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

@article{Dixit2013SelfreciprocalFP, title={Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations}, author={Atul Dixit and Victor H. Moll}, journal={Journal of Number Theory}, year={2013}, volume={147}, pages={211-249} }

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, Koshlyakov, Ferrar and others. An integral containing the square of the Riemann Ξ-function and involving an extra parameter z, whose type naturally extends that of the afore-mentioned integrals, was studied by Ramanujan. This integral implicitly involves squaring of the functional equation of ζ ( s ) . A… Expand

#### 19 Citations

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Twisted second moments and explicit formulae of the Riemann zeta-function

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Several aspects connecting analytic number theory and the Riemann zeta-function are studied and expanded. These include:
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Koshliakov zeta functions I: Modular Relations

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- 2021

Abstract. We examine an unstudied manuscript of N. S. Koshliakov over 150 pages long and containing the theory of two interesting generalizations ζp(s) and ηp(s) of the Riemann zeta function ζ(s),… Expand

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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

On squares of odd zeta values and analogues of Eisenstein series

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A Ramanujan-type formula involving the squares of odd zeta values is obtained by studying a more general analogue of the Eisenstein series, consisting of one more parameter N . Expand

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

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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

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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… Expand

A RAMANUJAN-TYPE FORMULA FOR ζ2(2m+ 1) AND ITS GENERALIZATIONS

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A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in… Expand

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We consider some properties of integrals considered by Hardy and Koshliakov, and which have also been further extended recently by Dixit. We establish a new general integral formula from some… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

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

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

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

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

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

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

Asymptotics and Mellin-Barnes Integrals

- Mathematics
- 2001

Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically… Expand

SOME FORMULAS OF RAMANUJAN INVOLVING BESSEL FUNCTIONS

- 2010

— We generalize a number of summation formulas involving the K-Bessel function, due to Ramanujan, Koshliakov, and others. Résumé. — Nous généralisons des formules sommatoires faisant intervenir les… Expand

Table of Integrals, Series, and Products

- Mathematics, Engineering
- 1943

Introduction. Elementary Functions. Indefinite Integrals of Elementary Functions. Definite Integrals of Elementary Functions. Indefinite Integrals of Special Functions. Definite Integrals of Special… Expand

RAMANUJAN’S INGENIOUS METHOD FOR GENERATING MODULAR-TYPE TRANSFORMATION FORMULAS

- 2013

We discuss an ingenious method of Ramanujan for generating modulartype transformations of the form F (α) = F (β), αβ = 1, having its origins in one of his published papers and in his Lost Notebook.… Expand