1,721,005 research outputs found
Eisenstein Series of Weight One, q-Averages of the 0-Logarithm and Periods of Elliptic Curves
Get PDFFor any elliptic curve E over k ⊂ R with E(C) = C^×/q^Z, q = e^(2πiz),Im(z) >, we study the q-average D_(0,q), defined on E(C), of the function D_0(z) = Im(z/(1−z)). Let Ω+(E) denote the real period of E. We show that there is a rational function R ∈ Q(X_1(N)) such that for any non-cuspidal real point s ∈ X_1(N) (which defines an elliptic curve E(s) over R together with a point P(s) of order N), πD_(0,q)(P(s)) equals Ω+(E(s))R(s). In particular, if s is Q-rational point of X_1(N), a rare occurrence according to Mazur, R(s) is a rational number
Transcendental Zeros Of Certain Modular Forms
No full textKohnen showed that the zeros of the Eisenstein series Ek in the standard fundamental domain other than i and ρ are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz
ON LARGE PRIME FACTORS OF FOURIER COEFFICIENTS OF NEWFORMS (Zeta functions and their representations)
No full textThis is an expository article showcasing some existing results about large Fourier coefficients of normalized Hecke eigenforms which are non CM forms. We also allude to some very recent works in this direction
ON LARGE PRIME FACTORS OF FOURIER COEFFICIENTS OF NEWFORMS (Zeta functions and their representations)
No full textThis is an expository article showcasing some existing results about large Fourier coefficients of normalized Hecke eigenforms which are non CM forms. We also allude to some very recent works in this direction
On the Zeros of Certain Cusp Forms
No full textF. K. C. Rankin and H. P. F. Swinnerton–Dyer proved that all the zeros of the Eisenstein Series Ek contained in the standard fundamental domain F lie on the arc A ={eiθ π/3 ≤ θ ≤ π/2}. Recently, J. Getz has generalized the method of Rankin and Swinnerton–Dyer to show that modular forms under certain conditions have similar properties. In this paper we prove similar results for certain types of cusp forms, motivated by the work of R. A. Rankin. Further, we give a closed formula for the zeros of a class of cusp forms in terms of the Fourier coefficients following the method of Kohnen
On zeros of quasi-modular forms
No full textAbstractSeveral authors have studied the nature and location of zeros of modular forms for the full modular group Γ and other congruence subgroups. In this paper, we investigate the zeros of certain quasi-modular forms for Γ. In particular, we study the transcendence and existence of infinitely many Γ-inequivalent zeros of these quasi-modular forms. We also estimate the number of such zeros in Siegel sets, motivated by a recent work of Ghosh and Sarnak
On the Ramanujan–Petersson conjecture for modular forms of half-integral weight
No full textWe investigate the (still unknown) Ramanujan-Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space
On The Meromorphic Continuation Of The Multiple Lerch Zeta Functions And An Idea Of Ramanujan
No full textIn this note, we obtain the meromorphic continuation of the multiple Lerch zeta functions. We also determine the location of their polar hyperplanes. An idea of Ramanujan and its subsequent generalisation by Ecalle constitutes the nucleus of this work
On special values of certain Dirichlet L-functions
No full textLet r k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469–490, [2002]), conjectured a formula for r 9(t), t≡5 (mod 8), r 11(t), t≡7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17:281–314, [1949]; Acta Arith. 68(3):245–253, [1994]). Further we express r 9(t), r 11(t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols
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