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Example text

For a < -1 the function ~'- ( -s) has a representation as an absolutely convergent Dirichlet series. Therefore, we can interchange sum and integral. 1. , a+ 1, s). ,a,v)da. , a) da. 3 as It I ~ oo. , a) converges uniformly in a E [1, 2]. Thus, we may interchange the order of integration. _(-s) ds. 9) 2rri L r(u) U + V + S- 1 = _1 11 Now we suppose that Re u > 1 and Re v < 1. The contour L can be taken as a straight line from c0 - ioo to c0 + ioo with -Reu < c0 < min(-1, 1Re(u + v)). Then there exists b such that max(- Re u - c0 , -1) < b < 0.

1 is, for a ~ 1 2 2a _ 1 + 2a _ 1 Re {A(O)- 2y0 Re {A(O) 4+0, 47 Moments + 2Re(s{(O)- sA(O) r'(u + r(u ~t)) + lt) - 2Re 1 1 . (e-21riAsA(u +it)- 1) + o(l). 17) Let 0 < x < 1. +2rrik, k E Z. 18) converges uniformly to a limit in every bounded part of the s-plane not containing a point n, n e N, as x ~ 1. By the analytic continuation principle this limit is SA (s). For positive integers n this is trivial. e. < 1, is summable in the Abel sense to the function sA(s). m m=l is summable in the Abel sense to the function Re SA (0).

8) was obtained for 0 < u < 1. Hence, by analytic continuation, it is valid for -1 < u < 1. Now let -1 < u < 0. 6) we replaces by s + 1. Then we have 1 00 1/f(x)exp{2n'il.. )-s f(1- s) s Moreover, we observe that ).. ) m m ->.. ).. ) m m+>.. a} x ~ (exp{2nima} _ exp{-nis- 2nimaJ). 10) equals -a-s- (~-a) exp{2rriA(l -a)}. , 1- s) - exp{-rris}L(-a, A, 1- s)), and by analytic continuation it remains valid for all s. 2. Notes The function {(s, a) was introduced by A. Hurwitz (1882). He also proved the functional equation for {(s, a).

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