Exploring Riemann’s functional equation
Open Access
- 14 June 2016
- journal article
- research article
- Published by Taylor & Francis Ltd in Cogent Mathematics
Abstract
An equivalent, but variant form of Riemann’s functional equation is explored, and several discoveries are made. Properties of Riemann’s zeta function , from which a necessary and sufficient condition for the existence of zeros in the critical strip, are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for on the critical line , the consequences of which are explored, and the “LogZeta" function is introduced. A singular linear transform between the real and imaginary components of and on the critical line is derived, and an implicit relationship for locating a zero ( ) on the critical line is found between the arguments of and . Notably, the Volchkov criterion, a Riemann Hypothesis (RH) equivalent, is analytically evaluated and verified to be half equivalent to RH, but RH is not proven. Numerical results are presented, some of which lead to the identification of anomalous zeros, whose existence in turn suggests that well-established, traditional derivations such as the Volchkov criterion and counting theorems require re-examination. It is proven that the derivative will never vanish on the perforated critical line ( ). Traditional asymptotic and counting results are obtained in an untraditional manner, yielding insight into the nature of as well as very accurate asymptotic estimates for distribution bounds and the density of zeros on the critical line.
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This publication has 9 references indexed in Scilit:
- On Equalities Involving Integrals of the Logarithm of the Riemann -Function with Exponential Weight Which Are Equivalent to the Riemann HypothesisInternational Journal of Analysis, 2015
- On the exact location of the non-trivial zeros of Riemann's zeta functionActa Arithmetica, 2014
- On equalities involving integrals of the logarithm of the Riemann ζ-function and equivalent to the Riemann hypothesisUkrainian Mathematical Journal, 2012
- The Riemann HypothesisPublished by Springer Science and Business Media LLC ,2008
- On an equality equivalent to the Riemann hypothesisUkrainian Mathematical Journal, 1995
- Zeros of Derivatives Of the Riemann Zeta-Function Near the Critical LinePublished by Springer Science and Business Media LLC ,1990
- Zeros of derivatives of Riemann's xi-function of the critical line. IIJournal of Number Theory, 1983
- Zeros of derivatives of Riemann's ξ-function on the critical lineJournal of Number Theory, 1983
- Zeros of $\zeta^{\prime} (s)$ and the Riemann hypothesisIllinois Journal of Mathematics, 1973