On a generalization of a formula of Ser and applications to the Riemann zeta function and to Dirichlet L -series

摘要

Let s_n = 1 + 1/2 +...+ l/(n - 1) - logn. In 1995 and 2006, the author has foundrnseries transformations of the type Σ_k=0~n μn,k, t_1 ~Sk+t_2 with integer coefficients μ_n,k,t_1, from which follows geometric convergence to Euler's constant γ for t_1, t_2 = O(n). In recently published papers, T. Rivoal and Kh. & T. Hessami Pilehrood have generalized these results. One starting point for such investigations is the formula of J. Ser, which expresses the remainder γ - s_n by a rational series. In this paper we generalize Ser's formula to Stieltjes constants of order zero and to the values of the Riemann zeta function and Dirichlet L-series at integer points 2, 3,... We investigate a linear series transformation Σ_k=0~n μn,k, t Σ_m=1~k+t l/m~r which converges quickly to ζ (r) for any fixed positive parameter t when n tends to infinity.