# A double character sum of Conrey-Iwaniec and Petrow-Young

## TL;DR

This paper demonstrates that a specific double character sum related to Weyl bounds for certain L-functions is essentially a hypergeometric sum, providing a simplified proof for its upper bound.

## Contribution

It identifies the double character sum as a hypergeometric sum, offering a new, streamlined proof of its upper bound compared to previous methods.

## Key findings

- The double character sum is equivalent to a hypergeometric sum.
- A simplified proof of the upper bound for this sum is established.
- The work connects character sums with hypergeometric functions in a novel way.

## Abstract

We show that a double character sum, appearing in the work of Conrey-Iwaniec and Petrow-Young on Weyl bound for certain $L$-functions, is essentially a hypergeometric sum introduced by Katz. This produces a simple proof of the upper bound for this sum.