Explicit improvements of the Brun-Titchmarsh theorem for arbitrary intervals

Tomohiro Yamada

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arXiv · 2312.16090

Explicit improvements of the Brun-Titchmarsh theorem for arbitrary intervals

Tomohiro Yamada

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Abstract

We give an explicit version of Brun-Titchmarsh theorem applicable for arbitrary moduli and arbitrary intervals. For example, we show that $\pi(x+y; k, a)-\pi(x; k, a)<2y/(\varphi(k)(\log (y/k)+0.8601))$ for any relatively prime positive integers $k$ and $a$ and any positive real numbers $x$ and $y$ such that $y>k$.

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Tomohiro Yamada. 2023-12-26. Explicit improvements of the Brun-Titchmarsh theorem for arbitrary intervals. https://arxiv.org/abs/2312.16090

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