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September 2008 A Test for the Riemann Hypothesis
Juan Arias de Reyna
Funct. Approx. Comment. Math. 38(2): 159-170 (September 2008). DOI: 10.7169/facm/1229696537

Abstract

We prove that the Riemann Hypothesis holds if and only if $$I=\int_1^{+\infty}\bigl\{\Pi(x)-\Li(x)\bigr\}^2x^{-2}\,dx<+\infty$$ with $I=J$, where $J$ is some definite, computable real number ($1.266<J<1.273$). This provides us with a numerical test for the Riemann Hypothesis. The main interest of our test lies in the fact that it can also supply a \emph{goal}. Namely, having computed $J(a):=\int_1^a \bigl\{\Pi(x)-\Li(x)\bigr\}^2x^{-2}\,dx< J$ for a number of values of $a=a_n$, we can estimate a value $a$ for which, within our precision, we will have $J(a)\approx J$.

Citation

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Juan Arias de Reyna. "A Test for the Riemann Hypothesis." Funct. Approx. Comment. Math. 38 (2) 159 - 170, September 2008. https://doi.org/10.7169/facm/1229696537

Information

Published: September 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1235.11081
MathSciNet: MR2492854
Digital Object Identifier: 10.7169/facm/1229696537

Subjects:
Primary: 11M26

Keywords: Fourier transform , prime numbers , Riemann hypothesis

Rights: Copyright © 2008 Adam Mickiewicz University

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Vol.38 • No. 2 • September 2008
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