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

E. I. Deza ; L. V. Varukhina

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 319-333.

Résumé

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\left(\frac{x\ln^{2}x}{T}\right), $ 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 $0 \Re s1$. We obtain similar representations over non-trivial zeros of zeta function for two arithmetic functions, relative to the Chebyshev function: $$\psi_{1}(x)=\sum\limits_{n\leq x}(x-n)\Lambda(n), \, \text{ and } \, \psi_{2}(x)=\sum\limits_{n \leq x}\Lambda(n)\ln\frac{x}{n}.$$ Similar results can be received also for some other functions, related to the Chebyshev function, if to use logarithmic derivatives of Dirichlet $L$-functions.

Détail

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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