An exact formula for a Lambert series associated to a cusp form and the Möbius function

Abstract

In 1981, Zagier conjectured that the constant term of the automorphic form $y^{12}|\Delta (z)|^2$, that is, $a_0(y):=y^{12} \sum _{n=1}^{\infty } \tau ^2(n) \exp ({-4 \pi ny}),$ where $\tau (n)$ is the nth Fourier coefficient of the Ramanujan cusp form $\Delta (z)$, has an asymptotic expansion when $y \rightarrow 0^{+}$ and it can be expressed in terms of the non-trivial zeros of the Riemann zeta function $\zeta (s)$. This conjecture was settled by Hafner and Stopple, and later Chakraborty, Kanemitsu, and the second author have extended this result for any Hecke eigenform over the full modular group. In this paper, we investigate a Lambert series associated to the Fourier coefficients of a cusp form and the Möbius function $\mu (n)$. We present an exact formula for the Lambert series and interestingly the main term is in terms of the non-trivial zeros of $\zeta (s)$, and the error term is expressed as an infinite series involving the generalized hypergeometric series ${}_2F_1(a,b;c;z)$. As an application of this exact form, we also establish an asymptotic expansion of the Lambert series.