An infinite series associated to the Rankin-Selberg L-Function

Abstract

Around four decades ago, Don Zagier speculated that the constant term of an automorphic form associated to the Ramanujan delta function has an asymptotic expansion. Moreover, he observed that it has a connection with the complex zeros of ⇣(s). This speculation was finally proved by Hafner and Stopple in 2000. Later in 2017, Chakraborty, Kanemitsu and Maji protracted this observation by taking any cusp form over SL2(Z). This thesis examine a similar infinite sum, namely,1 n=1 c2 f (n)n⌫/2 K⌫( pnx)where cf (n) represents nth Fourier coefficient of a cusp form f(z) and K⌫ represents the modified Bessel function of second kind with order ⌫. Interestingly, we also observe that this series has a connection with the complex zeros of ⇣(s).