Kloosterman paths of prime powers moduli

Abstract

In [KS16], the authors proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S a, b 0 ; p /p 1/2 converge in the sense of finite distributions to a specific random Fourier series, as a varies over Z/pZ × , b 0 is fixed in Z/pZ × and p tends to infinity among the odd prime numbers. This article considers the case of S a, b 0 ; p n /p n/2 , as a varies over Z/p n Z × , b 0 is fixed in Z/p n Z × , p tends to infinity among the odd prime numbers and n 2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0, 1] is also established, as (a, b) varies over Z/p n Z × × Z/p n Z × , p tends to infinity among the odd prime numbers and n 2 is a fixed integer. This is the analogue of the result obtained in [KS16] in the prime moduli case.In memory of Kevin Henriot.

Kloosterman paths of prime powers moduli, II

“…where as usual x stands for the inverse of x modulo p n and we also define e(z) := exp (2i πz) for any complex number z. Recall that its absolute value is less than 2 by its explicite formula (see [RR18,Lemma 4.6] for instance). For a and b in Z/p n Z × , the associated partial sums are the ϕ(p n ) = p n−1 (p − 1) complex numbers This is the polygonal path obtained by concatenating the closed segments Kl j 1 ;p n (a, b), Kl j 2 ;p n (a, b)…”

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Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S a, b; p n /p n/2 converge in law in the Banach space of complex-valued continuous function on [0, 1] to an explicit random Fourier series as (a, b) varies over Z/p n Z × × Z/p n Z × , p tends to infinity among the odd prime numbers and n 2 is a fixed integer. This is the analogue of the result obtained by E. Kowalski and W. Sawin (2016) in the prime moduli case.

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Squarefree Integers in Arithmetic Progressions to Smooth Moduli

“…An important work treating such correlations, with 𝐾 2 sums again replaced by Kloosterman sums, was undertaken by Ricotta and Royer [20]. They used their estimates to establish distribution theorems for Kloosterman sums modulo 𝑝 𝑛 , as 𝑝 → ∞.…”

“…As we will not be able to extract cancellation when 𝑃 𝝐 ,𝑑 has degree 0 or 1, we will need to check that the number of d (satisfying ( )) for which this happens is small. To this end, we need the following lemma, which is a modification of Proposition 4.8 of [20]. Proof.…”

“…For larger degree polynomials we need bounds for Weyl sums. In contrast to the work in [20], where Weyl differencing is used to obtain cancellation, we will instead apply Vinogradov's method in order to obtain a stronger Weyl sum estimate. For this, we recall the Vinogradov main conjecture, proved in the groundbreaking work of Bourgain, Demeter and Guth [2] (and independently in the work of Wooley [25]).…”