On Zeros of Certain Cusp Forms of Integral Weight for Full Modular Group

Abstract

We prove that the cusp form $\sum ^{\prime }\;(c_1z+d_1)^{-(k-\sigma )} (c_2z+d_2)^{-\sigma }$ of weight $k\ge 280$ for the full modular group, where the sum is taken over the nonzero pairs of integers $(c_{j},d_{j})$ for $j=1,2$ with $c_{2}d_{1}\not =c_{!}d_{2}$, and $\frac{55}{4}< \sigma < \frac{k}{20}$, has at least $\lfloor \frac{k-2\sigma }{6\sigma }\rfloor -2$ zeros on the arc A = $\{e^{i \theta }\,|\, \dfrac{\pi }{2}< \theta < \dfrac{2\pi }{3}\}$.

Dedicated to Professor Vijay Kumar Murty on the occasion of his 60th birthday