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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 DingOpen 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

Information

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

Let $\omega (n)$ be the number of distinct prime divisors of n. In about 100 years ago, Hardy and Ramanujan [Reference Hardy and Ramanujan3] found out that $\omega (n)$ has normal order $\log \log n$ , which means that for almost all integers n we have $\omega (n)\sim \log \log n$ . Later, Turán [Reference Turán6] provided a quite elegantly simplified proof by establishing

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

In 1955, Prachar [Reference Prachar5] considered a variant arithmetic function of $\omega $ . Let $\omega ^*(n)$ be the number of primes p such that $p-1$ divides n. Prachar proved that

$$ \begin{align*}\sum_{n\le x}\omega^*(n)=x\log\log x+Bx+O(x/\log x)\end{align*} $$

and

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2=O\left(x(\log x)^2\right),\end{align*} $$

where B is a constant. Motivated by Prachar’s work, Erdős and Prachar [Reference Erdős and Prachar2] proved that the number of pairs of primes p and q so that the least common multiple $[p-1, q-1]\le x$ is bounded by $O(x\log \log x)$ . Following a remark of Erdős and Prachar, M. R. Murty and V. K. Murty [Reference Murty and Murty4] improved this to $O(x)$ . By this improvement, they reached the nice bounds

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

With these in hands, M. R. Murty and V. K. Murty 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 note, the author shall give a slight improvement of the result due to M. R. Murty and V. K. Murty toward the correct direction of their conjecture.

Theorem 1 There are two absolute constants $a_1$ and $a_2$ such that

$$ \begin{align*}a_1x\log x\le \sum_{n\le x}\omega^*(n)^2\le a_2x\log x.\end{align*} $$

Proof We only need to prove the lower bound as the upper bound is displayed by M. R. Murty and V. K. Murty. Throughout our proof, the number x is sufficiently large. From the paper of M. R. Murty and V. K. Murty [Reference Murty and Murty4, equation (4.10)], we have

(1) $$ \begin{align} \sum_{n\le x}\omega^*(n)^2=x\sum_{d\le x}\varphi(d)\left(\sum_{\substack{p\le x\\p\equiv 1 {\ (\mathrm{{mod}}\ d)}}} \frac{1}{p-1}\right)^2+O(x). \end{align} $$

Integrating by parts gives

(2) $$ \begin{align} \sum_{\substack{p\le x\\p\equiv 1 {\ (\mathrm{{mod}}\ d)}}} \frac{1}{p}=\frac{\pi(x;d,1)}{x}+\int_{2}^{x}\frac{\pi(t;d,1)}{t^2}dt\ge \int_{x^{3/4}}^{x}\frac{\pi(t;d,1)}{t^2}dt, \end{align} $$

where $\pi (t;d,1)$ is the number of primes $p\equiv 1 {\ (\mathrm {{mod}}\ d)}$ up to t. Thus, from equations (1) and (2), we obtain

(3) $$ \begin{align} \sum_{n\le x}\omega^*(n)^2\ge x\sum_{d\le x^{1/3}}\varphi(d)\left(\int_{x^{3/4}}^{x}\frac{\pi(t;d,1)}{t^2}dt\right)^2+O(x). \end{align} $$

For any integer $0\le j\le \left \lfloor \frac {\log x}{13\log 2}\right \rfloor $ , let $Q_j=2^jx^{1/4}$ . Then $Q_j<x^{1/3}$ for all integers j. From a weak form of the Bombieri–Vinogradov theorem (see, for example, [Reference Davenport1]), we have

$$ \begin{align*} \sum_{Q_j<d\le 2Q_j}\max_{y\le z}\left|\pi(y;d,1)-\frac{\text{li}~y}{\varphi(d)}\right|\ll \frac{z}{(\log z)^5}, \end{align*} $$

for any $0\le j\le \left \lfloor \frac {\log x}{13\log 2}\right \rfloor $ and $x^{3/4}\le z\le x$ , where the implied constant is absolute. It follows immediately that

(4) $$ \begin{align} \max_{y\le z}\left|\pi(y;d,1)-\frac{\text{li}~y}{\varphi(d)}\right|<\frac{\text{li}~z}{\varphi(d)\log z} \end{align} $$

hold for all $Q_j<d\le 2Q_j$ but at most $O\left (Q_j/(\log x)^2\right )$ exceptions. From equation (4), we have

$$ \begin{align*} \pi(y;d,1)>\frac{\text{li}~y}{2\varphi(d)} \quad \left(z/2<y\le z\right) \end{align*} $$

for all $Q_j<d\le 2Q_j$ with at most $O\left (Q_j/(\log x)^2\right )$ exceptions. A little thought with the dichotomy of z between the interval $[x^{3/4},x]$ leads to the fact

(5) $$ \begin{align} \pi(y;d,1)>\frac{\text{li}~y}{2\varphi(d)}> \frac{y}{3\varphi(d)\log y} \quad \left(\forall~ x^{3/4}\le y\le x\right) \end{align} $$

for all $Q_j<d\le 2Q_j$ except for $O\left (Q_j/\log x\right )$ exceptions. For any integer $0\le j\le \left \lfloor \frac {\log x}{13\log 2}\right \rfloor $ , let $S_j$ be the set of all integers $Q_j<d\le 2Q_j$ such that equation (5) holds. Thus, from the analysis above and equations (3) and (5), we conclude that

$$ \begin{align*} \sum_{n\le x}\omega^*(n)^2&\ge \frac{x}{9}\sum_{0\le j\le \left\lfloor \frac{\log x}{13\log 2}\right\rfloor}\sum_{\substack{d\in S_j}}\varphi(d)\left(\int_{x^{3/4}}^{x}\frac{1}{\varphi(d)t\log t}dt\right)^2+O(x)\nonumber\\ &\gg x\sum_{0\le j\le \left\lfloor \frac{\log x}{13\log 2}\right\rfloor}\sum_{\substack{d\in S_j}}\frac1{\varphi(d)}+x\gg x\log x.\\[-36pt] \end{align*} $$

It is worth here mentioning that we have the following corollary:

$$ \begin{align*}\sum_{p,q\le x}\frac{1}{[p-1,q-1]}\asymp \log x\end{align*} $$

due to (see [Reference Murty and Murty4, p. 6, last line])

$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2=\sum_{p,q\le x}\frac{x}{[p-1,q-1]}+O(x).\end{align*} $$

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

Davenport, H., Multiplicative number theory. 2nd ed., Graduate Texts in Mathematics, 74, Springer, New York, 1980.CrossRefGoogle Scholar
Erdős, P. and Prachar, K., Über die Anzahl der Lösungen von $\left[p-1,q-1\right]\le x$ . Monatsh. Math. 59(1955), 318–319.CrossRefGoogle Scholar
Hardy, G. H. and Ramanujan, S., The normal number of prime factors of a number $n$ . Quart. J. Math. 48(1917), 76–92.Google Scholar
Murty, M. R. and Murty, V. K., A variant of the Hardy–Ramanujan theorem . Hardy–Ramanujan J. 44(2021), 32–40.Google Scholar
Prachar, K., Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form $p-1$ haben . Monatsh. Math. 59(1955), 91–97.CrossRefGoogle Scholar
Turán, P., On a theorem of Hardy and Ramanujan . J. Lond. Math. Soc. 9(1934), 274–276.CrossRefGoogle Scholar