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arXiv · 2308.04458

The number of primes in short intervals and numerical calculations for Harman's sieve

Runbo Li
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Abstract

The author gives nontrivial upper and lower bounds for the number of primes in the interval $[x - x^{\theta}, x]$ for some $0.52 \leqslant \theta \leqslant 0.525$, showing that the interval $[x - x^{0.52}, x]$ contains prime numbers for all sufficiently large $x$. This refines a result of Baker, Harman and Pintz (2001) and gives an affirmative answer to Harman and Pintz's argument. New arithmetic information, a delicate sieve decomposition, various techniques in Harman's sieve and accurate estimates for integrals are used to good effect.

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Runbo Li. 2023-08-07. The number of primes in short intervals and numerical calculations for Harman's sieve. https://arxiv.org/abs/2308.04458

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