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By Greco S., Strano R.

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Both holomorphic and Maass forms can be most convincingly put into a single framework through the study of the representation theory of G L(2, R) (or of the adele group G L(2, A) in the arithmetic case). Using the definition above, one can impose more regularity conditions at the cusps. Definition. , an automorphic function). -periodic function fa = f lk a a is of moderate growth at infinity. , Maass forms). , sJ. , Maass cusp forms). 2. Other equivalent formulations can be given. ,).. =F 0) and in L 2 (fo(q)\H) (with respect to the hyperbolic measure).

Then A (s; a) admits analytic continuation to a meromorphic function on C with simple poles at s = 1 and at s = 0 and it satisfies the functional equation · A(s; a)= IDI 112-s A(l- s; (aD)- 1), where D is the ideal class of the different of K /Q. JIDT . Summing over a E H(K), this proposition implies the analytic continuation and functional equation of {K (s) as stated in Section 4. 2), are absolutely convergent for Re(s) > 1, and since all partial zeta functions have the same residue at s = 1.

2. , [Wa]). C. 4] prove that the denominator of -bk/ k contains all the primes p such that p - 1 I 2k, and only them, so that for a p in the denominator we have xk- 1 = x- 1 (mod p). Hence even the "poles" modulo primes of ~(1 - k) are still "explained" by the divergence of the harmonic series! In the other direction, maybe it is not so surprising that the numerator of Bernoulli numbers should remain so mysterious. We finally only mention that the congruence properties of the values of the zeta function at negative integers were the motivation for the discovery by Leopoldt of the p-adic zeta function, later much generalized by others to p-adic L-functions of various kinds.

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Complete Intersections by Greco S., Strano R.

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