On the nature of e^γ and non-vanishing of derivatives of L-series at s = 1/2

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• 作者：M. Ram Murty ; ¹ ; ^{murty@mast.queensu.ca" class="auth_mail" title="E-mail the corresponding author} ; Naomi Tanabe² ; ^{naomi@mast.queensu.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11J81 ; 11J91 ; 11M41
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：161
• 期：Complete
• 页码：444-456
• 全文大小：337 K

文摘

In 2011, M.R. Murty and V.K. Murty [10] proved that if L(s,χ_D)$L(s,{\chi }_D)$ is the Dirichlet L  -series attached a quadratic character χ_D${\chi }_D$, and L^′(1,χ_D)=0$L^{\prime }(1,{\chi }_D)=0$, then e^γ$e^{\gamma }$ is transcendental. This paper investigates such phenomena in wider collections of L-functions, with a special emphasis on Artin L  -functions. Instead of s=1$s=1$, we consider s=1/2$s=1/2$. More precisely, we prove that is transcendental with some rational number α  . In particular, if we have L(1/2,χ)≠0$L(1/2,\chi )\ne 0$ and L^′(1/2,χ)=0$L^{\prime }(1/2,\chi )=0$ for some Artin L  -series, we deduce the transcendence of e^γ$e^{\gamma }$.