Selected questions of the theory of adding functions of Dirichlet series

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 forms one of central domains of classical Number Theory.
Under some special conditions for
$$f(s)=\sum_{n=1}^{+\infty} a_nn^{-s},$$
the function $\Phi(x)$ can be represented using $f(s)$. This connection is given by the famous Perron formula
$$\sum_{n\leq x} a_n = \frac{1}{2\pi i }\int_{c_0-i\infty}^{c_0+i\infty} f(s)\frac{x^s}{s}ds, \quad c_0>\sigma_0, $$
where
$$\sum_{n=1}^{\infty} a_nn^{-s}$$
for $f(s)$ is absolutely convergenting for $\sigma>\sigma_0$. More precisely, classical scheme of a research of the adding function
$$\Phi(x)=\sum_{n\leq x}a_n$$
of coefficients of
$$f(s)=\sum_{n=1}^{\infty}a_nn^{-s}$$
leans on a formula, which (under certain conditions) is expressing the function $\Phi(x)$ through the integral
$$\frac{1}{2\pi i}\int_{b-iT}^{b+iT} \frac{f(s)x^s}{s}ds.$$

In 1972, A. A. Karatsuba received an "integrated" formula of such kind, which connects
$$\int_{1}^x\Phi(y)dy$$
with the integral

$$\frac{1}{2\pi i}\int_{b-iT}^{b+iT} \frac{f(s)x^{s+1}}{s(s+1)}ds.$$
This formula allows to receive new results in the research of corresponding number-theoretical questions.
In this paper we presernt a new formula, expressing the adding function
$$\Phi(x)=\sum_{n\leq x}a_n$$
of
$$f(s)=\sum_{n=1}^{+\infty}a_nn^{-s}$$
through $f(s),$ related to the Perron formula and to the integrated formula of A. A. Karatsuba. In fact, the following statement is proved.

• Let the row
  $$f(s)=\sum_{n=1}^{+\infty}a_nn^{-s}$$
  is absolutely convergenting for $\mathrm{Re}\, s >1$, $a_n=O(n^{\varepsilon})$, where $\varepsilon>0$ is any small real number, and for $\sigma\rightarrow 1+$ there is an estimation
  $$\sum_{n=1}^{\infty}|a_n|n^{-\sigma}=O((\sigma-1)^{-\alpha}), \,\, \alpha>0.$$
  Then for any $b \geq b_0>1$, $T\geq 1$, $x=N+0,5$, $H>b$ we have the formula
  $$\Phi(x)=\sum_{n\leq x}a_n = \frac{1}{2\pi i}\int_{b-iT}^{b+iT} f(s)\frac{x^sH}{s(H-s)}ds +$$
  
  $$ +O\left(\frac{x^bH}{T^2(b-1)^{\alpha}}\right) +O\left(\frac{x^{1+\varepsilon}H\log x}{T^2}\right)+O\left(x^{\varepsilon}e^{H\log\frac{x}{x+0,5}}\left(1+\frac{x}{H}\right)\right).$$
  

The most famous Dirichlet series is the Riemann zeta function $\zeta(s)$, defined for any $s=\sigma+it$ with $\mathrm{Re}\, s=\sigma> 1$ as
$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.$$

The square of zeta function is connected with the divisor function
$$\tau (n)=\sum_ { d | n } 1,$$
giving the number of positive integer divisors of positive integer number $n$. Generally,
$$ \zeta^{k}(s)=\sum_{n=1}^{\infty}\frac{\tau_k(n)}{n^s}, \quad \mathrm{Re}\, s > 1, $$
where function
$$\tau_k (n)=\sum_{n=n_1\cdot \ldots\cdot n_k} 1$$
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 Dirichlet divisor problem.
In this article we prove two new asymptotic formulas for the functions
$$\sum_{n \leq x} \tau_{k_{1}}(n) \cdot \ldots \cdot \tau_{k_{l}}(n)$$
и
$$ \sum_{n\leq x}\tau_{k}(n^{2}),$$
connected with $D_k(x)$.

• 
  $$\sum_{n \leq x} \tau_{k_{1}}(n) \cdot\ldots \cdot \tau_{k_{l}}(n)=x P_{m}(\log x) + \theta x^{1 - \frac{1}{13}m^{-2/3}}(C\log x)^{m},$$
  where $l \geq 1$, $k_{1}, \dots, k_{l} \geq 2$, $m=k_{1}\cdot \ldots \cdot k_{l}$, $P_{m}$ is a polinomial of degree $m-1$, $\theta$ is a complex number, $|\theta|\leq 1$, $m\ll \log^{\frac{5}{6}}x$, $C>0$ is an absolute constant.
  
• 
  $$\sum_{n\leq x}\tau_{k}(n^{2})=xP_{m}(\log x) + \theta x^{1 - \frac{1}{13}m^{-2/3}}(C\log x)^{m},$$
  where $k \geq 2$, $m=\frac{k(k+1)}{2}$, $P_{m}$ is a polinomial of degree $m-1$, $\theta$ is a complex number, $|\theta|\leq 1$, $m\ll \log^{\frac{5}{6}}x$, $C>0$ is an absolute constant.

Other well-known example of Dirichlet series is given by Dirichlet $L$-function
$$L(s, \chi)=\sum_{n=1}^{\infty} \chi(n)n^{-s}, \mathrm{Re}\, s>1,$$
where $\chi$ is a Dirichlet character modulo $D$. A product of several $L$-functions gives for $\mathrm{Re}\, s>1$ a row
$$L_1(s,\chi_1)\cdot \ldots\cdot L_k(s,\chi_k)=\sum_{n=1}^{\infty}c_nn^{-s},$$
with adding function
$$C_k(x)=\sum_{n\leq x}c_n=\sum_{n_1\cdot \ldots\cdot n_k\leq x}\chi_1(n_1)\cdot \ldots\cdot \chi_k(n_k).$$
The problem of asymptotic behavior of $C_k(x)$ is a generalisation of the Dirichlet divisor problem. It is connected with the divisor problem in number fields, in particular, in quadratic fields and in сyclotomic fields.
In this article we give new asymptotic formulas for the mean values of the two functions $\tau^{K}_{k_{1}}(n) \cdot \ldots \cdot \tau^{K}_{k_{l}}(n)$ and $\tau^K_k(n^2)$, connected with the function $\tau_k(n)$, in the quadratic field $K=Q(\sqrt{D})$, $D$ is squarefree integer number, and in cyclotomic field $K=Q(\varsigma)$, $\varsigma^{t}=1$, with the constant $c=\frac{1}{13}$ in the exponent of the error term.