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Let z ∈ C . We consider the following product-to-sum representation: ∏ m = 1 ∞ (1 - q m) - z = ∑ n = 0 ∞ P n (z) q n. The term P n (z) introduced by D'Arcais was found to be a polynomial in z of degree n. In this article, we revisit the Kostant's representation for the coefficients of the polynomial P n (z) . This polynomial representation of P n (z) helps in obtaining some congruence properties when z ∈ Z . Moreover, for any given n, we obtain a set of integers (possibly of small size) that contains the integral roots of P n (z) . [ABSTRACT FROM AUTHOR]

A central limit theorem is established for the absolute value of the modular Fourier-coefficient function defined by Ramanujan, and for that of the error term in the formula counting representations of integers as sums of twenty-four squares, in which the function appears. [ABSTRACT FROM AUTHOR]

We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals. [ABSTRACT FROM AUTHOR]

We prove a conjecture of Zagier, that the inverse Mellin transform of the symmetric square L-function attached to Ramanujan's tau function has an asymptotic expansion in terms of the zeros of the Riemann function. [ABSTRACT FROM AUTHOR]

Given a commutative ring R with identity, let HR be the set of sequences of elements in R. We investigate a novel isomorphism between (HR,+) and (H~R,∗), where + is the componentwise sum, ∗ is the convolution product (or Cauchy product) and H~R the set of sequences starting with 1R. We also define a recursive transform over HR that, together to the isomorphism, allows to highlight new relations among some well studied integer sequences. Moreover, these connections allow to introduce a family of polynomials connected to the D’Arcais numbers and the Ramanujan tau function. In this way, we also deduce relations involving the Bell polynomials, the divisor function and the Ramanujan tau function. Finally, we highlight a connection between Cauchy and Dirichlet products. [ABSTRACT FROM AUTHOR]