Dirichlet series associated with polynomials and applications |
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Authors: | Yoshio Tanigawa Wenguang Zhai |
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Institution: | a Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan b Department of Mathematics, Shandong Normal University, Jian, 250014 Shandong, PR China |
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Abstract: | We consider the possibility of the analytic continuation of the Dirichlet series SP;Z(s) associated with a polynomial P(x) and auxiliary series Z(s). In fact, we derive a certain criterion for the analytic continuation for some class of polynomials and give examples such that SP;Z(s) cannot be continued meromorphically to the whole plane C. We also study the asymptotic behaviors of the sum MP(x)=∑P(n1,…,nk)?xΛ(n1)?Λ(nk) considered first by M. Peter. Generalizations of this sum are also considered. |
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Keywords: | 11M26 11M41 11N37 |
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