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On a conjecture of M. R. Murty and V. K. Murty
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AbstractLet
$\omega ^*(n)$
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
$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\asymp x\log x.\end{align*} $$
Title: On a conjecture of M. R. Murty and V. K. Murty
Description:
AbstractLet
$\omega ^*(n)$
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
$$ \begin{align*}\sum_{n\le x}\omega^*(n)^2\asymp x\log x.
\end{align*} $$.
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