Sign changes of Fourier coefficients of Hilbert modular forms

Date: 2014-01-01

Creator: Jaban Meher, Naomi Tanabe

Access: Open access

Sign changes of Fourier coefficients of various modular forms have been studied. In this paper, we analyze some sign change properties of Fourier coefficients of Hilbert modular forms, under the assumption that all the coefficients are real. The quantitative results on the number of sign changes in short intervals are also discussed. © 2014 Elsevier Inc.

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

Date: 2014-04-30

Creator: M. Ram Murty, Naomi Tanabe

Access: Open access

In 2011, M.R. Murty and V.K. Murty [10] proved that if L(s, χD) is the Dirichlet L-series attached a quadratic character χD, and L'(1, χD)=0, then eγ 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, we consider s=1/2. More precisely, we prove thatexp (L'(1/2,χ)L(1/2,χ)-αγ) is transcendental with some rational number α. In particular, if we have L(1/2, χ)≠0 and L'(1/2, χ)=0 for some Artin L-series, we deduce the transcendence of eγ.