Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions
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DOI：
10.1017/s0305004100021095
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发表时间：
1939-07
影响因子：
0.8
通讯作者：
R. Rankin
中科院分区：
文献类型：
--
作者：
R. Rankin
在这篇和随后的文章中，我解决了Hardy教授提出的两个问题，即(1)证明Ramanujan函数在直线上没有零和(2)找到一个A为常数的渐近公式。关于一般模形式的系数，我也证明了类似的结果。我非常感谢Hardy教授和Ingham先生的各种建议，特别是Ingham先生的论文“关于黎曼的ζ函数和狄利克雷的l函数的注释”。
In this and the succeeding paper I solve two problems suggested by Prof. Hardy, namely (1) that of proving that Ramanujan's function has no zeros on the line and (2) that of finding an asymptotic formula where 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”.