By Courtieu M., Panchishkin A.A.

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D) χ(d)· d S(n3 )⊂S The proof of the theorem is accomplished by noting that µ(d)ds−1 χ(d)α(d)−1 = d|(M/Cχ ) (1 − χ(q)α(q)−1 q s−1 ), q∈S\S(χ) χ(n ¯ 3 )α(n3 )n−s 3 −s −1 (1 − χ(q)α(q)q ¯ ) = S(n3 )⊂S q∈S\S(χ) −s −1 −s Fq (χ(q)q ¯ ) Hq (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.

86) transforms to the following Ms G(χ) α(M )Cχ · = −s Fq (χ(q)q ¯ ) q∈S\S(χ) χ(n ¯ 3 )α S(n3 )⊂S Cχs−1 α(Cχ ) n3 M Cχ d n3 M cχ d d|(M/Cχ ) −s = −s Fq (χ(q)q ¯ ) G(χ)D(s, χ) ¯ q∈S\(χ) µ(d)ds−1 χ(d)α(d)−1 · d|(M/Cχ ) χ(n ¯ 3 )α(n3 )n−s 3 . · µ(d) χ(d)· d S(n3 )⊂S The proof of the theorem is accomplished by noting that µ(d)ds−1 χ(d)α(d)−1 = d|(M/Cχ ) (1 − χ(q)α(q)−1 q s−1 ), q∈S\S(χ) χ(n ¯ 3 )α(n3 )n−s 3 −s −1 (1 − χ(q)α(q)q ¯ ) = S(n3 )⊂S q∈S\S(χ) −s −1 −s Fq (χ(q)q ¯ ) Hq (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.

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 ¯ ). 86) transforms to the following Ms G(χ) α(M )Cχ · = −s Fq (χ(q)q ¯ ) q∈S\S(χ) χ(n ¯ 3 )α S(n3 )⊂S Cχs−1 α(Cχ ) n3 M Cχ d n3 M cχ d d|(M/Cχ ) −s = −s Fq (χ(q)q ¯ ) G(χ)D(s, χ) ¯ q∈S\(χ) µ(d)ds−1 χ(d)α(d)−1 · d|(M/Cχ ) χ(n ¯ 3 )α(n3 )n−s 3 . · µ(d) χ(d)· d S(n3 )⊂S The proof of the theorem is accomplished by noting that µ(d)ds−1 χ(d)α(d)−1 = d|(M/Cχ ) (1 − χ(q)α(q)−1 q s−1 ), q∈S\S(χ) χ(n ¯ 3 )α(n3 )n−s 3 −s −1 (1 − χ(q)α(q)q ¯ ) = S(n3 )⊂S q∈S\S(χ) −s −1 −s Fq (χ(q)q ¯ ) Hq (χ(q)q ¯ ).

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


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