# Modular-type transformations and integrals involving the Riemann ?-function

@inproceedings{Dixit2018ModulartypeTA, title={Modular-type transformations and integrals involving the Riemann ?-function}, author={A. Dixit}, year={2018} }

A survey of various developments in the area of modular-type transformations (along with their generalizations of different types) and integrals involving the Riemann Ξ-function associated to them is given. We discuss their applications in Analytic Number Theory, Special Functions and Asymptotic Analysis.

#### 3 Citations

On Hurwitz zeta function and Lommel functions

- Mathematics
- 2019

We obtain a new proof of Hurwitz's formula for the Hurwitz zeta function $\zeta(s, a)$ beginning with Hermite's formula. The aim is to reveal a nice connection between $\zeta(s, a)$ and a special… Expand

Ramanujan's Beautiful Integrals

- Mathematics
- 2021

Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the Journal of the Indian Mathematical Society, notebooks,… Expand

Superimposing theta structure on a generalized modular relation

- Mathematics
- 2020

A generalized modular relation of the form $F(z, w, \alpha)=F(z, iw,\beta)$, where $\alpha\beta=1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann… Expand

#### References

SHOWING 1-10 OF 29 REFERENCES

Transformation formulas associated with integrals involving the Riemann Ξ-function

- Mathematics
- 2011

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β),… Expand

Series transformations and integrals involving the Riemann Ξ-function

- Mathematics
- 2010

The transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar are unified, in the sense that all these formulas come from the same source, namely, a general formula involving an integral of… Expand

Zeros of combinations of the Riemann ξ-function on bounded vertical shifts

- Mathematics
- 2015

In this paper we consider a series of bounded vertical shifts of the Riemann ξ-function. Interestingly, although such functions have essential singularities, infinitely many of their zeros lie on the… Expand

Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations

- Mathematics, Physics
- 2013

Abstract Integrals containing the first power of the Riemann Ξ-function as part of the integrand that lead to modular-type transformations have been previously studied by Ramanujan, Hardy,… Expand

A First Course in Modular Forms

- Mathematics
- 2008

Modular Forms, Elliptic Curves, and Modular Curves.- Modular Curves as Riemann Surfaces.- Dimension Formulas.- Eisenstein Series.- Hecke Operators.- Jacobians and Abelian Varieties.- Modular Curves… Expand

A transformation formula involving the gamma and riemann zeta functions in Ramanujan's lost notebook

- Mathematics
- 2010

Two proofs are given for a series transformation formula involving the logarithmic derivative of the Gamma function found in Ramanujan’s lost notebook. The transformation formula is connected with a… Expand

Riesz-type criteria and theta transformation analogues

- Mathematics
- 2016

Abstract We give character analogues of a generalization of a result due to Ramanujan, Hardy and Littlewood, and provide Riesz-type criteria for Riemann Hypotheses for the Riemann zeta function and… Expand

Koshliakov kernel and identities involving the Riemann zeta function

- Mathematics
- 2015

Some integral identities involving the Riemann zeta function and functions reciprocal in a kernel involving the Bessel functions $J_{z}(x), Y_{z}(x)$ and $K_{z}(x)$ are studied. Interesting special… Expand

Analogues of the general theta transformation formula

- Mathematics
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2013

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and… Expand

Analogues of a transformation formula of Ramanujan

- Mathematics
- 2009

We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz… Expand