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

This article is cited in 2 scientific papers (total in 2 papers)

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.


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UDC: 517
Received: 27.04.2018

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
\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

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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  mathnet  crossref
    2. L. V. Varukhina, “Izbrannye voprosy teorii summatornykh funktsii ryadov Dirikhle”, Chebyshevskii sb., 20:2 (2019), 55–81  mathnet  crossref
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