Large values of L-functions from the Selberg class

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• 作者：Christoph Aistleitner^a ; ^{aistleitner@math.tugraz.at" class="auth_mail" title="E-mail the corresponding author} ; Łukasz Pańkowski^b ; ^c ; ^{lpan@amu.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Selberg class ; L-function ; Large values ; Resonance method ; Diophantine approximation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：446
• 期：1
• 页码：345-364
• 全文大小：446 K

文摘

In the present paper we prove lower bounds for L-functions from the Selberg class, by this means improving earlier results obtained by the second author together with Jörn Steuding. We formulate two theorems which use slightly different technical assumptions, and give two totally different proofs. The first proof uses the “resonance method”, which was introduced by Soundararajan, while the second proof uses methods from Diophantine approximation which resemble those used by Montgomery. Interestingly, both methods lead to roughly the same lower bounds, which fall short of those known for the Riemann zeta function and seem to be difficult to be improved. Additionally to these results, we also prove upper bounds for L-functions in the Selberg class and present a further application of a theorem of Chen which is used in the Diophantine approximation method mentioned above.