@phdthesis{Schmeller2022,
  author    = {Schmeller, Christof},
  title     = {Uniform distribution of zero ordinates of Epstein zeta-functions},
  doi       = {10.25972/OPUS-25199},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-251999},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2022},
  abstract  = {The dissertation investigates the wide class of Epstein zeta-functions in terms of uniform distribution modulo one of the ordinates of their nontrivial zeros. Main results are a proof of a Landau type theorem for all Epstein zeta-functions as well as uniform distribution modulo one for the zero ordinates of all Epstein zeta-functions asscoiated with binary quadratic forms.},
  subject      = {Zetafunktion},
  language  = {en}
}
@phdthesis{Rehberg2020,
  author    = {Rehberg, Martin},
  title     = {Weighted uniform distribution related to primes and the Selberg Class},
  doi       = {10.25972/OPUS-20925},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-209252},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {In the thesis at hand, several sequences of number theoretic interest will be studied in the context of uniform distribution modulo one. <br> <br> In the first part we deduce for positive and real \(z\not=1\) a discrepancy estimate for the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \), where \(\gamma_a\) runs through the positive imaginary parts of the nontrivial \(a\)-points of the Riemann zeta-function. If the considered imaginary parts are bounded by \(T\), the discrepancy of the sequence \( \left((2\pi )^{-1}(\log z)\gamma_a\right) \) tends to zero like \( (\log\log\log T)^{-1} \) as \(T\rightarrow \infty\). The proof is related to the proof of Hlawka, who determined a discrepancy estimate for the sequence containing the positive imaginary parts of the nontrivial zeros of the Riemann zeta-function. <br> <br> The second part of this thesis is about a sequence whose asymptotic behaviour is motivated by the sequence of primes. If \( \alpha\not=0\) is real and \(f\) is a function of logarithmic growth, we specify several conditions such that the sequence \( (\alpha f(q_n)) \) is uniformly distributed modulo one. The corresponding discrepancy estimates will be stated. The sequence \( (q_n)\) of real numbers is strictly increasing and the conditions on its counting function \( Q(x)=\\#\lbrace q_n \leq x \rbrace \) are satisfied by primes and primes in arithmetic progessions. As an application we obtain that the sequence \( \left( (\log q_n)^K\right)\) is uniformly distributed modulo one for arbitrary \(K>1\), if the \(q_n\) are primes or primes in arithmetic progessions. The special case that \(q_n\) equals the \(\textit{n}\)th prime number \(p_n\) was studied by Too, Goto and Kano. <br> <br> In the last part of this thesis we study for irrational \(\alpha\) the sequence \( (\alpha p_n)\) of irrational multiples of primes in the context of weighted uniform distribution modulo one. A result of Vinogradov concerning exponential sums states that this sequence is uniformly distributed modulo one. An alternative proof due to Vaaler uses L-functions. We extend this approach in the context of the Selberg class with polynomial Euler product. By doing so, we obtain two weighted versions of Vinogradov's result: The sequence \( (\alpha p_n)\) is \( (1+\chi_{D}(p_n))\log p_n\)-uniformly distributed modulo one, where \( \chi_D\) denotes the Legendre-Kronecker character. In the proof we use the Dedekind zeta-function of the quadratic number field \( \Bbb Q (\sqrt{D})\). As an application we obtain in case of \(D=-1\), that \( (\alpha p_n)\) is uniformly distributed modulo one, if the considered primes are congruent to one modulo four. Assuming additional conditions on the functions from the Selberg class we prove that the sequence \( (\alpha p_n) \) is also \( (\sum_{j=1}^{\nu_F}{\alpha_j(p_n)})\log p_n\)-uniformly distributed modulo one, where the weights are related to the Euler product of the function.},
  subject      = {Zahlentheorie},
