Koshliakov kernel and identities involving the Riemann zeta function

  title={Koshliakov kernel and identities involving the Riemann zeta function},
  author={Atul Dixit and Nicolas Robles and Arindam Roy and Alexandru Zaharescu},
  journal={arXiv: Number Theory},
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 cases of these identities are derived, one of which is connected to a well-known transformation due to Ramanujan, and Guinand. 

Figures from this paper

Twisted second moments and explicit formulae of the Riemann zeta-function
Several aspects connecting analytic number theory and the Riemann zeta-function are studied and expanded. These include: 1. explicit formulae relating the Mobius function to the non-trivial zeros ofExpand
Zeros of combinations of the Riemann $\Xi$-function and the confluent hypergeometric function on bounded vertical shifts
In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the firstExpand
Modular-type transformations and integrals involving the Riemann ?-function
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 isExpand
A generalized modified Bessel function and a higher level analogue of the theta transformation formula
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 asymptoticExpand
Superimposing theta structure on a generalized modular relation
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 RiemannExpand
Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory
Closed-form evaluations of certain integrals of $$J_{0}(\xi )$$ J 0 ( ξ ) , the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating currentExpand
Correlation kernels for sums and products of random matrices
Let $X$ be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of $GX$Expand
A generalized modified Bessel function and explicit transformations of certain Lambert series
Abstract. An exact transformation, which we call a master identity, is obtained for the series ∑∞ n=1 σa(n)e −ny for a ∈ C and Re(y) > 0. As corollaries when a is an odd integer, we derive theExpand


Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations
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 transformation formula involving the gamma and riemann zeta functions in Ramanujan's lost notebook
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 aExpand
Transformation formulas associated with integrals involving the Riemann Ξ-function
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
The Theory of the Riemann Zeta-Function
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 aspectsExpand
Analogues of a transformation formula of Ramanujan
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 HurwitzExpand
Zeros of combinations of the Riemann ξ-function on bounded vertical shifts
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 theExpand
Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions
A new bijection, involving the new concept of the parity sequence of a partition, is used to prove one of Ramanujan's fascinating identities for a partial theta function. Expand
Koshliakov's formula and Guinand's formula in Ramanujan's lost notebook
On two pages in his lost notebook, Ramanujan recorded several theorems involving the modified Bessel function Kν(z). These include Koshliakov’s formula and Guinand’s formula, both connected with theExpand
Asymptotics and Mellin-Barnes Integrals
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 typicallyExpand
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