On the representation of integers as sums of an odd number of squares

In a recent work, S. Cooper (J. Number Theory 103:135–162, [1988]) conjectured a formula for r _2k+1(p ²), the number of ways p ² can be expressed as a sum of 2k+1 squares. Inspired by this conjecture, we obtain an explicit formula for r _2k+1(n ²),n≥1. Dedicated to Srinivasa Ramanujan.     

Linear independence of digamma function and a variant of a conjecture of Rohrlich

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.     

Remarks on some zero-sum theorems

In the present paper, we give a new proof of a weighted generalization of a result of Gao in a particular case. We also give new methods for determining the weighted Davenport constant and another similar constant for some particular weights.     

Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

The Ramanujan Journal - Let F and G be Siegel cusp forms for $${\mathrm{Sp}}_4({{\mathbb {Z}}})$$ and weights $$k_1, k_2$$ , respectively. Also let F and G be Hecke eigenforms lying in distinct...     

On the zeros of weakly holomorphic modular forms

In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen (Comment Math Univ St Pauli 52:55–57, 2003). Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.     

Transcendence of the log gamma function and some discrete periods

We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number x with 0<x<1, the number logΓ(x)+logΓ(1−x) is transcendental 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 , where P(x) and Q(x) are polynomials with algebraic coefficients.     

A canonical subspace of modular forms of half-integral weight

We characterise the space of newforms of weight k + 1/2 on Γ₀(4N), N odd and square-free (studied by the second and third authors with Vasudevan) under the Atkin-Lehner W(4) operator. As an application, we show that the (±1)-eigensubspaces of the W(4) operator on the space of modular forms of weight k + 1/2 on Γ₀(4N) is mapped to modular forms of weight 2k on Γ₀(N), under a class of Shimura maps. The existence of such subspaces having this mapping property was conjectured by Zagier in a private communication. One of the special features of the (±1)-eigensubspaces is that the (2k + 1)-th power of the classical theta series of weight 1/2 belongs to the +  eigensubspace and hence this gives interesting congruences for r _2k+1(p ²).     

On zeros of quasi-modular forms

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 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.