Journal of the Ramanujan Mathematical Society
Volume 39, Issue 4, December 2024 pp. 337–347.
On a sum involving the von Mangoldt and the integral part function
Authors:
Ya-Fang Feng
Author institution:Graduate School of Hohai University, Nanjing 210024, People's Republic of China
Summary:
For any real number y, let ⌊ y ⌋ be the largest integer not exceeding y. As usual, let Λ(n) be the von
Mangoldt function. Recently, Liu, Wu and Yang obtain the following variant of the prime number theorem
{∑}
Λ n∈x
(⌊ x/n ⌋) = λx + O
(x{9/19+ε}),
where λ =
∑{∞} {d = 1}
{Λ(d)}/{d(d+1)} is an absolute constant and ε is an arbitrarily small number. In this article, we give a slight
generalization of their formula by showing
{∑}
{n≤x{1/c}}
Λ(⌊x/n{c}⌋) = λ{c}x{1/c} + O(x{γ}{c})
for any real number c > 0, where λc =
∑{∞}{d = 1} Λ(d)
({1}/{d{1/c}} − {1}/{(d+1){1/c}})
is a constant and γ{c} < 1/c is a real number. This can
be viewed as the Pjateckii – Sapiro type result in the variant prime number theorem given by Liu, Wu and Yang.
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