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On the mean square gap between primes

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arxiv 2212.10867 v1 pith:WTSA4SXX submitted 2022-12-21 math.NT

classification math.NT
keywords varepsilonboundheath-brownmeanprimesaverageconsecutivedifferences
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abstract

We prove that the average size of the squares of differences between consecutive primes less than $x$ is $O(x^{0.23+\varepsilon})$ for any fixed $\varepsilon>0$. This improves on a result of Peck, who gave bound $O(x^{0.25+\varepsilon})$ in the place of $O(x^{0.23+\varepsilon})$. Key ingredients are Harman's sieve, Heath-Brown's mean value theorem for sparse Dirichlet polynomials and Heath-Brown's $R^*$ bound.

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  1. On the number of exceptional intervals to the prime number theorem in short intervals

    math.NT 2025-05 accept novelty 7.0 of 10

    An explicit formula expresses the exceptional-set exponent for the short-interval prime number theorem in terms of zero density estimates, yielding improved numerical bounds such as μ(17/30) ≤ 7/12.

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