On a Dirichlet series associated with a polynomial

Abstract

Let P ( x ) = ∏ j = 2 k ( x + δ j ) P(x) = \prod \nolimits _{j = 2}^k {(x + {\delta _j})} be a polynomial with real coefficients and Re ⁡ δ j > − 1 ( j = 1 , … , k ) \operatorname {Re} {\delta _j} > - 1(j = 1, \ldots ,k) . Define the zeta function Z p ( s ) {Z_p}(s) associated with the polynomial P ( x ) P(x) as \[ Z P ( s ) = ∑ n = 1 ∞ 1 P ( n ) s , Re ⁡ s > 1 / k . {Z_P}(s) = \sum \limits _{n = 1}^\infty {\frac {1}{{P{{(n)}^s}}}} ,\operatorname {Re} s > 1/k. \] Z P ( s ) Z_P(s) is holomorphic for Re ⁡ s > 1 / k \operatorname {Re} s > 1/k and it has an analytic continuation in the whole complex s s -plane with only possible simple poles at s = j / k ( j = 1 , 0 , − 1 , − 2 , − 3 , … ) 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 ) {Z_P}( - m) for any non-negative integer m m , the asymptotic formula of Z P ( s ) {Z_P}(s) at s = 1 / k s = 1/k , the value Z P ′ ( 0 ) {Z’_P}(0) and its application to the determinants of elliptic operators.