@phdthesis{Srichan2015,
  author    = {Srichan, Teerapat},
  title     = {Discrete Moments of Zeta-Functions with respect to random and ergodic transformations},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-118395},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2015},
  abstract  = {In the thesis discrete moments of the Riemann zeta-function and allied Dirichlet series are studied. In the first part the asymptotic value-distribution of zeta-functions is studied where the samples are taken from a Cauchy random walk on a vertical line inside the critical strip. Building on techniques by Lifshits and Weber analogous results for the Hurwitz zeta-function are derived. Using Atkinson's dissection this is even generalized to Dirichlet L-functions associated with a primitive character. Both results indicate that the expectation value equals one which shows that the values of these zeta-function are small on average. The second part deals with the logarithmic derivative of the Riemann zeta-function on vertical lines and here the samples are with respect to an explicit ergodic transformation. Extending work of Steuding, discrete moments are evaluated and an equivalent formulation for the Riemann Hypothesis in terms of ergodic theory is obtained. In the third and last part of the thesis, the phenomenon of universality with respect to stochastic processes is studied. It is shown that certain random shifts of the zeta-function can approximate non-vanishing analytic target functions as good as we please. This result relies on Voronin's universality theorem.},
  subject      = {Riemannsche Zetafunktion},
  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}
}