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Search: All articles in the CMB digital archive with keyword zeta function

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1. CMB Online first

Sebbar, Abdellah; Al-Shbeil, Isra
Elliptic Zeta functions and equivariant functions
In this paper we establish a close connection between three notions attached to a modular subgroup. Namely the set of weight two meromorphic modular forms, the set of equivariant functions on the upper half-plane commuting with the action of the modular subgroup and the set of elliptic zeta functions generalizing the Weierstrass zeta functions. In particular, we show that the equivariant functions can be parameterized by modular objects as well as by elliptic objects.

Keywords:modular form, equivariant function, elliptic zeta function
Categories:11F12, 35Q15, 32L10

2. CMB 2009 (vol 52 pp. 186)

Broughan, Kevin A.
Extension of the Riemann $\xi$-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
If $K$ is a number field with $n_k=[k:\mathbb{Q}]$, and $\xi_k$ the symmetrized Dedekind zeta function of the field, the inequality $$\Re\,{\frac{ \xi_k'(\sigma + {\rm i} t)}{\xi_k(\sigma + {\rm i} t)}} > \frac{ \xi_k'(\sigma)}{\xi_k(\sigma)}$$ for $t\neq 0$ is shown to be true for $\sigma\ge 1+ 8/n_k^\frac{1}{3}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k=\mathbb{Q}$ the inequality is extended to $\si\ge 1$ for all $t$ sufficiently large or small and to the region $\si\ge 1+1/(\log t -5)$ for all $t\neq 0$. This answers positively a question posed by Lagarias.

Keywords:Riemann zeta function, xi function, zeta zeros
Categories:11M26, 11R42

3. CMB 2008 (vol 51 pp. 627)

Vidanovi\'{c}, Mirjana V.; Tri\v{c}kovi\'{c}, Slobodan B.; Stankovi\'{c}, Miomir S.
Summation of Series over Bourget Functions
In this paper we derive formulas for summation of series involving J.~Bourget's generalization of Bessel functions of integer order, as well as the analogous generalizations by H.~M.~Srivastava. These series are expressed in terms of the Riemann $\z$ function and Dirichlet functions $\eta$, $\la$, $\b$, and can be brought into closed form in certain cases, which means that the infinite series are represented by finite sums.

Keywords:Riemann zeta function, Bessel functions, Bourget functions, Dirichlet functions
Categories:33C10, 11M06, 65B10

4. CMB 2008 (vol 51 pp. 334)

Ascah-Coallier, I.; Gauthier, P. M.
Value Distribution of the Riemann Zeta Function
In this note, we give a new short proof of the fact, recently discovered by Ye, that all (finite) values are equidistributed by the Riemann zeta function.

Keywords:Nevanlinna theory, deficiency, Riemann zeta function

5. CMB 2008 (vol 51 pp. 100)

Petkov, Vesselin
Dynamical Zeta Function for Several Strictly Convex Obstacles
The behavior of the dynamical zeta function $Z_D(s)$ related to several strictly convex disjoint obstacles is similar to that of the inverse $Q(s) = \frac{1}{\zeta(s)}$ of the Riemann zeta function $\zeta(s)$. Let $\Pi(s)$ be the series obtained from $Z_D(s)$ summing only over primitive periodic rays. In this paper we examine the analytic singularities of $Z_D(s)$ and $\Pi(s)$ close to the line $\Re s = s_2$, where $s_2$ is the abscissa of absolute convergence of the series obtained by the second iterations of the primitive periodic rays. We show that at least one of the functions $Z_D(s), \Pi(s)$ has a singularity at $s = s_2$.

Keywords:dynamical zeta function, periodic rays
Categories:11M36, 58J50

6. CMB 2007 (vol 50 pp. 11)

Borwein, David; Borwein, Jonathan
van der Pol Expansions of L-Series
We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series.

Keywords:Dirichlet series integrals, Hurwitz zeta functions, Plancherel theorems, L-series
Categories:11M35, 11M41, 30B50

7. CMB 2003 (vol 46 pp. 95)

Gauthier, P. M.
Cercles de remplissage for the Riemann Zeta Function
The celebrated theorem of Picard asserts that each non-constant entire function assumes every value infinitely often, with at most one exception. The Riemann zeta function has this Picard behaviour in a sequence of discs lying in the critical band and whose diameters tend to zero. According to the Riemann hypothesis, the value zero would be this (unique) exceptional value.

Keywords:cercles de remplissage, Riemann zeta function

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