explicit lower bounds on $|L(1,\chi)|$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:39:06Z http://mathoverflow.net/feeds/question/82296 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82296/explicit-lower-bounds-on-l1-chi explicit lower bounds on $|L(1,\chi)|$ Matt Johnson 2011-11-30T18:15:01Z 2011-11-30T18:32:48Z <p>Does anyone know of an explicit effective lower bound for $|L(1,\chi)|$, where $\chi$ is an odd complex (primitive) Dirichlet character?</p> <p>I know of Landau's paper Uber Dirichletsche Reihen mit komplexen Charakteren, where he bounds $$|L(1,\chi)|>\frac{1}{c \log(q)},$$</p> <p>where $q$ is the conductor of $\chi$, but the constant $c$ he gets is on the order of $e^{50}$, and is totally useless for computations.</p> <p>I know of many papers dealing with quadratic characters but very few that address complex characters (explicitly).</p> http://mathoverflow.net/questions/82296/explicit-lower-bounds-on-l1-chi/82299#82299 Answer by Anonymous for explicit lower bounds on $|L(1,\chi)|$ Anonymous 2011-11-30T18:32:48Z 2011-11-30T18:32:48Z <p>This is discussed on page 47 of Narkiewicz's new book (Rational Number Theory in the 20th Century); see</p> <p><a href="http://books.google.ca/books?id=3SWNZaDM6iMC&amp;lpg=PP1&amp;dq=rational%20number%20theory%20in%20the&amp;pg=PA47#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.ca/books?id=3SWNZaDM6iMC&amp;lpg=PP1&amp;dq=rational%20number%20theory%20in%20the&amp;pg=PA47#v=onepage&amp;q&amp;f=false</a></p> <p>Reference [4268] is to </p> <p>Metsankyla, T.: Estimations for L-functions and the class numbers of certain imaginary cyclic fields, Ann. Univ. Turku, Ser. AI 140, 1--11 (1970)</p> <p>[3995] is</p> <p>Louboutin, Stéphane(F-CAEN) Minoration au point 1 des fonctions L et détermination des corps sextiques abéliens totalement imaginaires principaux. (French) [Lower bound at the point 1 of L-functions and determination of the principal totally imaginary abelian sextic fields] Acta Arith. 62 (1992), no. 2, 109–124. </p> <p>and [338] is</p> <p>Barrucand, Pierre; Louboutin, Stéphane(F-CAEN) Minoration au point 1 des fonctions L attachées à des caractères de Dirichlet. (French) [Lower bound at the point 1 of L-functions associated with Dirichlet characters] Colloq. Math. 65 (1993), no. 2, 301–306. </p>