# Superimposing theta structure on a generalized modular relation

@article{Dixit2020SuperimposingTS, title={Superimposing theta structure on a generalized modular relation}, author={Atul Dixit and Rahul Kumar}, journal={arXiv: Number Theory}, year={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 $\Xi$-function. It is a two-variable generalization of a transformation found on page $220$ of Ramanujan's Lost Notebook. This modular relation involves a surprising generalization of the Hurwitz zeta function $\zeta(s, a)$, which we denote by $\zeta_w(s, a)$. While $\zeta_w(s, 1)$ is essentially a product… Expand

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#### References

SHOWING 1-10 OF 104 REFERENCES

A generalized modified Bessel function and a higher level analogue of the theta transformation formula

- Mathematics
- 2017

A new generalization of the modified Bessel function of the second kind $K_{z}(x)$ is studied. Elegant series and integral representations, a differential-difference equation and asymptotic… Expand

Contributions to the theory of the Hurwitz zeta-function

- Mathematics
- 2007

We give various contributions to the theory of Hurwitz zeta-function. An elementary part is the argument relating to the partial sum of the defining Dirichlet series for it; how much can we retrieve… Expand

Error functions, Mordell integrals and an integral analogue of partial theta function

- Mathematics
- 2016

A new transformation involving the error function $\textup{erf}(z)$, the imaginary error function $\textup{erfi}(z)$, and an integral analogue of a partial theta function is given along with its… 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

Period functions for Maass wave forms. I.

- Mathematics
- 2001

Recall that a Maass wave form on the full modular group Gamma=PSL(2,Z) is a smooth gamma-invariant function u from the upper half-plane H = {x+iy | y>0} to C which is small as y \to \infty and… 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

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

Fourier transforms with only real zeros

- Mathematics
- 1976

The class of even, nonnegative, finite measures p on the real line such that for any b > 0 the Fourier transform of exp(bt2) dp(t) has only real zeros is completely determined. This result is then… Expand

The Theory of the Riemann Zeta-Function

- Mathematics
- 1987

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects… Expand

New pathways and connections in Number Theory and Analysis motivated by two incorrect claims of Ramanujan

- Mathematics
- 2016

We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -- each is… Expand