Chowla and Sarnak conjectures for Kloosterman sums

Abstract

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

Kloosterman sums do not correlate with periodic functions

“…Recently, it was conjectured by El Abdalaoui-Shparlinski-Steiner [10] that Kloosterman sums should in general not correlate with low-complexity sequences. Whilst this was shown to be true for (vertical) averages over the entries of the Kloosterman sums, see [24,Example 10.4] or [10,Theorem 2.8], it is wide open for (horizontal) averages over the modulus in this generality. In this paper, we make progress towards this conjecture by completely resolving the case of periodic functions.…”