On a Dirichlet series associated with a polynomial
Author:
Min King Eie
Journal:
Proc. Amer. Math. Soc. 110 (1990), 583590
MSC:
Primary 11M41; Secondary 11F66
DOI:
https://doi.org/10.1090/S00029939199010372060
MathSciNet review:
1037206
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: Let $P(x) = \prod \nolimits _{j = 2}^k {(x + {\delta _j})}$ be a polynomial with real coefficients and $\operatorname {Re} {\delta _j} >  1(j = 1, \ldots ,k)$. Define the zeta function ${Z_p}(s)$ associated with the polynomial $P(x)$ as \[ {Z_P}(s) = \sum \limits _{n = 1}^\infty {\frac {1}{{P{{(n)}^s}}}} ,\operatorname {Re} s > 1/k.\] $Z_P(s)$ is holomorphic for $\operatorname {Re} s > 1/k$ and it has an analytic continuation in the whole complex $s$plane with only possible simple poles at $s = j/k(j = 1,0,  1,  2,  3, \ldots )$ other than nonpositive integers. In this paper, we shall obtain the explicit value of ${Z_P}(  m)$ for any nonnegative integer $m$, the asymptotic formula of ${Z_P}(s)$ at $s = 1/k$, the value ${Zâ€™_P}(0)$ and its application to the determinants of elliptic operators.

W. L. Baily, Jr., Introductory lectures on automorphic forms, Princeton Univ. Press, 1973.
 Billie Chandler Carlson, Special functions of applied mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1977. MR 0590943
 Min King Eie, A zetafunction associated with zero ternary forms, Proc. Amer. Math. Soc. 94 (1985), no. 3, 387â€“392. MR 787878, DOI https://doi.org/10.1090/S00029939198507878789
 Akira Kurihara, On the values at nonpositive integers of Siegelâ€™s zeta functions of ${\bf Q}$anisotropic quadratic forms with signature $(1,\,n1)$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 567â€“584 (1982). MR 656037
 Yukihiko Namikawa, Toroidal compactification of Siegel spaces, Lecture Notes in Mathematics, vol. 812, Springer, Berlin, 1980. MR 584625
 Yukihiko Namikawa, A new compactification of the Siegel space and degeneration of Abelian varieties. I, Math. Ann. 221 (1976), no. 2, 97â€“141. MR 480537, DOI https://doi.org/10.1007/BF01433145
 I. Satake, Special values of zeta functions associated with selfdual homogeneous cones, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, BirkhĂ¤user, Boston, Mass., 1981, pp. 359â€“384. MR 642867
 Mikio Sato and Takuro Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131â€“170. MR 344230, DOI https://doi.org/10.2307/1970844
 Takuro Shintani, On zetafunctions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 22 (1975), 25â€“65. MR 0384717
 Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields at nonpositive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393â€“417. MR 427231
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11M41, 11F66
Retrieve articles in all journals with MSC: 11M41, 11F66
Additional Information
Article copyright:
© Copyright 1990
American Mathematical Society