Coefficients of modular forms
type: Item
instance of: scholarly article
title: Coefficients of modular forms
zbMATH Open document ID: 0283.10011
DOI: 10.1016/0022-247X(74)90074-2
published in: Journal of Mathematical Analysis and Applications
publication date: 1974-01-01T00:00:00Z
review text: Ramanujan's \(\tau\)-function is defined by \[ \sum_{n=1}^\infty \tau(n)x^n =x \prod_{n=1}^\infty (1 - x^n)^{24} \quad\text{ for } \vert x \vert < 1. \] The ``Ramanujan hypothesis'' is the conjecture that \(\vert \tau(n)\vert \le n^{11/2} d(n)\), where \(d(n)\) is the number of divisors of \(n\). In two earlier papers [J. Math. Anal. Appl. 30, 335--352 (1970; Zbl 0172.10601); ibid. 35, 285--311 (1971; Zbl 0188.34605)], the author gave a ``proof'' of the Ramanujan hypothesis, and a discussion of the Riemann hypothesis for certain modular forms based on his results on the Ramanujan hypothesis. Unfortunately there is a gap in his proof of the Ramanujan hypothesis. In the present paper, a formal proof of the Ramanujan hypothesis is studied, and the difficulties in the way of a rigorous argument are discussed. An estimate of the coefficients of modular forms is obtained which falls short of the Ramanujan hypothesis. [Note: The main result of this paper is now superseded by Deligne's proof of the Ramanujan hypothesis. ]
reviewed by: Leonard Y. H. Yap
Mathematics Subject Classification ID: 11F12
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 3443713
zbMATH Keywords: formal proof of Ramanujan hypothesis
zbMATH Keywords: Ramanujan's \(\tau\)-function
zbMATH Keywords: coefficients of modular forms
MaRDI profile type: Publication
cites work: A proof of the Ramanujan hypothesis
cites work: The Riemann hypothesis for modular forms
cites work: Jacobi spaces of entire functions
cites work: Modular expansions
cites work: Modular spaces of entire functions
author: Louis de Branges de Bourcia
description: scientific article; zbMATH DE number 3443713