On a Dirichlet series associated with a polynomial

Author:
Min King Eie

Journal:
Proc. Amer. Math. Soc. **110** (1990), 583-590

MSC:
Primary 11M41; Secondary 11F66

DOI:
https://doi.org/10.1090/S0002-9939-1990-1037206-0

MathSciNet review:
1037206

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Abstract: Let be a polynomial with real coefficients and . Define the zeta function associated with the polynomial as

In this paper, we shall obtain the explicit value of for any non-negative integer , the asymptotic formula of at , the value and its application to the determinants of elliptic operators.

**[1]**W. L. Baily, Jr.,*Introductory lectures on automorphic forms*, Princeton Univ. Press, 1973.**[2]**B. C. Carlson,*Special functions of applied mathematics*, Academic Press, 1977. MR**0590943 (58:28707)****[3]**Minking Eie,*A zeta-function associated with zero ternary forms*, Proc. Amer. Math. Soc.**94**(1985), 387-392. MR**787878 (86g:11022)****[4]**A. Kurihara,*On the values at non-positive integers of Siegel's zeta functions of**-anisotroptc quadratic forms with signature*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**28**(1981), 567-584. MR**656037 (84a:10021)****[5]**Y. Namikawa,*Toroidal compactification of Siegel spaces*, Lecture Notes in Math., vol. 812, Springer-Verlag, Berlin and New York. MR**584625 (82a:32034)****[6]**-,*A new compactification of the Siegel space and degeneration of abelian varieties*. I, Math. Ann.**221**(1976), 97-141. MR**0480537 (58:697a)****[7]**I. Satake,*Special values of zeta functions associated with self dual homogeneous cones*, manuscript, 1981. MR**642867 (83h:10051)****[8]**M. Sato and T. Shintani,*On zeta functions associated with prehomogeneous vector spaces*, Ann. of Math.**100**(1974), 131-170. MR**0344230 (49:8969)****[9]**T. Shintani,*Zeta-functions associated with the vector of quadratic forms*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**22**(1975), 25-65. MR**0384717 (52:5590)****[10]**-,*On evaluation of zeta functions of totally real algebraic number fields at non-positive integers*, J. Fac. Sci. Univ. Tokyo**23**(1976), 393-417. MR**0427231 (55:266)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1037206-0

Article copyright:
© Copyright 1990
American Mathematical Society