Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions

Abstract

In this and the succeeding paper I solve two problems suggested by Prof. Hardy, namely (1) that of proving that Ramanujan's functionhas no zeros on the line and (2) that of finding an asymptotic formulawhere A is a constant. I also prove similar results concerning the coefficients of general modular forms. I am indebted to Prof. Hardy and Mr Ingham for various suggestions, and in particular to Mr Ingham's paper, “A note on Riemann's ζ-function and Dirichlet's L-functions”.

Ω-Result on Coefficients of Automorphic L-Functions Over Sparse Sequences

“…Hafner and Ivić [4] obtained an O-estimate and Ω ± -results for n≤x λ f (n). The second moment n≤x |λ f (n)| 2 was treated in Rankin [16] and Selberg [21]. Subsequently, Rankin [17,18,19] initiated the theme of lower and upper estimates for the power moments n≤x |λ f (n)| 2β for β > 0.…”

Multiplicative functions and Ramanujan's τ-function

“…It was conjecture by Ramanujan and proved by Mordell (1917) that r{n) is multiplicative. With g(n) = (|T(«)|/T 1 1 / 2 ) 2 we may deduce Theorem 1 from Theorem 2 and Rankin's (1934) Let us for the moment assume the validity of the Sato-Tate conjecture that as p varies the 9 p are distributed over this interval with a probability density 2(Sin 0 ) 7 77. Then…”

О Нулях Линейных Комбинаций $L$-Функций Гекке На Критической Прямой

“…[34]). Отметим, что в упомянутой работе [34] доказано утверждение о том, что функция ( ) имеет мероморфное продолжение на всю комплексную плос-кость с полюсом первого порядка в точке = 1. § 5.…”