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On the average value of $\pi (t)-\operatorname {\textrm {li}}(t)$

Part of: Multiplicative number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  14 March 2022

Daniel R. Johnston Open the ORCID record for Daniel R. Johnston [Opens in a new window]
Daniel R. Johnston*
Affiliation:
School of Science, The University of New South Wales Canberra, Northcott Drive, Campbell, ACT 2612, Australia
*
e-mail: daniel.johnston@adfa.edu.au
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Abstract

We prove that the Riemann hypothesis is equivalent to the condition $\int _{2}^x\left (\pi (t)-\operatorname {\textrm {li}}(t)\right )\textrm {d}t<0$ for all $x>2$ . Here, $\pi (t)$ is the prime-counting function and $\operatorname {\textrm {li}}(t)$ is the logarithmic integral. This makes explicit a claim of Pintz. Moreover, we prove an analogous result for the Chebyshev function $\theta (t)$ and discuss the extent to which one can make related claims unconditionally.

Keywords

Distribution of primesprime-counting functionlogarithmic integralRiemann hypothesisRiemann zeta function

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11N05: Distribution of primes
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 1 , March 2023 , pp. 185 - 195
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Copyright
© Canadian Mathematical Society, 2022

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