# On special Riemann xi function formulae of Hardy involving the digamma function

## TL;DR

This paper explores properties of integrals related to Hardy and Koshliakov, deriving a new integral formula involving the digamma function and establishing bounds for Hardy's integral.

## Contribution

It introduces a new general integral formula involving the digamma function and provides bounds for Hardy's integral based on digamma properties.

## Key findings

- Derived a new integral formula involving the digamma function
- Established bounds for Hardy's integral using digamma properties
- Extended previous work by Hardy, Koshliakov, and Dixit

## Abstract

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 observations about the digamma function. We also obtain lower and upper bounds for Hardy's integral through properties of the digamma function.