Functiones et Approximatio Commentarii Mathematici

A Test for the Riemann Hypothesis

Juan Arias de Reyna

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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 $J(a)\approx J$.

Article information

Funct. Approx. Comment. Math., Volume 38, Number 2 (2008), 159-170.

First available in Project Euclid: 19 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses

Riemann hypothesis prime numbers Fourier Transform


de Reyna, Juan Arias. A Test for the Riemann Hypothesis. Funct. Approx. Comment. Math. 38 (2008), no. 2, 159--170.

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