 2.6.Series manipulations.In this subsection, we set ξ = 0 in eq.(2.56).We will manipulate the right-hand side of this equation to prove Theorem 1.8.First of all, we have (2.62) We will rewrite the sum of the β k (s) using the substitution (d 1 , d 2 ) = ( k n 2 , n 2 ).The following manipulation is legal by Proposition 2.8.
Split up the series into four pieces.
e(d 2 p 2 ) (2.67) Now, move the contour integral outside the sums, and rewrite the series as We want to write this sum of logarithms as a logarithm of a product, but there is the issue of the choice of branch.In order to make a clear choice, let for ξ ∈ H.This is a function on the upper half-plane which is never zero, and the upper half-plane is simply connected, so it has a choice of continuous logarithm.Let (Log ϕ p 1 ,p 2 ) (ξ) be the branch such that lim ξ→i∞ Here log(1 − e(p 2 )) is the standard principal branch.Thus, (2.75) 2.7.Collapsing the contour onto the branch cuts.We could declare ourselves done at this point.Equation (2.73) is a formula for ζ c 1 ,c 2 p,0 (Ω, 1), as we desired, and it appears very difficult to evaluate or simplify the contour integral in any way.However, eq.(2.73) is not a useful formula for computation because the integral converges slowly.The integrand decays polynomially as ξ → ±∞ along the horizontal contour C + .We will obtain a Kronecker limit formula with rapid convergence by shifting the contour so that the integrand decays exponentially.In doing so, we will also split up the formula as a difference of a c 1 -piece and a c 2 -piece.The movement of the contour is shown in Section 2.7.
Let Λ 2 by Lemma 3.6 of [5], we know by Lemma 2.2 that we may choose τ 1 (c) to be in the upper half-plane and τ 2 (c) in the lower half-plane.
The complex function ξ → κ c Ω ξ 1 has branch cuts along the vertical ray from τ 1 (c) to i∞ and the vertical ray from τ 2 (c) to −i∞.We check that this function is holomorphic away from these branch cuts.Since κ c  .These residues do not depend on c, so they cancel, and κ c 1 ,c 2 Ω ξ 1 is holomorphic at r 1 and r 2 .Move the countours of integration above the zeros of Q Ω ±ξ 1 .Now we may safely split up the integral into a term for c 1 and a term for c 2 .Now we retract the integral onto the branch cut.As ξ = ±τ ± + ε and ε → 0, the denominator of the integrand blows up like ε 1/2 , so the integral converges.The integrand changes sign when we cross the branch cut.Thus, eq.This agrees (to 30 decimal digits) with the computations described in Section 7.1 of [5].The conjectural Stark unit is exp(Z I (0)) ≈ 3.89086171394307925533764395962.This number is appears to be the root of the polynomial

Theorem 1 . 6 (
Analytic continuation and functional equation for ζ c 1 ,c 2 and a is positive at each place in S} .(1.22)For Re(s) > 1, define the ray class zeta function of A to be ζ(s, A) := a∈A N (a) −s .(1.23)This function has a simple pole at s = 1 with residue independent of A. The pole may be eliminated by considering the function Z A (s), defined as follows.Definition 1.13 (Differenced ray class zeta function).Let R be the element of Cl cS (O K ) defined by R := {aO K : a ≡ −1 (mod c) and a is positive at each place in S}. (1.24)For Re(s) > 1, define the differenced ray class zeta function of A to be Z A (s) := ζ(s, A) − ζ(s, RA).(1.25)