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Search: MSC category 11M26 ( Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses )

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1. 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

2. CMB 2008 (vol 51 pp. 561)

Kuznetsov, Alexey
Expansion of the Riemann $\Xi$ Function in Meixner--Pollaczek Polynomials
In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner--Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

Categories:41A10, 11M26, 33C45

3. CMB 2004 (vol 47 pp. 468)

Soundararajan, K.
Strong Multiplicity One for the Selberg Class
We investigate the problem of determining elements in the Selberg class by means of their Dirichlet series coefficients at primes.

Categories:11M41, 11M26, 11M06

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