摘要
Dirichlet L-函数是Riemannζ-函数的目然推厂.在本文中我们首先给出关于ζ-函数与一些L-函数特殊值的Apery型级数方面的综述.然后建立了关于几类同余式的两个转换定理.基于这些结果与相关的计算,我们猜出了涉及这类特殊值与相关常数的48个新的级数(其中许多与调和数有关).例如,我们猜测以及其中(k/3)为Legendre符号.
Dirichlet's L-functions are natural extensions of the Riemann zeta function. In this paper we first give a brief survey of Apery-like series for some special values of the zeta function and certain L-functions. Then, we establish two theorems on transformations of certain kinds of congruences. Motivated by the results and based on our computation, we pose 48 new conjectural series (most of which involve harmonic numbers) for such special values and related constants. For example, we conjecture that
∞∑k=1 1/k4(2kk)(1/k+2k∑j=k1/j)=11/9ζ(5),
∞∑k=1 (-1)k-1/k3(2kk)(1/5k3+k∑j=11/j3)=2/5ζ(3)2
and ∞∑k=1 48k/k(2k-1)(4k2k)(2kk)=15/2∞∑k=1(k3)/k2,
where (k) is the Legendre symbol.
出处
《南京大学学报(数学半年刊)》
2015年第2期189-218,共30页
Journal of Nanjing University(Mathematical Biquarterly)
基金
Supported by the National Natural Science Foundation(grant 11171140)of China