publication . Preprint . 2016

Local Extrema of the $\Xi(t)$ Function and The Riemann Hypothesis

Kobayashi, Hisashi;
Open Access English
  • Published: 04 Mar 2016
Abstract
In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function $\Xi(t)=\xi(\tfrac{1}{2}+it)$. First, we prove that positivity of all local maxima and negativity of all local minima of $\Xi(t)$ form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of $\Xi(t)$ is a saddle point of the function $\Re\{\xi(s)\}$, we prove that the above properties of local extrema of $\Xi(t)$ are also a sufficient condition for the Riemann hypothesis to hold at $t\gg 1$. We present a numerical example to illustrate our approach towards...
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free text keywords: Mathematics - Number Theory
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[1] Edwards, H. M. (1974), Riemann's Zeta Function, Originally published by Academic Press in 1974. Republished by Dover Publications in 2001.

[2] Iwaniec, H. (2014) Lectures on the Riemann Zeta Function, University Lecture Series, Volume 62, American Mathematical Society. [OpenAIRE]

[4] Matsumoto, K. (2005) Riemann's Zeta Function (in Japanese), Asakura Shoten, Japan.

[5] Riemann, B. (1859), “ U¨ber die Anzahl der Primizahlen under einer gegebener Gr¨osse,” Motatsberichte. der Berliner Akademie, November 1859, pp. 671-680. Its English translation “On the Number of Primes Less Than a Given Magnitude,” can be found in Appendix of Edwards, pp. 299-305.

[6] Titchmarsh, E.C. (1951), Revised by D. R. Heath-Brown (1986), The Theory of the Riemann Zetafunction (2nd Edition), Oxford University Press.

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