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On a conjecture of M. R. Murty and V. K. Murty

Part of: Multiplicative number theory

Published online by Cambridge University Press:  25 October 2022

Yuchen Ding Open the ORCID record for Yuchen Ding [Opens in a new window]
Yuchen Ding*
Affiliation:
School of Mathematical Science, Yangzhou University, Yangzhou 225002, P.R. China
*
e-mail: ycding@yzu.edu.cn
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Abstract

Let $\omega ^*(n)$ be the number of primes p such that $p-1$ divides n. Recently, M. R. Murty and V. K. Murty proved that

$$ \begin{align*}x(\log\log x)^3\ll\sum_{n\le x}\omega^*(n)^2\ll x\log x.\end{align*} $$

They further conjectured that there is some positive constant C such that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\sim Cx\log x,\end{align*} $$

as $x\rightarrow \infty $ . In this short note, we give the correct order of the sum by showing that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\asymp x\log x.\end{align*} $$

Keywords

PrimesBombieri–Vinogradov theoremDichotomy

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Secondary: 11N05: Distribution of primes
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 2 , June 2023 , pp. 679 - 681
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

The author is supported by the National Natural Science Foundation of China under Grant No. 12201544, the Natural Science Foundation of Jiangsu Province of China under Grant No. BK20210784, and the China Postdoctoral Science Foundation under Grant No. 2022M710121. He is also supported by the foundation numbers JSSCBS20211023 and YZLYJF2020PHD051.

References

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