Abstract
We obtain a lower bound for \(\#\{x/2<p_n\leq x \colon\, \, p_n\equiv\dots\equiv p_{n+m}\equiv a\pmod{q}\), \(p_{n+m} - p_n\leq y\}\), where \(p_n\) is the \(n\)th prime.
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Acknowledgments
The author is deeply grateful to Sergei Konyagin and Maxim Korolev for their attention to this work and useful comments. The author also expresses his gratitude to Mikhail Gabdullin and Pavel Grigor’ev for useful comments and suggestions.
Funding
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).
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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 314, pp. 152–210 https://doi.org/10.4213/tm4163.
On the occasion of the 130th anniversary of I. M. Vinogradov’s birth
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Radomskii, A.O. Consecutive Primes in Short Intervals. Proc. Steklov Inst. Math. 314, 144–202 (2021). https://doi.org/10.1134/S008154382104009X
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DOI: https://doi.org/10.1134/S008154382104009X