Around the Lipschitz Summation Formula

Boundary behavior of important functions has been an object of intensive research since the time of Riemann. Kurokawa, Kurokawa-Koyama, and Chapman studied the boundary behavior of generalized Eisenstein series which falls into this category. The underlying principle is the use of the Lipschitz summation formula. Our purpose is to show that it is a form of the functional equation for the Lipschitz–Lerch transcendent (and in the long run, it is equivalent to that for the Riemann zeta-function) and that this being indeed a boundary function of the Hurwitz–Lerch zeta-function, one can extract essential information. We also elucidate the relation between Ramanujan’s formula and automorphy of Eisenstein series.

Ramanujan’s convolution sum twisted by Dirichlet characters

We find formulas for convolutions of sum of divisor functions twisted by the Dirichlet character [Formula: see text], which are analogous to Ramanujan’s formula for convolution of usual sum of divisor functions. We use the theory of modular forms to prove our results.

GENERALIZED LAMBERT SERIES, RAABE’S COSINE TRANSFORM AND A GENERALIZATION OF RAMANUJAN’S FORMULA FOR

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$, $N\in \mathbb{N}$ and $h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for $\unicode[STIX]{x1D701}(2m+1)$ for $m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for $\unicode[STIX]{x1D701}(1/N),N$ even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series $E_{2}(z)$ and $E_{2}(-1/z)$. An identity relating $\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. In particular, this gives a beautiful relation between $\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$ and $\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with $n^{2}$ in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.

On Jackson’s proof of Ramanujan’s 1ψ1 summation formula

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

Small values of coefficients of a half Lerch sum

Andrews, Dyson and Hickerson proved many interesting properties of coefficients for a Ramanujan’s [Formula: see text]-hypergeometric series by relating it to real quadratic field [Formula: see text] and using the arithmetic of [Formula: see text] to solve a conjecture of Andrews on the distributions of its Fourier coefficients. Motivated by Andrews’s conjecture, we discuss an interesting [Formula: see text]-hypergeometric series which comes from a Lerch sum and rank and crank moments for partitions and overpartitions. We give Andrews-like conjectures for its coefficients. We obtain partial results on the distributions of small values of its coefficients toward these conjectures.