dc.contributor.author Elma, Ertan dc.date.accessioned 2020-07-27 17:17:48 (GMT) dc.date.available 2020-07-27 17:17:48 (GMT) dc.date.issued 2020-07-27 dc.date.submitted 2020-07-23 dc.identifier.uri http://hdl.handle.net/10012/16079 dc.description.abstract In this thesis, we obtain several results in number theory. en 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 $\Re(s)>1/2$. dc.language.iso en en dc.publisher University of Waterloo en dc.subject Dirichlet L-functions en dc.subject error term in the Prime Number Theorem en dc.subject number of prime factors en dc.subject Dirichlet characters en dc.title Some Problems in Multiplicative and Additive Number Theory en dc.type Doctoral Thesis en dc.pending false uws-etd.degree.department Pure Mathematics en uws-etd.degree.discipline Pure Mathematics en uws-etd.degree.grantor University of Waterloo en uws-etd.degree Doctor of Philosophy en uws.contributor.advisor Liu, Yu-Ru uws.contributor.advisor Kuo, Wentang uws.contributor.affiliation1 Faculty of Mathematics en uws.published.city Waterloo en uws.published.country Canada en uws.published.province Ontario en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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