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

Published online by Cambridge University Press:  14 March 2022

Daniel R. Johnston*
Affiliation:
School of Science, The University of New South Wales Canberra, Northcott Drive, Campbell, ACT2612, Australia

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.

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Article
Copyright
© Canadian Mathematical Society, 2022

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