ON ANALYTIC CONTINUATION AND FUNCTIONAL EQUATION OF CERTAIN OIRICHLET SERIES

Analytic continuation and functional equation of Riemann's type are proved for a 
class of Dirichlet series associated to rational functions.


hi-(s) y
Oj>O (nj-l)!h=OE (2,) () h!T) h-[(-i) s-h(s+h,l_Oj)+ where'he last sum appears if Q(I)=0 with multiplicity n o, g(s,a) is the Hurwitz zeta function with 0 < a < and T are suitable constants computed in 53, then ,I,(,) is an entire function and the function satisfies the functional equation (s)= (I-s).
We note that the class of Dirichlet series which satisfy our hypotheses contains strictly that of linear combination of shifted Dirichlet L-series L(s-k, X), k non-negative integer.
2. SOME LEMMAS.LEMMA 2.1.Let G(z)= lan zn be a complex power series with radius of convergence p,= 1.If z is a pole of order n o _> for G(z), then the Dirich]et series L()= .=Elann has a meromorphic continuation over c with simple poles at s n o and possibly at s 1,...,%-1.If z is a regular point, then L(s) is continuable as an entire function.PROOF.From the classical integral representation of r(,) one gets .''--Jlan[-nt ts-1 The function G(e-t) is 0(e-Kt) for a suitable K > 0, as t-+ oo, and is infinite of order %, as t-,0 +, provided z is a pole.So we get from (2.1)I I G(e-t) tS-dt ( large).
Let G(e-t)= ant n be the Laurent expansion at 0; we can suppose that its outer radius A is greater than (for otherwise we write A instead in (2.2)).Then The last series in (2.3) defines a meromorphic function with simple poles at s n o and possibly at s n < n o, since it converges uniformly on any compact subset of { e C: Is + n[ >_ C,n > no}, C being fixed positive constant.
The second claim follows from the above argument with n o 0. Z E Res(II(z)( z)-s ,zj, l), where H(z)=G(e -z) and zj, k =2ri(-Oj+k) are the non zero complex numbers such that exp( zj, k) rj, < j <. m.
PROOF.Let CN, and r r be the contours drawn below.We put CN, r Here (-z) s-is defined to be ezp((s-1)log(-z)), where the logarithm is real on the positive real axis.If is sufficiently small and N # I'-'j, k for each j,k, then the above functions are analytic functions of s, because they are expressed as sums of absolutely convergent integrals with parameter of univalent holomorphic functions.
N,r l-r.
<2rLe[tlra-n where L sup H(z)( z) n I: z r} and s a + it.It follows that, for a > n o,