Distribution of short sums of classical Kloosterman sums of prime powers moduli

Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article considers the case of short sums of normalized classical Kloosterman sums of prime powers moduli $\mathsf{Kl}_{p^n}$, as $p$ tends to infinity among the prime numbers and $n\geq 2$ is a fixed integer. A convergence in law towards a real-valued standard Gaussian random variable is proved under some very natural conditions.

Cet article est mis à disposition selon les termes de la licence C C -4.0 F .

Introduction and statement of the results
Let p be an odd prime number. For F q the finite field of cardinality q and of characteristic p, t q a complex-valued function on F q and I q a subset of F q , the normalized partial sum of t q over I q is defined by where as usual |I q | stands for the cardinality of I q . Such sums have a long history in analytic number theory, confer [5,Chapter 12]. The normalization is explained by the fact that in a number theory context one expects the square-root cancellation philosophy. One can define a complex-valued random variable on F q endowed with the uniform measure by ∀ x ∈ F q , S(t q , I q ; x) S(t q , I q + x) where as usual I q + x stands for the translate of I q by x for any x in F q . Given a sequence t q of -adic trace functions over F q and a sequence I q of subsets of F q , C. Perret-Gentil got interested in [13] in the distribution as q and |I q | tend to infinity of the sequence of complex-valued random variables S(t q , I q ; * ) and proved a deep general result under very natural conditions. Let us mention that his general result is not only a generalization but also an improvement over previous works such as [2], [10], [11] and [12].
Let us state the case of the normalized Kloosterman sums of rank 2 given by where as usual e(z) exp (2iπz) for any complex number z. C. Perret-Gentil proved the following qualitative result. Theorem 1.1 (C. Perret-Gentil (Qualitative result)). As q and |I q | tend to infinity with log (|I q |) = o(log (q)) then the sequence of real-valued random variables S(Kl 2,F q , I q ; * ) converges in law to a real-valued standard Gaussian random variable.
He also proved the following quantitative result. Theorem 1.2 (C. Perret-Gentil (Quantitative result)). As q and |I q | tend to infinity with log (|I q |) = o(log (q)) then for any real numbers α < β and for any 0 < ε < 1/2.
The main purpose of this work is to consider the case of Kloosterman sums of prime powers moduli, namely to replace finite fields by finite rings, and to give a probabilistic meaning to the histogram given in Figure 1 The normalized Kloosterman sum of modulus p n is the real number given by for any integer a and where as usual x stands for the inverse of x modulo p n . For any subset I p n of (Z/p n Z) × , let be the normalized partial sum over I p n . Given a sequence of sets I p n of Z/p n Z, we are interested in the distribution of the sequence of real random variables over (Z/p n Z) × endowed with the uniform measure given by ∀ x ∈ (Z/p n Z) × , S Kl p n , I p n ; x S Kl p n , I p n + x . (1.1) for any prime number p. If p and |I p n | tend to infinity with then the sequence of real-valued random variables S Kl p n , I p n ; * converges in law to a standard Gaussian real-valued random variable. Remark 1.4. This theorem is the analogue of Theorem 1.1. The condition (1.1) is new and comes from the context of finite rings in this work instead of finite fields in [13] whereas the condition (1.2) is exactly the same and is inherent to the method of proof itself namely the method of moments. Note that the condition (1.1) requires that |I p n | < p holds, which is automatically satisfied by (1.2).
Let us state the quantitative result of this work. Assume that for any prime number p. If p and |I p n | tend to infinity with for any real numbers α < β and for any 0 < ε < β n /3. Notations.
• The main parameter in this paper is an odd prime number p, which tends to infinity. Thus, if f and g are some C-valued function of the real variable then the mean that | f (p)| is smaller than a "constant", which only depends on A, times g(p) at least for p large enough.
• n 2 is a fixed integer.
• For any real number x and integer k, e k (x) exp 2iπ x k .
• For any finite set S, |S| stands for its cardinality.
• We will denote by ε an absolute positive constant whose definition may change from one line to the next one.
• The notation × means that the summation is over a set of integers coprime with p.
• Finally, if P is a property then δ P is the Kronecker symbol, namely 1 if P is satisfied and 0 otherwise.

Moments of products of additively shifted Kloosterman sums
The crucial ingredient in the proof of Theorem 1.3 is the asymptotic evaluation of the complete sums of products of shifted Kloosterman sums S p n (µ) defined by for µ = (µ(τ)) τ ∈Z/p n Z a sequence of p n -tuples of non-negative integers different from the 0-tuple. Let us define for such sequence µ, The following proposition, which contains an asymptotic formula for the sums S p n (µ), is an improvement of [14,Proposition 4.10] in the sense that the dependency in the tuple µ in the error term has been made explicit. Proposition 2.1. Let µ = (µ(τ)) τ ∈Z/p n Z be a sequence of p n -tuples of non-negative integers satisfying for some absolute positive constant M. If for any ε > 0 and where the implied constant only depends on ε.

Proof of Proposition 2.1. By [14, p. 511], the error term to bound is given by
4πs a+τ, p n p n + θ p n where • means that the summation is over the u τ 's satisfying In the previous equation s a+τ, p n stands for any square-root modulo p n of a + τ for any relevant a and τ. Obviously, cos (µ(τ) − 2u τ ) 4πs a+τ, p n p n + θ p n .
The following proposition, which heavily relies on A. Weil's proof of the Riemann hypothesis for curves over finite fields and is [14,Proposition 4.8], states an asymptotic formula for the cardinality of the sets A p n (µ). Proposition 2.3 (G. Ricotta-E. Royer). Let µ = (µ(τ)) τ ∈Z/p n Z be a sequence of p n -tuples of non-negative integers. If p is odd then where the implied constant is absolute.

Various approximation results
The following lemma, which enables us to approximate characteristic functions of random variables from their moments, is a reformulation of [13, Lemma 5.1].

Lemma 2.4.
Let X 1 and X 2 be real-valued random variables. If for any non-negative integer k and for some function h : R → R then for any even integer k 1 and any real number u.
The following lemma, which allows us to approximate joint distributions of random variables via their characteristic functions, follows from [9, Section 4]. Lemma 2.5. Let X 1 and X 2 be real-valued random variables and α < β be real numbers. If E e 2iπuX 1 = E e 2iπuX 2 + O (g(|u|)) for any real number u and some continuous function g : R → R + then for any real number t > 0.
Finally, the following lemma is an explicit version of the Berry-Esseen theorem in dimension one (see [1,Theorem 13.2]). Lemma 2.6. Let α < β be two real numbers. Let X 1 , . . . , X h be centered independent identically distributed real-valued random variables of variance 1 satisfying E(|X 1 | 3 ) < ∞ and where H I p n . Obviously, H depends on p and n but such dependency has been removed for clarity. With these notations, By the multinomial formula, for any ε > 0 since T(µ k ) = T(µ k ) by (3.2). The obvious fact that has been used. One has for any ε > 0 and where S H is defined in (3.1) and since for any real number u. Let α < β be two real numbers and t 1 be a real number determined later. By Lemma 2.5 and (4.2), one gets for any non-negative real number u. Let us bound the second error term in (4.3). By the independence of the random variables U 1 , . . . , U H , for any real number u. The random variable U 1 being 4-subgaussian, since it is centered and bounded by 2 (see [15, p. 11 where γ = γ(k) > 0 will be chosen later. Thus, the first error term in (4.3) is bounded by k γ(k+1)−3k/4 + p −β n +2ε k γ(k+1) .