Abstract
Let us denote by \(\pi (x)\) the number of primes \(\leqslant x\), by \({{\mathrm{li}}}(x)\) the logarithmic integral of x, by \(\theta (x)=\sum _{p\leqslant x} \log p\) the Chebyshev function and let us set \(A(x)={{\mathrm{li}}}(\theta (x))-\pi (x)\). Revisiting a result of Ramanujan, we prove that the assertion “\(A(x) > 0\) for \(x\geqslant 11\)” is equivalent to the Riemann Hypothesis.
To Krishna Alladi for his sixtieth birthday
Research partially supported by CNRS, UMR 5208.
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Acknowledgements
I am pleased to thank Marc Deléglise for his computations and for several discussions about this paper.
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Nicolas, JL. (2017). Estimates of \({{\mathrm{li}}}(\theta (x))-\pi (x)\) and the Riemann Hypothesis. In: Andrews, G., Garvan, F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham. https://doi.org/10.1007/978-3-319-68376-8_32
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DOI: https://doi.org/10.1007/978-3-319-68376-8_32
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