• Volume 103, Issue 1

April 1993,   pages  1-103

• A. Ramanathan (1946–1993)

• Transformation formula for exponential sums involving fourier coefficients of modular forms

In 1984 Jutila [5] obtained a transformation formula for certain exponential sums involving the Fourier coefficients of a holomorphic cusp form for the full modular groupSL(2, ℤ). With the help of the transformation formula he obtained good estimates for the distance between consecutive zeros on the critical line of the Dirichlet series associated with the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper we follow Jutila to obtain a transformation formula for exponential sums involving the Fourier coefficients of either holomorphic cusp forms or certain Maass forms for congruence subgroups ofSL(2, ℤ) and prove similar estimates for the corresponding Dirichlet series.

• DirichletL-function and power series for Hurwitz zeta function

For 0 &lt; α &lt; 1, letζ(s, α) be the Hurwitz zeta function and let ζ1 (s, α) = ζ(s, α) -α− s. For a fixeds, we developζ1(s,α) as a power series in α in the complex circle ¦α¦ &lt; 1. If$$\sum\limits_{\chi \left( {\bmod q} \right)} {L\left( {s,\chi } \right)L\left( {s',\bar \chi } \right)} = \frac{{\phi \left( q \right)}}{{q^{s + s'} }}\sum\limits_{k/q} \mu \left( {\frac{q}{k}} \right)\left( {\sum\limits_{a = 1}^k {\left( {\frac{k}{a}} \right)^{\operatorname{Re} s + \operatorname{Re} s'} + Q\left( {s,s',k} \right)} } \right)$$, we obtain an asymptotic expansion for Q(k) =Q(s,s′,k) using the power series forζ1(s,α)

• Determinants of parabolic bundles on Riemann surfaces

LetX be a compact Riemann surface andMsp(X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural Kähler metric onMsp(X). We obtain a natural metrized holomorphic line bundle onMsp(X) whose Chern form equalsmr times the Kähler form, wherem is the common denominator of the weights andr the rank.

• Convolution properties of some classes of meromorphic univalent functions

Convolution technique and subordination theorem are used to study certain class of meromorphic univalent functions in the punctured unit disc.

• Simultaneous operational calculus involving a product of a general class of polynomials, Fox’sH-function and the multivariableH-function

New operational relations between the original and the image for two-dimensional Laplace transforms involving a general class of polynomials, Fox’sH-function and the multivariableH-function are obtained. The result provides a unification of the bivariate Laplace transforms for theH-functions given by Chaurasia [2, 3].

• Some applications of Briot—Bouquet differential subordination

Some applications of Briot—Bouquet differential subordination are obtained which improve or extend a number of classical results in the univalent function theory.

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019