Chowla and Sarnak Conjectures for Kloosterman Sums

TL;DR

This paper extends the Chowla and Sarnak conjectures to Kloosterman sums, providing new unconditional results in some cases, thereby advancing understanding of their pseudorandomness properties.

Contribution

It formulates analogues of the Chowla and Sarnak conjectures for Kloosterman sums and proves some cases unconditionally, a novel extension from the M"obius function setting.

Findings

Certain conjectures hold unconditionally for Kloosterman sums

New analogues of Chowla and Sarnak conjectures are established

Advances understanding of pseudorandomness in exponential sums

Abstract

We formulate several analogues of the Chowla and Sarnak conjectures, which are widely known in the setting of the M\"obius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.