An exact formula for a Lambert series associated to a cusp form and the Möbius function
type: Item
instance of: scholarly article
title: An exact formula for a Lambert series associated to a cusp form and the Möbius function
zbMATH Open document ID: 1496.11113
author: Abhishek Juyal
author: Bibekananda Maji
author: Sumukha Sathyanarayana
published in: The Ramanujan Journal
publication date: 2022-02-11T00:00:00Z
review text: In 1916, Hardy and Littlewood, correcting an identity due to Ramanujan involving the Möbius function \(\mu\), published the following reciprocity formula. Let \(\alpha\) and \(\beta\) be positive real numbers such that \(\alpha\beta=\pi\). Then, assuming that the nontrivial zeros \(\rho\) of the Riemann zeta function \(\zeta(s)\) are simple, \[ \sqrt{\alpha}\sum_{n=1}^{\infty} \frac{\mu(n)}{n}\exp\left(-\left(\frac{\alpha}{n}\right)^2\right) - \sqrt{\beta}\sum_{n=1}^{\infty} \frac{\mu(n)}{n}\exp\left(-\left(\frac{\beta}{n}\right)^2\right) = -\frac{1}{2\sqrt{\beta}} \sum_{\rho} \frac{\Gamma\left(\frac{1-\rho}{2}\right)\beta^{\rho}}{\zeta'(\rho)}, \] provided the terms in the sum on the right-side are bracketed according to a certain technical condition. Such reciprocity formulas are often of interest. Closely-related formulas appear in proofs of functional equations of \(L\)-functions, for example, but without a sum over zeros. The introduction of the paper under review contains a summary of analogues of this Hardy-Littlewood formula that have been derived over the years in various settings. These analogues, still assuming the simplicity of zeros and under additional convergence conditions when necessary, include an analogue when \(\mu\) is twisted by a primitive Dirichlet character \(\chi\), when \(\mu\) is twisted by a certain cosine function, and when \(\mu\) is replaced with \(\mu*\chi\) (the Dirichlet convolution), as well as an analogue in the setting of normalized Hecke eigenforms. The authors of the paper under review give a new analogue of this Hardy-Littlewood formula, for when \(\mu\) is replaced with \(\mu*a_f\) where \(a_f\) is the Fourier coefficient of a Hecke eigenform \(f\) of weight \(k\) over \(\textrm{SL}_2(\mathbb{Z})\). Assuming the simplicity of zeta zeros, and letting \(\mu_k(n)\) denote \(\mu(n)n^{k-1}\) and \({}_2F_1\) denote the usual hypergeometric series, Theorem 2.1 states \[\sum_{n=1}^{\infty} (\mu*a_f)(n) \exp(-ny) = \mathcal{P}(y) + 2…
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M26
Mathematics Subject Classification ID: 11N37
zbMATH DE Number: 7472670
zbMATH Keywords: cusp forms
zbMATH Keywords: Lambert series
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: non-trivial zeros
zbMATH Keywords: Rankin-Selberg \(L\)-function
reviewed by: Ghaith A. Hiary
MaRDI profile type: Publication
full work available at URL: s11139-020-00375-7
OpenAlex ID: W3126744507
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DOI: 10.1007/S11139-020-00375-7
describes a project that uses: Digital Library of Mathematical Functions
description: scientific article; zbMATH DE number 7472670