Math 139 050225 Dirichlet's Theorem Part IV: L(1,X) nonzero for nontrivial real Dirichlet characters скачать в хорошем качестве

Math 139 050225 Dirichlet's Theorem Part IV: L(1,X) nonzero for nontrivial real Dirichlet characters

Recall: we want to show that L(1,X) is nonzero for X a non-trivial real Dirichlet character (mod q). First: two asymptotic estimates on the partial sums of two p-series (p = 1, p = 1/2). Construction of Hyperbolic sum S_N. Strategy: it suffices to show the proposition: i) S_N is bounded below by log N. ii) Further, S_N = 2N^{1/2} L(1,X) + O(1). For if L(1,X) vanishes, then we have a contradiction. Lemma: the sum of X(n) over all divisors of k is greater than or equal to 1 for k a square. Lemma: the truncated "p-series for X" are of O(a^{-1}) for p=1, O(a^{-1/2}) for p = 1/2 (where the sum starts from a). Lemma part i) implies the lower bound; lemma part ii) yields that the the hyperbolic sum is basically 2N^{1/2} L(1,X).