FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES

In this paper we study the special Dirichlet series 
L(s)=23∑n=1∞sin(2πn3)n−s,  s∈C 
This series converges uniformly in the half-plane Re(s)>1 and thus represents a holomorphic function there. We show that the function L can be extended to a holomorphic function in the whole complex-plane. The values of the function L at the points 0,±1,−2,±3,−4,±5,… are obtained. The values at the positive integers 1,3,5,… are determined by means of a functional equation satisfied by L.

By a Dirichlet series we mean a series of the form n:l where the coefficients a are any given numbers, and s is a complex variable [1], n [2].
In this paper we study the special Dirichlet series _2 2=n -s L(s) sin(---)n s C n=l which converges uniformly in the half-plane Re(s) > and thus represents an analytic function there.In section we study the analytic behaviour of the function L beyond the half-plane Re(s) > I, and prove that the function L can be extended to a holomorphic function in the whole complex-plane.Moreover values of L at the points -m (m=0,I,2,3,...) are obtained at the end of this section.The values of L at the positive integers 1,3,5,... are determined by means of the functional equation L() is uniformly convergent in the half-plane Re(s) and so it represents an analytic function there.The aim of this section is to extend L to the whole complex plane and to prove that L is holomorphic in C. LEMMA ,where E e 2rI/3  can be extended to a holomorphic function in the whole complex plane.
PROOF.Let us define P and Q for Re(s) > by The integral f G(t)t s-ldt exists and converges uniformly in any finite region of the s-plane, is bounded for all values of Re(s), and we can compare the integral with that of i/t2.Thus Q is an entire function.Recall from Lemma 2.2 that g(t) =-a t 2n n ,t ,[ o,1] n=o the convergence being uniform on [0,i].We deduce for Re(s) that P(s) =' a t2n+s-ldt n n=o n=o Thus P is a meromorphic function on C with simple poles at 0,-2,-4,-6 Since I/F is an entire function we may now extend L to the whole of C by L(s) __P(s) Since Q and I/F are entire functions, the singularities of L can only be those of P/F We have seen that P has simple poles at 0,-2,-4,-6 Since I/F has simple zeros at 0,-2,-4,... it follows that L is regular for all values of s in the complex plane.This completes the proof of the theorem.LEMMA 2.3.(i) L has zeros at -I,-3,-5 (ii) The values of L at 0,-2,-4,-6 are given by L(-2m) (2m)!a ,m 0 ,1,2,5,4 PROOF.(i) This follows immediately from the fact that I/r has zeros at 0,-I, -2,-3,..., and thus P(1-2m)  According to Cauchy's theorem, the integral around C is zero.Thus (-t)s-  We have seen in the proof of theorem 2.1 that the function defined by the integral ts_i t -t e +e +1 is a meromorphic function with simple poles at the points 0,-2,-4, Since the function sin(s) has simple zeros at 0,-2,-4 it follows that I is regular for FUNCTIONAL EQUATION OF A SPECIAL DIRICHLET SERIES all values of s in the complex plane.If we assume s x is a negative real number, then we have (-t)x-1 e(X-1)lg (-t)   It follows that x-1 x-1 I(-t)i

P
li) As in (i) we use the partial fraction (2.3) of L to get >-2m r s) 2m/s m Since F has simple poles at the points -m (m=0,I,2,3,...) with residues (-l)m/m!, the values of L at 1,3,5,..., by the use of the functional equation obtained above.LEMMA 3.1.There exists an integral function I such that L(s) =--P(1-s)I(s) ,s C PROOF.Let 0 < r < i, and let Cr be the contour consisting of the paths C the contour in figure (b).
positive real axis from R to r, a circle radius r and center n at the origin oriented in the positive direction, the positive real axis from r to R and finally a circle of radius R with center at the origin oriented in the n and D (0) the integrand has poles at the points r n 2=i 2=i(3m-I), m=1,2,3, +/--5---+ --5 -(3m+I) and +/ section give the proof of the lemma.
2.1.For all values of s in the half-plane Re(s) >