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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\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 $0< \Re s<1$.
We obtain similar representations over non-trivial zeros
of zeta function for two arithmetic functions, relative to the Chebyshev function:
$$\psi_{1}(x)=\sum\limits_{n\leq x}(x-n)\Lambda(n), \, \text{ and } \,
\psi_{2}(x)=\sum\limits_{n \leq x}\Lambda(n)\ln\frac{x}{n}.$$
Similar results can be received also for some other functions, related to the
Chebyshev function, if to use logarithmic derivatives of Dirichlet
$L$-functions.
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.
UDC:
517 Received: 27.04.2018
Accepted: 17.08.2018
DOI:
10.22405/2226-8383-2018-19-2-319-333
Fulltext:
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