Một kết quả cổ điển liên quan đến \zeta(k) với k chẵn


Hôm rồi loạng quạng gặp phải kết quả này, họ bảo cổ điển nhưng mình chưa bao giờ nghe thấy và cũng không tài nào tìm thấy! 😀  Mình trích cả nó ra đây, ai biết thì giúp mình cái nhé! Cho luôn file hoặc tên sách thì tốt quá! 😛

…Let p be an odd prime number, \mu_p the group of p-th roots of unity, and \mathbb{Q}_p the field of p-adic numbers. Write U'_0 for the group of local units in \mathbb{Q}_p(\mu_p), which are \equiv 1 and have norm 1 to \mathbb{Q}_p. Let C_0 be the classical group of cyclotomic units of \mathbb{Q}(\mu_p) , which are \equiv 1 modulo the unique prime \mathfrak{p}_0 above p, and let \overline{C_0} be the closure of C_0 in the \mathfrak{p}_0-adic topology. Denote by G_0  the Galois group of \mathbb{Q}_p(\mu_p) over \mathbb{Q}_p, and by \chi the canonical character giving the action of G_0 on \mu_p . Let \zeta (s) denote the Riemann zeta function. Then it is classical that, for each even integer k with 1<k<p-1, the \chi^k-th eigenspace for the action of G_0 on U'_0/\overline{C_0} is non-trivial if and only if p divides (2\pi i)^{-k}\zeta (k)

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