  language  = {en}
}
@phdthesis{Sourmelidis2020,
  author    = {Sourmelidis, Athanasios},
  title     = {Universality and Hypertranscendence of Zeta-Functions},
  doi       = {10.25972/OPUS-19369},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-193699},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2020},
  abstract  = {The starting point of the thesis is the {\it universality} property of the Riemann Zeta-function \$\zeta(s)\$ which was proved by Voronin in 1975: {\it Given a positive number \$\varepsilon>0\$ and an analytic non-vanishing function \$f\$ defined on a compact subset \$\mathcal{K}\$ of the strip \$\left\{s\in\mathbb{C}:1/2 < \Re s< 1\right\}\$ with connected complement, there exists a real number \$\tau\$ such that \begin{align}\label{continuous} \max\limits_{s\in \mathcal{K}}|\zeta(s+i\tau)-f(s)|<\varepsilon. \end{align} } In 1980, Reich proved a discrete analogue of Voronin's theorem, also known as {\it discrete universality theorem} for \$\zeta(s)\$: {\it If \$\mathcal{K}\$, \$f\$ and \$\varepsilon\$ are as before, then \begin{align}\label{discretee} \liminf\limits_{N\to\infty}\dfrac{1}{N}\sharp\left\{1\leq n\leq N:\max\limits_{s\in \mathcal{K}}|\zeta(s+i\Delta n)-f(s)|<\varepsilon\right\}>0, \end{align} where \$\Delta\$ is an arbitrary but fixed positive number. } We aim at developing a theory which can be applied to prove the majority of all so far existing discrete universality theorems in the case of Dirichlet \$L\$-functions \$L(s,\chi)\$ and Hurwitz zeta-functions \$\zeta(s;\alpha)\$, where \$\chi\$ is a Dirichlet character and \$\alpha\in(0,1]\$, respectively. Both of the aforementioned classes of functions are generalizations of \$\zeta(s)\$, since \$\zeta(s)=L(s,\chi_0)=\zeta(s;1)\$, where \$\chi_0\$ is the principal Dirichlet character mod 1. Amongst others, we prove statement (2) where instead of \$\zeta(s)\$ we have \$L(s,\chi)\$ for some Dirichlet character \$\chi\$ or \$\zeta(s;\alpha)\$ for some transcendental or rational number \$\alpha\in(0,1]\$, and instead of \$(\Delta n)_{n\in\mathbb{N}}\$ we can have: \begin{enumerate} \item \textit{Beatty sequences,} \item \textit{sequences of ordinates of \$c\$-points of zeta-functions from the Selberg class,} \item \textit{sequences which are generated by polynomials.} \end{enumerate} In all the preceding cases, the notion of {\it uniformly distributed sequences} plays an important role and we draw attention to it wherever we can. Moreover, for the case of polynomials, we employ more advanced techniques from Analytic Number Theory such as bounds of exponential sums and zero-density estimates for Dirichlet \$L\$-functions. This will allow us to prove the existence of discrete second moments of \$L(s,\chi)\$ and \$\zeta(s;\alpha)\$ on the left of the vertical line \$1+i\mathbb{R}\$, with respect to polynomials. In the case of the Hurwitz Zeta-function \$\zeta(s;\alpha)\$, where \$\alpha\$ is transcendental or rational but not equal to \$1/2\$ or 1, the target function \$f\$ in (1) or (2), where \$\zeta(\cdot)\$ is replaced by \$\zeta(\cdot;\alpha)\$, is also allowed to have zeros. Until recently there was no result regarding the universality of \$\zeta(s;\alpha)\$ in the literature whenever \$\alpha\$ is an algebraic irrational. In the second half of the thesis, we prove that a weak version of statement \eqref{continuous} for \$\zeta(s;\alpha)\$ holds for all but finitely many algebraic irrational \$\alpha\$ in \$[A,1]\$, where \$A\in(0,1]\$ is an arbitrary but fixed real number. Lastly, we prove that the ordinary Dirichlet series \$\zeta(s;f)=\sum_{n\geq1}f(n)n^{-s}\$ and \$\zeta_\alpha(s)=\sum_{n\geq1}\lfloor P(\alpha n+\beta)\rfloor^{-s}\$ are hypertranscendental, where \$f:\mathbb{N}\to\mathbb{C}\$ is a {\it Besicovitch almost periodic arithmetical function}, \$\alpha,\beta>0\$ are such that \$\lfloor\alpha+\beta\rfloor>1\$ and \$P\in\mathbb{Z}[X]\$ is such that \$P(\mathbb{N})\subseteq\mathbb{N}\$.},
  subject      = {Analytische Zahlentheorie},
  language  = {en}
}