Cotangent sums, quantum modular forms, and the generalized Riemann hypothesis

John Lewis,Don Zagier

doi:10.1007/s40687-018-0159-8

Abstract

We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.

Highlights

To answer immediately a question that the reader may be asking, we should say from the outset that the generalized Riemann hypothesis of the title refers only to the L-series associated with odd real Dirichlet characters and that we do not aim to prove it, but merely to show its equivalence to an asymptotic statement about the determinants of certain matrices whose entries cm,n are given by finite sums of cotangents of rational multiples of π
In the course of that talk he mentioned the homogeneity property c m, n = cm,n, leading the second author to ask whether the function C : Q → R defined by C(m/n) = cm,n/n might be related to a quantum modular form in the sense of [16]
We define the matrices CN and CN exactly as we did in the special case D = −4 (except for replacing the lower right-hand entry in CN in (3) by |D|N /π ), and from the scalar product calculations ξ, ξ = 1, ξ, Ga = L(1) a, Ga, Gb = a C(a/b), which are proved exactly as before, we deduce the same connection as in Theorems 1 and 3 between the unboundedness of the function R(N ) and the Riemann hypothesis for L(s, χ)

Summary

Introduction and main results

To answer immediately a question that the reader may be asking, we should say from the outset that the generalized Riemann hypothesis (just GRH) of the title refers only to the L-series associated with odd real Dirichlet characters and that we do not aim to prove it, but merely to show its equivalence to an asymptotic statement about the determinants of certain matrices whose entries cm,n are given by finite sums of cotangents of rational multiples of π. In the course of that talk he mentioned the homogeneity property c m, n = cm,n, leading the second author to ask whether the function C : Q → R defined by C(m/n) = cm,n/n might be related to a quantum modular form in the sense of [16]. The answer to this question turned out to be positive. The number defined by (7) is given as the following half-integral linear combination of cotangents of odd multiples of π/4n: 2n−1 hm,n = χ (k) S km n cot πk 4n (14).

The square-wave function G and its multiplicative shifts

Proof of Proposition 7

The function H on the real line

Generalization to other odd Dirichlet characters

Findings

Modular forms are everywhere