On the spectral large sieve inequality for symmetric-squares

TL;DR

This paper advances the spectral large sieve inequality for symmetric-squares by providing improved bounds and establishing a lower bound that refutes the most optimistic upper bound.

Contribution

It introduces tighter bounds for the spectral large sieve inequality for symmetric-squares and demonstrates the limitations of existing upper bounds.

Findings

Improved spectral large sieve inequality bounds for symmetric-squares.

Proved a lower bound showing the most optimistic upper bound does not hold.

Refuted the conjecture of the tightest possible upper bound for this family.

Abstract

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.