Related papers: The $(-\beta)$-shift and associated Zeta Function

For any $a\in\mathbb{C}$, the zeros of $\zeta(s)-a$, denoted by $\rho_a=\beta_a+i\gamma_a$, are called $a$-points of the Riemann zeta function $\zeta(s)$. In this paper, we reformulate some basic results about the $a$-points of $\zeta(s)$…

We consider a family of isometric extensions of the full shift on p symbols (for p a prime) parametrized by a probability space. Using Heath-Brown's work on the Artin conjecture, it is shown that for all but two primes p the set of limit…

Let $0<a\leq1, s\in\mathbb{C}$, and $\zeta(s,a)$ be the Hurwitz zeta-function. Recently, T.~Nakamura showed that $\zeta(\sigma,a)$ does not vanish for any $0<\sigma<1$ if and only if $1/2\leq a \leq1$. In this paper, we show that…

We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by the Sine$_\beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power…

In this article, we establish an additive decomposition of the discrete zeta function (for $s \in \mathbb{N}^*$, $s > 1$), more precisely of the function $4(\zeta(s)-1)$, as a series whose general term is of the form $1/x_n(s) + 1/y_n(s) +…

For $\beta>1$, let $T_\beta:[0,1]\rightarrow [0,1)$ be the $\beta$-transformation. We consider an invariant $T_\beta$-orbit closure contained in a closed interval with diameter $1/\beta$, then define a function $\Xi(\alpha,\beta)$ by the…

For a finite group $G$, we consider the zeta function $\zeta_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for…

We formulate a parametrized uniformly absolutely globally convergent series of $\zeta$(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s -- 1)$\zeta$(s) + 1 x Li s z z -- 1 dz, where Li s (x) is the…

Let $p,x$ be real numbers, and $s$ be a complex number, with $\Re(s)>1-r$, $p\geq 1$, and $x+1>0$. The zeta function $Z^{\bf\alpha}_p(s;x)$ is defined by $$ Z^{\bf\alpha}_p(s;x) =\frac{1}{\Gamma(s)}\int^\infty_0 \frac{e^{-xt}}…

These notes are based on three lectures given by the second author at Copenhagen University (October 2009) and at Aarhus University, Denmark (December 2009). We mostly present here a survey of results of Dieter Mayer on relations between…

We introduce a class of coded systems that we construct from sofic systems and Dyck shifts and we study a class of subshifts that we obtain by excluding words of length two from Dyck shifts. We derive expressions for zeta functions and…

The Fischer, Krieger, and fiber product covers of sofic beta-shifts are constructed and used to show that every strictly sofic beta-shift is $2$-sofic. Flow invariants based on the covers are computed, and shown to only depend on an single…

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short…

In the paper, we shall establish the existence of a meromorphic continuation of the Global Zeta Function $\zeta(f,\chi)$ of a Global Number Field $K$ and also deduce the functional equation for the same, using different properties of the…

The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical…

This paper begins with a re-examination of the Riemann-Siegel Integral, which first discovered amongst by Bessel-Hagen in 1926 and expanded upon by C. L. Siegel on his 1932 account of Riemanns unpublished work on the zeta function. By…

This paper shows that, in the critical strip, the Riemann zeta function $\zeta(s)$ have the same set of zeros as $F(s):=\int_0^\infty t^{s-1}(e^t+1)^{-1}dt$, and then discusses the behavior of $F(s)$.

In this paper we investigate analytic inequalities related to a conjecture of Henry involving the difference between the Riemann zeta function and the digamma function. By treating $\zeta(s)-\psi(1-s)$ as a unified analytic object, we…