1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1 Analytic properties of individual £-functions 1.1. Automorphic L-functions 1.1.1. Principal £-functions Let 7r = ®7rp ...

188 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS where q11: ~ 1 is an integer (the arithmetic conductor of n) ...

LECTURE 1. ANALYTIC PROPERTIES OF INDIVIDUAL £-FUNCTIONS 189 absolutely convergent for ~es large enough. This is an Euler produc ...

190 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS £-functions of pairs are defined more generally for automorph ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 191 Lemma 1.1.1. For n ~ 1, ftr®ir ( n) ~ O; in particular, for any n ~ 1 ...

192 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £ -FUNCTIONS The bound Hd(l/2) is probably far from the truth: assuming t ...

LECTURE 1. ANALYI'IC PROPERTIES OF INDMDUAL £-FUNCTIONS 193 by subtracting the N - 1 sum from the N sum. Taking N =pk (with k -- ...

194 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Remark 1.5. This zero-free region gives the PNT as above with ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 195 Lemma 1.2.1. Let D( s) be a Dirichlet series with non-negative coeffi ...

196 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS this method (this is also accessible to the Hadamard/ de la V ...

LECTURE 1. ANALYTIC PROPERTIES OF INDIVIDUAL £-FUNCTIONS 197 (althought still small) range q «A (log x )A for any A? 1. Finally, ...

198 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Another property of the Landau/ Siegel zero, when it gets too ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL L -FUNCTIONS 199 Remark 1. 7. Although Siegel's Theorem does not strictly eliminate t ...

200 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS solution comes from a weak (but effective) lower bound for L( ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £ -FUNCTIONS 201 Hoffstein/ Lockhart proved the analog of Siegel's theorem for the s ...

202 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS Convexity Bound. For any c > 0, and 3tes = 1 /2, we have ( ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 203 where S is a (sufficiently large) product of primes containing the fi ...

204 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS variants are called the Subconvexity Problem in the t-aspect, ...

LECTURE 1. ANALYTIC PROPERTIES OF INDMDUAL £-FUNCTIONS 205 the following integral: let G(u) be an even function, normalized by G ...

206 PH. MICHEL, ANALYTIC NUMBER THEORY AND FAMILIES OF £-FUNCTIONS the implied constant depending on rJ, j , A. When A is large, ...