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 Chebyshevskii Sb., 2018, Volume 19, Issue 2, Pages 319–333 (Mi cheb657)

Problems of the summation of arithmetical sums, relative to Chebyshev function

E. I. Deza, L. V. Varukhina

Moscow State Pedagogical University

Abstract: Many problems of Number Theory are connected with investigation of Dirichlet series $f(s)=\sum\limits_{n=1}^{\infty} a_nn^{-s}$ and the adding functions $\Phi(x)=\sum\limits_{n\leq x} a_n$ 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 $\zeta(s)=\sum\limits_{n=1}^{\infty}\frac{1}{n^s}.$
The square of zeta function $\zeta^{2}(s)=\sum\limits_{n=1}^{\infty}\frac{\tau(n)}{n^s}, \Re s > 1,$ is connected with the divisor function $\tau (n)=\sum\limits_ { d | n } 1$, 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 $D (x)=\sum\limits_ { n\leq x}\tau(n)$; the questions of the asymptotic behavior of this function are known as Dirichlet divisor problem. Generally, $\zeta^{k}(s)=\sum\limits_{n=1}^{\infty}\frac{\tau_k(n)}{n^s}, \Re s > 1,$ where function $\tau_k (n)=\sum\limits_{n=n_1\cdot...\cdot n_k} 1$ 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 $D_k (x)=\sum\limits_ { n\leq x}\tau_k(n)$; 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 $\frac{\zeta^{'}(s)}{\zeta(s)}=-\sum\limits_{n=1}^{\infty} \frac{\Lambda(n)}{n^s},$ $\Re s >1.$ 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 $\psi(x)=\sum\limits_{n\leq x}\Lambda(n)$ is the adding function of the coefficients of the Dirichlet series $\sum\limits_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}$, 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: $\psi(x)=x-\sum\limits_{|\Im \rho|\leq T}\frac{x^{\rho}}{\rho}+O(\frac{x\ln^{2}x}{T}),$ 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 two arithmetic functions, relative to the Chebyshev function:
$$\psi_{1}(x)=\sum\limits_{n\leq x}(x-n)\Lambda(n), and \psi_{2}(x)=\sum\limits_{n \leq x}\Lambda(n)\ln\frac{x}{n}.$$
Similar results can be received also for some other functions, related to the Chebyshev function, if to use logarithmic derivatives of Dirichlet $L$-functions.

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, contour integration.

DOI: https://doi.org/10.22405/2226-8383-2018-19-2-319-333

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UDC: 517
Accepted:17.08.2018

Citation: E. I. Deza, L. V. Varukhina, “Problems of the summation of arithmetical sums, relative to Chebyshev function”, Chebyshevskii Sb., 19:2 (2018), 319–333

Citation in format AMSBIB
\Bibitem{DezVar18} \by E.~I.~Deza, L.~V.~Varukhina \paper Problems of the summation of arithmetical sums, relative to Chebyshev function \jour Chebyshevskii Sb. \yr 2018 \vol 19 \issue 2 \pages 319--333 \mathnet{http://mi.mathnet.ru/cheb657} \crossref{https://doi.org/10.22405/2226-8383-2018-19-2-319-333} \elib{https://elibrary.ru/item.asp?id=37112157} 

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This publication is cited in the following articles:
1. S. A. Gritsenko, E.I. Deza, L. V. Varukhina, “O povedenii funktsii, rodstvennykh funktsii Chebysheva”, Chebyshevskii sb., 20:3 (2019), 154–164
2. L. V. Varukhina, “Izbrannye voprosy teorii summatornykh funktsii ryadov Dirikhle”, Chebyshevskii sb., 20:2 (2019), 55–81
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