Functiones et Approximatio Commentarii Mathematici

Changes of sign of the error term in the prime number theorem

Hugh L. Montgomery and Ulrike M.A. Vorhauer

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Abstract

We assume the Riemann Hypothesis (RH). It is classical that there is an absolute constant $C > 1$ such that $\psi(x)-x$ changes sign in every interval $[x, Cx]$ for $x\ge 1$. We prove that $\psi(x)-x$ changes sign in $[x,19x]$ for all $x\ge 1$, and also that for $x\ge x_0$, $\psi(x)-x$ changes sign in the interval $[x,Cx]$ where $C = 2.02$.

Article information

Source
Funct. Approx. Comment. Math., Volume 35 (2006), 235-247.

Dates
First available in Project Euclid: 16 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229442626

Digital Object Identifier
doi:10.7169/facm/1229442626

Mathematical Reviews number (MathSciNet)
MR2271616

Zentralblatt MATH identifier
1196.11125

Subjects
Primary: 11N05: Distribution of primes

Keywords
Prime Number Theorem Riemann Hypothesis

Citation

Montgomery, Hugh L.; Vorhauer, Ulrike M.A. Changes of sign of the error term in the prime number theorem. Funct. Approx. Comment. Math. 35 (2006), 235--247. doi:10.7169/facm/1229442626. https://projecteuclid.org/euclid.facm/1229442626


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