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On Witten multiple zeta-functions associated with semisimple Lie algebras I

Kohji Matsumoto, Hirofumi Tsumura (2006)

Annales de l’institut Fourier

We define Witten multiple zeta-functions associated with semisimple Lie algebras $𝔰𝔩 (n)$ , $(n = 2, 3, ...)$ of several complex variables, and prove the analytic continuation of them. These can be regarded as several variable generalizations of Witten zeta-functions defined by Zagier. In the case $𝔰𝔩 (4)$ , we determine the singularities of this function. Furthermore we prove certain functional relations among this function, the Mordell-Tornheim double zeta-functions and the Riemann zeta-function. Using these relations, we prove...

On zeros of approximate functions of the Rankin-Selberg L-functions

Masatoshi Suzuki (2009)

Acta Arithmetica

On Zeros of Dirirchlet's L Series.

H. Iwaniec (1974)

Inventiones mathematicae

On zeta-functions associated to certain cusp forms. I

A. Laurinčikas, J. Steuding (2004)

Open Mathematics

In the paper the asymptotics for Dirichlet polynomials associated to certain cusp forms are obtained.

On zeta-functions associated to certain cusp forms. II

Antanas Laurinčikas, Joern Steuding, Darius Šiaučiūnas (2009)

Open Mathematics

A formula for the mean value of multiplicative functions associated to certain cusp forms is obtained. The paper is a continuation of [4].

Operator representation of fermi-Dirac and Bose-Einstein integral functions with applications.

Chaudhry, M.Aslam, Qadir, Asghar (2007)

International Journal of Mathematics and Mathematical Sciences

Opět o Riemannově dzeta funkci

Břetislav Novák (1993)

Pokroky matematiky, fyziky a astronomie

Ordre, convergence et sommabilité de produits de séries de Dirichlet

Jean-Pierre Kahane, Hervé Queffélec (1997)

Annales de l'institut Fourier

L’article donne des réponses optimales ou presque optimales aux questions suivantes, qui remontent à Stieltjes, Landau et Bohr, et concernent des séries de Dirichlet $A_{j} = \sum_{n = 1}^{\infty} a (j, n) n^{- s}$ $(j = 1, 2 ...$

Ordre de grandeur de $L (1, χ)$ et de $L^{'} (1, χ)$

Jean-René Joly, Claude Moser (1979)

Annales de l'institut Fourier

On étudie sommairement la distribution des valeurs de $L^{'} (1, χ)$ ( $χ$ : caractère de Dirichlet primitif réel) et on constate qu’on a en général $L^{'} (1, χ) < π^{2} / 6$ ; on démontre par ailleurs que si $L^{'} (1, χ) < (π^{2} / 6) - ϵ$ , alors $L (1, χ) > c (ϵ) / log k$ ( $k$ : conducteur de $χ$ ; $c (ϵ)$ : constante positive effectivement calculable.

Oscillation of Mertens’ product formula

Harold G. Diamond, Janos Pintz (2009)

Journal de Théorie des Nombres de Bordeaux

Mertens’ product formula asserts that $\prod_{p \leq x} (1 - \frac{1}{p}) log x \to e^{- γ}$ as $x \to \infty$ . Calculation shows that the right side of the formula exceeds the left side for $2 \leq x \leq 10^{8}$ . It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $π (x) - li x$ , this and a complementary inequality might change their sense for sufficiently large values of $x$ . We show this to be the case.