Chebyshevskii Sbornik General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Chebyshevskii Sb.: Year: Volume: Issue: Page: Find

 Personal entry: Login: Password: Save password Enter Forgotten password? Register

 Chebyshevskii Sb., 2019, Volume 20, Issue 3, Pages 154–164 (Mi cheb805)  On behavior of arithmetical functions, related to Chebyshev function

S. A. Gritsenkoa, E. Dezab, L. V. Varukhinab

a Lomonosov Moscow state University (Moscow)
b Moscow Pedagogical State University (Moscow)

Abstract: Many problems of Number Theory are connected with investigation of Dirichlet series $f(s)=\sum_{n=1}^{\infty} a_nn^{-s}$ and the adding functions $\Phi(x)=\sum_{n\leq x} a_n$ of their coefficients. The most famous Dirichlet series is the Riemann zeta function $\zeta(s)$, defined for any $s=\sigma+it$ with $\Re s=\sigma> 1$ as $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.$
The square of zeta function $\zeta^{2}(s)=\sum_{n=1}^{\infty}\frac{\tau(n)}{n^s}, \Re s > 1,$ is connected with the divisor function $\tau (n)=\sum_ { d | n } 1$, giving the number of a positive integer divisors of positive integer number $n$. The adding function of the Dirichlet series $\zeta^2(s)$ is the function $D (x)=\sum_ { n\leq x}\tau(n)$; the questions of the asymptotic behavior of this function are known as Dirichlet divisor problem. Generally, $\zeta^{k}(s)=\sum_{n=1}^{\infty}\frac{\tau_k(n)}{n^s}, \Re s > 1,$ where function $\tau_k (n)=\sum_{n=n_1\cdot...\cdot n_k} 1$ gives the number of representations of a positive integer number $n$ as a product of $k$ positive integer factors. The adding function of the Dirichlet series $\zeta^k (s)$ is the function $D_k (x)=\sum_ { n\leq x}\tau_k(n)$; its research is known as the multidimensional Dirichlet divisor problem.
The logarithmic derivative $\frac{\zeta^{'}(s)}{\zeta(s)}$ of zeta function can be represented as $\frac{\zeta^{'}(s)}{\zeta(s)}=-\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s},$ $\Re s >1.$ Here $\Lambda(n)$ is the Mangoldt function, defined as $\Lambda(n)=\log p$, if $n=p^{k}$ for a prime number $p$ and a positive integer number $k$, and as $\Lambda(n)=0$, otherwise. So, the Chebyshev function $\psi(x)=\sum_{n\leq x}\Lambda(n)$ is the adding function of the coefficients of the Dirichlet series $\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}$, corresponding to logarithmic derivative $\frac{\zeta^{'}(s)}{\zeta(s)}$ of zeta function. It is well-known in analytic Number Theory and is closely connected with many important number-theoretical problems, for example, with asymptotic law of distribution of prime numbers.
In particular, the following representation of $\psi(x)$ is very useful in many applications: $\psi(x)=x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho}}{\rho}+O(\frac{x\ln^{2}x}{T}),$ where $x=n+0,5$, $n \in\mathbb{N}$, $2\leq T \leq x$, and $\rho=\beta+i\gamma$ are non-trivial zeros of zeta function, i.e., the zeros of $\zeta(s)$, belonging to the critical strip $0< \Re s<1$.
We obtain similar representations over non-trivial zeros of zeta function for an arithmetic function, relative to the Chebyshev function: $\psi_{1}(x)=\sum_{n\leq x}(x-n)\Lambda(n).$ In fact, we prove the following theorem: $\psi_1(x)=\frac{x^2}{2}-(\frac{\zeta^{'}(0)}{\zeta(0)})x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho+1}}{\rho(\rho+1)}+O(\frac{x^{2}}{T^2}\ln^2 x)+O(\sqrt{x}\ln^2x),$ where $x>2$, $T \geq 2$, and $\rho=\beta+i\gamma$ are non-trivial zeros of zeta function, i.e., the zeros of $\zeta(s)$, belonging to the critical strip $0< \Re s<1$.

Keywords: arithmetical functions, Dirichlet series, adding function of the coefficients of a Dirichlet series, the Riemann zeta function, the Chebyshev function, non-trivial zeros of the Riemann zeta function, Cauchy's residue theorem.

DOI: https://doi.org/10.22405/2226-8383-2018-20-3-154-164  Full text: PDF file (620 kB) References: PDF file   HTML file

UDC: 517
Accepted:12.11.2019

Citation: S. A. Gritsenko, E. Deza, L. V. Varukhina, “On behavior of arithmetical functions, related to Chebyshev function”, Chebyshevskii Sb., 20:3 (2019), 154–164 Citation in format AMSBIB
\Bibitem{GriDezVar19}
\by S.~A.~Gritsenko, E.~Deza, L.~V.~Varukhina
\paper On behavior of arithmetical functions, related to Chebyshev function
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 3
\pages 154--164
\mathnet{http://mi.mathnet.ru/cheb805}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-3-154-164}

 SHARE:       Contact us: math-net2022_01 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2022