Coefficients of modular forms
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 [Note: The main result of this paper is now superseded by Deligne's proof of the Ramanujan hypothesis. ]
scientific article; zbMATH DE number 3443713