Proves a density-1 set where (log n) times the probability that a Cramér model sum S_n is prime is bounded below by 1/sqrt(2 pi e), an asymptotic Gaussian integral formula involving the prime counting function pi(t), and related bounds for quasiprimes and interval lengths tied to Sturm-Liouville.
The Differences Between Consecutive Primes. V
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abstract
We show that \[\sum_{\substack{p_n\le x\\ p_{n+1}-p_n\ge\sqrt{p_n}}}(p_{n+1}-p_n)\ll_{\varepsilon} x^{3/5+\varepsilon}\] for any fixed $\varepsilon>0$. This improves a result of Matom\"{a}ki, in which the exponent was $2/3$.
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2021 1verdicts
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Critical probabilistic characteristics of the Cram\'er model for primes and arithmetical properties
Proves a density-1 set where (log n) times the probability that a Cramér model sum S_n is prime is bounded below by 1/sqrt(2 pi e), an asymptotic Gaussian integral formula involving the prime counting function pi(t), and related bounds for quasiprimes and interval lengths tied to Sturm-Liouville.