Some Problems in Multiplicative and Additive Number Theory

Abstract

In this thesis, we obtain several results in number theory. Let $k\geqslant 1$ be a natural number and $\omega_k(n)$ denote the number of distinct prime factors of a natural number $n$ with multiplicity $k$. We estimate the first and the second moments of the functions $\omega_k$, $k\geqslant 1$. Moreover, we prove that the function $\omega_1(n)$ has normal order $\log\log n$ and the functions $\omega_k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function $F$.

Let $\chi$ be a nonprincipal Dirichlet character modulo a prime number $p\geqslant 3$. Define \begin{align*} \mathcal{M}_{p}(-s,\chi)&:=\frac{2}{p-1}\sum_{\substack{\psi \pmod p\\\psi(-1)=-1}}L(1,\psi)L(-s,\chi\overline{\psi}), \\ \mathcal{A}_{p}(\chi)&:=\frac{1}{p-1}\sum_{{\substack{1\leqslant N \leqslant p-1}}}\sum_{\substack{1\leqslant n_1,n_2\leqslant N\\\chi(n_1)=\chi(n_2)}}1, \\\Delta(s,\chi)&:=\sum_{n=2}^{\infty}\frac{\chi(n)\Delta(n)}{n^s}, \quad \quad (\Re(s)>2) \end{align*} where $\Delta(n)$ is the error term in the Prime Number Theorem. We investigate the mean value $\mathcal{M}_{p}(-s,\chi)$ for $\Re(s)>-1$, give an exact formula for the average $\mathcal{A}_{p}(\chi)$ and obtain the meromorphic continuation of the function $\Delta(s,\chi)$ to the region $\Re(s)>1/2$.