Sharper bounds for the Chebyshev function 𝜃(𝑥)

Broadbent, Samuel; Kadiri, Habiba; Lumley, Allysa; Ng, Nathan; Wilk, Kirsten

doi:10.1090/mcom/3643

Sharper bounds for the Chebyshev function $\theta (x)$
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by Samuel Broadbent, Habiba Kadiri, Allysa Lumley, Nathan Ng and Kirsten Wilk HTML | PDF

Math. Comp. 90 (2021), 2281-2315

Abstract:

In this article, we provide explicit bounds for the prime counting functions $\theta (x)$ for all ranges of $x$. The bounds for the error term for $\theta (x)- x$ are of the shape $\varepsilon x$ and $\frac {c_k x}{(\log x)^k}$, for $k=1,\ldots ,5$. Tables of values for $\varepsilon$ and $c_k$ are provided.

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