1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 3. IARGE SIEVE INEQUALITIES 227 Thus (3. 7) shows quasiorthogonality of the Fourier coefficients on average over n ~ N « ...

228 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS In the case of Maass forms, one has a similar inequality (by ...

LECTURE 3. LARGE SIEVE INEQUALITIES 3.2.1. The large sieve inequality for forms of weight one Theorem 3.4. For any c: > 0, N; ...

230 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Theorem 3.5. Let F be a finite subset of A3(Q) and Q;:: = sup ...

LECTURE 3. LARGE SIEVE INEQUALITIES 231 3.3. Some applications of the large sieve As we have seen in section 3.1.1, one of the m ...

232 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Question. Given f and g two distinct primitive holomorphic fo ...

LECTURE 3. LARGE SIEVE INEQUALITIES 233 either to a dihedral group D2n, n ;:?: 1, or to one of the exotic groups A 4 , S 4 , A 5 ...

234 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS By (3.14) and the prime number theorem, one has for any f E 3 ...

LECTURE 3. LARGE SIEVE INEQUALITIES 235 density estimates have been applied by Linnik and then by others to study the following: ...

236 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS where Q:F = MaxQ"; here the exponents Band c and the constant ...

LECTURE 3. IARGE SIEVE INEQUALITIES 237 in [Mon]) by using an existing large sieve inequality for F; for instance, under the ass ...

238 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF L-FUNCTIONS has for such 11 's, a good approximation of L(11, 1y by a ver ...

LECTURE 3. LARGE SIEVE INEQUALITIES Proposition 3.3. there exists constants b, c > 0 such that for 3/4::::; o: < 1 L N(sym ...

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Lecture 4. The subconvexity problem In this lecture we describe the state of the art regarding the Subconvexity Prob- lem (ScP) ...

242 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS and the proof of the s aspect for general Dirichlet L-functio ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 243 4.1. Around Weyl's shift The shifting method was introduced by Weyl in his proof of ( 4 ...

244 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Proof. The method of Burgess consists of artificially adding ...

LECTURE 4. THE SUBCONVEXITY PROBLEM 245 l.J rl' ITr --b-u + bi. h h 1 rs not a k-t power (w ere we denote by k the order of x), ...

246 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS the trivial bound being (at least when the sequence (an) is n ...