Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion

Laurinčikas, Antanas

Journal de théorie des nombres de Bordeaux, Tome 8 (1996) no. 2, pp. 315-329.

Dans cet article on prouve un théorème limite dans l’espace des fonctions continues pour le polynôme de Dirichlet

$$\sum _{m\le T}\frac{d_{{\kappa }_T}\left(m\right)}{m^{{\sigma }_T+it}}$$

où $d_{{\kappa }_T}\left(m\right)$ sont les coefficients du développement en série de Dirichlet de la fonction ${\zeta }^{{\kappa }_T}\left(s\right)$ dans le demi-plan $\sigma >1$, ${\kappa }_T={(2^{-1}log{l}_T)}^{-\frac{1}{2}}$, ${\sigma }_T=\frac{1}{2}+\frac{{log}^2{l}_T}{l_T}$, $l_T>0$, $l_T\le logT$ et $l_T\to \infty$ lorsque $T\to \infty .$

A limit theorem in the space of continuous functions for the Dirichlet polynomial

$$\sum _{m\le T}\frac{d_{{\kappa }_T}\left(m\right)}{m^{{\sigma }_T+it}}$$

where $d_{{\kappa }_T}\left(m\right)$ denote the coefficients of the Dirichlet series expansion of the function ${\zeta }^{{\kappa }_T}\left(s\right)$ in the half-plane $\sigma >1$ ${\kappa }_T={(2^{-1}log{l}_T)}^{-\frac{1}{2}}$, ${\sigma }_T=\frac{1}{2}+\frac{1n^2{l}_T}{l_T}$ and $l_T>0$, $l_T\le$ 1n $T$ and $l_T\to \infty$ as $T\to \infty$, is proved.

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