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By Courtieu M., Panchishkin A.A.

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Now we transform the summation into that one modulo Cχ . 63): Gn,Cχ (χ) = χ(n)G(χ), ¯ establishing the lemma. 81) M s−1 B S (n)n−s an1 n−s 1 Gnn1 ,M (χ) α(M ) n n 1 Ms = G(χ) B S (n)n−s an1 n−s 1 · α(M )Cχ n n1 dnn1 dnn1 µ(d) χ(d)δ · χ ¯ . d (M/Cχ ) (M/Cχ ) = d|(M/Cχ ) From the last formula we see that non-vanishing terms in the sum over n and n1 must satisfy the condition (M/Cχ d)|nn1 . Let us now split n1 into two factors n1 = n1 · n1 so that S(n1 ) ⊂ S, and (n1 , S) = 1. 83) Fq (q −s )−1 . 84) since (n1 , M ) = 1.

83) Fq (q −s )−1 . 84) since (n1 , M ) = 1. 76) and of the polynomials Hq (X) which we rewrite here in the form B S (n)n−s n Fq (q −s )−1 = q∈S (1 − α(q)q −s )−1 . 6 Complex valued distributions, associated with Euler products 43 Consequently, B S (n)n−s n an2 n−s 2 an1 n1 −s = S(n1 )⊂S S(n2 )⊂S and for S(n2 ) ⊂ S we have that B S (n)an1 . 86) (n1 ,S)=1 χ ¯ nn1 (M/Cχ d) . e. we put n2 = (M/Cχ d)n3 , S(n3 ) ⊂ S). We also note that by the definition of our Dirichlet series we have B S (n)an1 (nn1 )−s = D(s, χ) ¯ n,n1 −s Fq (χ(q)q ¯ ).

2 Concluding remarks This construction admits a generalization [Pa4] to the case of rather general Euler products over prime ideals in algebraic number fields. These Euler products have the form an N (n)−s = D(s) = n Fp (N (p)−s )−1 , p where n runs over the set of integrals ideals, and p over the set of prime ideals of integers OK of a number field K, with N (n) denoting the absolute norm of an ideal n, and Fp ∈ C[X] being polynomials with the condition Fp (0) = 1. In [Pa4] we constructed certain distributions, which provide integral representations for special values of Dirichlet series of the type χ(n)an N (n)−s , D(s, χ) = n where χ denote a Hecke character of finite order, whose conductor consists only of prime ideals belonging to a fixed finite set S of non-Archimedean places of K.

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Non-Archimedean analytic functions, measures and distributions by Courtieu M., Panchishkin A.A.

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