Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 215416, 15 pages
doi:10.1155/2010/215416
Research Article

Applications of Wirtinger Inequalities on the Distribution of Zeros of the Riemann Zeta-Function

Samir H. Saker1,2

1Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia
2Department of Mathematics, Mansoura University, Mansoura 35516, Egypt

Received 10 October 2010; Accepted 17 December 2010

Academic Editor: Soo Hak Sung

Copyright © 2010 Samir H. Saker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

On the hypothesis that the ( 2 𝑘 ) th moments of the Hardy 𝑍 -function are correctly predicted by random matrix theory and the moments of the derivative of 𝑍 are correctly predicted by the derivative of the characteristic polynomials of unitary matrices, we establish new large spaces between the zeros of the Riemann zeta-function by employing some Wirtinger-type inequalities. In particular, it is obtained that Λ ( 1 5 ) ≥ 6 . 1 3 9 2 which means that consecutive nontrivial zeros often differ by at least 6.1392 times the average spacing.