Soient $P\left(\underset{\_}{x}\right)=P(x_1,...,x_n)$ et $Q\left(\underset{\_}{x}\right)=Q(x_1,...,x_n)$ deux polynômes à coefficients positifs vérifiant :$$\underset{\genfrac{}{}{0pt}{}{\left|\underset{\_}{x}\right|\to +\infty }{x_1,...,x_n\ge 1}}{lim}\frac{P\left(\underset{\_}{x}\right)}{Q\left(\underset{\_}{x}\right)}=+\infty .$$Soient $\underset{\_}{\eta }=({\eta }_1,...,{\eta }_n)\in {\mathbf{N}}^n$ et $R=P/Q$. On étudie la série de Dirichlet $Z(R,\underset{\_}{\eta };s)={\sum }_{{\eta }_1,...,{\eta }_n=1}^{\infty }{\underset{\_}{\eta }}^{\underset{\_}{\eta }}R(\underset{\_}{\eta }{)}^{-s}$ : abscisse de convergence absolue, existence et nature du prolongement méromorphe, ordre de grandeur dans les bandes verticales. On donne un procédé de construction du prolongement méromorphe de la fonction $s\mapsto Z(R,\underset{\_}{\eta };s)$ qui ne dépend que de $\underset{\_}{\eta }$ et de certains monômes de $P$ et $Q$: les monômes extrémaux.

In questo lavoro vengono studiati gli zeri reali di una classe di serie di Dirichlet, che generalizzano le funzioni $L(s,\chi )$, definite in [8], Combinando le tecniche elementari di Pintz [9] con alcuni metodi analitici si ottiene l’estensione dei classici teoremi di Hecke e Siegel.

The well-known estimate of the order of the Hurwitz zeta function      $\zeta (s,\alpha )-{\alpha }^{-s}\ll t^{c{(1-\sigma )}^{3/2}}lo{g}^{2/3}t$ 0.    The improvement of the constant c is a consequence of some technical modifications in the method of estimating exponential sums sketched by Heath-Brown ([11], p. 136).

We give a Chowla-Selberg type formula that connects a generalization of the eta-function to $GL\left(n\right)$ with multiple gamma functions. We also present some simple infinite product identities for certain special values of the multiple gamma function.