Horizontal Sato–Tate conjecture for Kloosterman sums

Prove that, for every fixed nonzero integer a, the values of the normalized Kloosterman sums Kl(a,p) over primes p not dividing a equidistribute on [-2,2] with respect to the Sato–Tate measure (1/(2π))√(4−t²) dt.

Background

The paper distinguishes between vertical distributions, in which the modulus is fixed and the parameter varies, and horizontal distributions, in which a is fixed and the prime modulus p varies. Katz’s vertical Sato–Tate theorem is known, but the analogous horizontal distribution is presented as substantially less understood.

The conjecture asserts that the normalized Kloosterman sums Kl(a,p), for a fixed nonzero integer a and varying primes p∤a, become equidistributed according to the Sato–Tate measure. The paper uses this conjecture heuristically to predict the average absolute value of Kloosterman sums over primes, but does not resolve the conjecture.

References

Let $a\neq0$ be a fixed integer. Then the set ${Kl(a,p):p\inP, p\nmid a}\subseteq[-2,2]$ equidistributes with respect to the Sato--Tate measure $\frac{1}{2\pi}\sqrt{4-t2}d t$.

— Horizontal moments of Kloosterman sums and a density version of sieves  (2610.11083 - Xi, 8 Oct 2026) in Conjecture 1.1, Section 1.1 (Background and main results)