Dirichlet series related to the Riemann zeta function
Dirichlet series related to the Riemann zeta function
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与黎曼 zeta 函数相关的狄利克雷级数
DOI：
10.1016/0022-314x(84)90094-5
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发表时间：
1984
影响因子：
0.7
通讯作者：
Thiennu H. Vu
中科院分区：
文献类型：
--
作者：
T. Apostol;Thiennu H. Vu
研究了由Dirichlet级数esh (s,z) =Σn=1∞n−sΣm=1nm−z的解析延拓所定义的函数h (s,z)，其中andzz为复变量。对于每一个固定的zit，证明了th （s,z）作为s的亚纯函数存在于整个平面上，并确定了其极点和残数。此外，对于每个定值≠1，证明了ath （s,z）作为z的亚纯函数存在于整个平面上，并确定了其极点和残数。给出了H（s,z）的两种不同的表示，并由此推导出了互易律H(s,z) +H(z,s) =ζ(s)ζ(z) +ζ(s+z)。对于每个整数q≥0，函数值esh （s, - q）和h （- q,s）用黎曼ζ函数表示。对于Dirichlet序列(s,z) =Σn=1∞n−sΣm=1nm−z(m+n)−1，也得到了类似的结果。应用包括Ramanujan， Williams， Rao和Sarma之前得到的恒等式。
A study is made of the functionH(s,z) defined by analytic continuation of the Dirichlet seriesH(s,z) =Σn=1∞n−sΣm=1nm−z, wheresandzare complex variables. For each fixedzit is shown thatH(s,z) exists in the entires-plane as a meromorphic function ofs, and its poles and residues are determined. Also, for each fixeds≠ 1 it is shown thatH(s,z) exists in the entirez-plane as a meromorphic function ofz, and again its poles and residues are determined. Two different representations ofH(s,z) are given from which a reciprocity law,H(s,z) +H(z,s) =ζ(s)ζ(z) +ζ(s+z), is deduced. For each integerq≥ 0 the function valuesH(s, −q) andH(−q,s) are expressed in terms of the Riemann zeta function. Similar results are also obtained for the Dirichlet seriesT(s,z) =Σn=1∞n−sΣm=1nm−z(m+n)−1. Applications include identities previously obtained by Ramanujan, Williams, and Rao and Sarma.