## 230 Citations

RAMANUJAN’S PAPER ON RIEMANN’S FUNCTIONS ξ(s) AND Ξ(t) AND A TRANSFORMATION FROM THE LOST NOTEBOOK

- Mathematics
- 2020

Here Γ(s) and ζ(s) are the Euler Gamma and the Riemann zeta functions respectively. In his review of Ramanujan’s work in [17], Hardy lists [24] as one of the four most important papers of Ramanujan…

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

- Mathematics
- 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…

Zeros of general L-functions on the critical line[HBNI Th52]

- Mathematics
- 2012

We study the gaps between consecutive zeros on the critical line for the Riemann zeta function ζ(s) and certain generalisations of ζ(s), namely, the Epstein zeta function and the Selberg class of…

On large gaps between consecutive zeros, on the critical line, of some zeta-functions

- Mathematics
- 2011

In this thesis we extend a method of Hall $[30, 34]$ which he used to show the existence of large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function $zeta(s)$. Our…

On the success and failure of Gram's Law and the Rosser Rule

- Mathematics
- 2011

This article contains a survey on results relating to Gram’s Law and the distribution of the zeroes of the Riemann zeta-function. It is shown that Gram’s Law and the Rosser Rule fail a positive…

Improvements to Turing's method

- MathematicsMath. Comput.
- 2011

This paper refines the argument of Lehman by reducing the size of the constants in Turing's method by giving in Theorem 1 and scope for further improvements is also given.

Aspects of Analytic Number Theory : The Universality of the Riemann Zeta-Function

- Mathematics, Philosophy
- 2009

Abstract. These notes deal with Voronin’s universality theorem which states, roughly speaking, that any non-vanishing analytic function can be uniformly approximated by certain shifts of the Riemann…

The Weak Gram Law for Hecke $L$-functions

- Mathematics
- 2022

From the functional equation (1) it follows that Zζ(t) is real-valued. Furthermore the ordinates of the zeros of ζ(s) on the critical line coincide with the zeros of Zζ(t). The function θζ(t)…

A Proof to the Riemann Hypothesis Using a Simplified Xi-Function

- Mathematics
- 2021

: The Riemann hypothesis has been of great interest in the mathematics community since it was proposed by Bernhard Riemann in 1859, and makes important implications about the distribution of prime…

The Julia Line of a Riemann-Type Functional Equation {\footnotesize -- Counting Formulae, Dirichlet Polynomial Approximations, and a Weak Gram Law --}

- Mathematics
- 2020

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, we improve upon results of Bombieri and Friedlander on…