On behavior of arithmetical functions, related to Chebyshev function

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: $$\psi(x)=x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho}}{\rho}+O\left(\frac{x\ln^{2}x}{T}\right), $$ 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 < Res < 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: $$\psi_1(x)=\frac{x^2}{2}-\left(\frac{\zeta^{'}(0)}{\zeta(0)}\right)x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho+1}}{\rho(\rho+1)}+O\left(\frac{x^{2}}{T^2}\ln^2 x\right)+O\left(\sqrt{x}\ln^2x\right), $$ 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 < Res < 1.

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

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