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On some new properties of the gamma function and the Riemann zeta function

✍ Scribed by Liangwen Liao; Chung-Chung Yang

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
126 KB
Volume
257
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

In this paper, we have exhibited, by utilizing value distribution theory, some new properties of the Gamma function Ξ“(z) and the Riemann zeta function ΞΆ(z). Specifically, we have proved that both of the two functions are prime and the Riemann zeta function, like Ξ“(z), does not satisfy any algebraic differential equation with coefficients in ℒ︁~0~. Moreover, the two functions do not satisfy any functional equation of the form P(Ξ“, ΞΆ, z) ≑ 0, where P(x, y, z) is a nonconstant polynomial in x, y and z.

πŸ“œ SIMILAR VOLUMES

Mean-Value Theorem of the Riemann Zeta-F
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✍ A. Sankaranarayanan; K. Srinivas πŸ“‚ Article πŸ“… 1993 πŸ› Elsevier Science 🌐 English βš– 139 KB

Let \(s=\sigma+i t\). Then, on the assumption of Riemann Hypothesis, we prove the Mean-Value Theorem for the square of the Riemann zeta-function over shorter intervals for \(1 / 2+A_{1} / \log \log T \leqslant \sigma \leqslant 1-\delta\). Here \(A_{1}\) is a large positive constant, \(\delta\) is a