We study the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al.The special values of such integrals at algebraic points in the upper half-plane are linked to Riemann zeta values at odd positive integers.

Several 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 quasimodular 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.

Abstract. In this paper, we investigate a conjecture due to S. and P. Chowla and its generalization by Milnor. These are related to the delicate question of non-vanishing of L-functions associated to periodic functions at integers greater than 1. We report on some progress in relation to these conjectures. In a different vein, we link them to a conjecture of Zagier on multiple zeta values and also to linear independence of polylogarithms.

Let ψ(x) denote the digamma function. We study the linear independence of ψ(x) at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series.

We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number xtranscendental with at most one possible exception. Assuming Schanuel's conjecture, this possible exception can be ruled out. Further, we derive a variety of results on the Γ -function as well as the transcendence of certain series of the form ∞ n=1 P (n)/Q (n), where P (x) and Q (x) are polynomials with algebraic coefficients.

ABSTRACT. In this article, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus were dealt in an earlier work of Choie, the first author and Kohnen.

Let τ (·) be the classical Ramanujan τ -function and let k be a positive integer such that τ (n) = 0 for 1 ≤ n ≤ k/2. (This is known to be true for k < 10 23 , and, conjecturally, for all k.) Further, let σ be a permutation of the set {1, ..., k}. We show that there exist infinitely many positive integers m such that |τ (m + σ(1))| < |τ (m + σ(2))| < ... < |τ (m + σ(k))|.We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

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