An exact formula for a Lambert series associated to a cusp form and the Möbius function
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 \[\sum_{n=1}^{\infty} (\mu*a_f)(n) \exp(-ny) = \mathcal{P}(y) + 2\Gamma(k)\left(\frac{i}{2\pi}\right)^k \sum_{n=1}^{\infty} \frac{(\mu_k * a_f)(n)}{n^k} \,\left[{}_2F_1\left(\frac{k}{2},\frac{k+1}{2};\frac{1}{2};-\frac{y^2}{4n^2\pi^2}\right)-1\right], \] where \[\mathcal{P}(y) = \sum_{\rho} \frac{L(f,\rho)\Gamma(\rho)}{\zeta'(\rho)}\frac{1}{y^{\rho}},\] and the last sum is bracketed according to a similar condition as in the Hardy-Littlewood formula. As an application of this identity, and under the additional assumption of the Riemann hypothesis, the authors establish the oscillatory behavior of the series on the left-side in the last display as \(y\to 0^+\). This oscillatory behavior is in analogy with previous such results involving the Ramanujan \(\tau\) function, and more generally in the setting of Hecke eigenforms.
scientific article; zbMATH DE number 7472670
scientific article