The limiting distribution of Legendre paths

Let $p$ be a prime number and $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol modulo $p$. The \emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n\leq j} \left(\frac{n}{p}\right)$ for $0\leq j\leq p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\leq p\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \to \infty$, these paths converge in law, in the space of real-valued continuous functions on $[0, 1]$, to a certain random Fourier series constructed using Rademacher random completely multiplicative functions. This was previously proved by the first author under the assumption of the Generalized Riemann Hypothesis.


Introduction
A central question in number theory is to understand the behavior and the size of the character sum (1.1) where • p is the Legendre symbol modulo an odd prime p and 1 ⩽ x ⩽ p.Such sums encode important information on the distribution of quadratic residues and non-residues modulo p.In particular, bounds for S p (x) lead to results on the size of the least quadratic non-residue modulo p (see for example the works of Ankeny [1]; Burgess [5]; Graham and Ringrose [8]; and Montgomery [24]).Moreover, a recent work of Granville and Mangerel [9] shows that cancellations in the sum S p (x) in short initial intervals are more or less equivalent to improved bounds for the value of the Dirichlet L-function L s, • p at s = 1, which is connected to class numbers of the quadratic extensions Q( √ ±p).The Legendre symbol appears to exhibit random behavior over short intervals.This is for example illustrated by the work of Davenport and Erdős [7], who studied the distribution of S p (x) over short moving intervals.More precisely, they showed that if p is large and x varies among the integers {0, 1, . . ., p − 1}, the short sum S p (x + H) − S p (x) tends to a normal distribution with mean zero and variance H, provided log H = o(log p) and H → ∞ as p → ∞.Recently, Harper [12] showed that this is no longer the case if H is much larger, namely when H ⩾ p/(log p) A , and A > 0 is a fixed constant.
Our goal in this paper is to gain some understanding on the distribution of S p (x) over long intervals.In particular we will attempt to answer the following question: do these character sums, viewed as functions of x, possess a limiting distribution as p varies? and if so, can we describe this distribution?
Asked this way, this question does not make sense since one has max 1⩽x⩽p |S p (x)| ≫ √ p (this is an easy consequence of Parseval's theorem applied to Pólya's Fourier expansion (1.4) below), and max 1⩽x⩽p |S p (x)| ≪ ε √ p for all but at most O(επ(Q)) primes p ⩽ Q (this was proved by Montgomery and Vaughan [25]), where π(Q) is the number of primes p ⩽ Q.Therefore, we need to normalize the sum S p (x) by √ p.
Moreover, since S p (x) is periodic with period p, we shall normalize the length of the sum by p, and consider the function t → 1 √ p S p (tp).However, a final complication remains: this function is discontinuous, with jumps at every t ∈ 1 p N. Instead, we shall consider the continuous function f p on [0, 1], where we concatenate the points where 1 √ p S p (tp) changes.We call such a function the Legendre path associated to p.Note that for t ∈ [0, 1] we have where {x} is the fractional part of x.
In this paper, we shall prove that these paths, viewed as random processes on [0, 1], have a nice limiting distribution.Before stating our result, we shall construct a probabilistic random model for the Legendre path f p .A good random model for the Legendre symbol, which was extensively studied in recent years, is a Rademacher random completely multiplicative function.This is defined as (1.3) X n := q k ||n X k q for n ∈ N, where X 1 = 1, and {X q } q prime is a sequence of I. I. D. random variables taking the values ±1 with probability 1/2 each.The distribution of sums of  Rademacher random multiplicative functions was extensively studied in recent years, in particular by Harper [10] and [11].In studying the distribution of the character sum S p (x), our first guess is to model the normalized sum 1 √ p S p (tp) by the following normalized sum of random variables However, it turns out that this is not a good model for 1 √ p S p (tp), since it does not account for the periodicity of the Legendre symbol modulo p.To exploit this periodicity, we shall use the Fourier series expansion of f p , which was first established by Pólya in the following quantitative form [26, eq. (9.19) for any real number Z ⩾ 1, where e(t) := e 2πit and τ ( • p ) is the Gauss sum associated to the Legendre symbol modulo p, more precisely -The value of the prime modulo 4 influences the shape of the Legendre path, as shown in Figure 1.In fact, a simple calculation implies that for all j ∈ {0, 1, . . ., p − 1} we have S p (p − 1 − j) = S p (j) if p ≡ 3 (mod 4), and S p (p − 1 − j) = −S p (j) if p ≡ 1 (mod 4).This shows that if we extend the Legendre path f p to R by periodicity, the periodic extension is an even function if p ≡ 3 (mod 4), and is an odd function if p ≡ 1 (mod 4).
Inserting the values of the Gauss sum (1.5) in the Fourier expansion (1.4) gives where ε p = 1 if p ≡ 1 (mod 4), and equals i if p ≡ 3 (mod 4).Let {X q } q prime be a sequence of I. I. D. random variables taking the values ±1 with equal probability 1/2.Since the primes split evenly into the residue classes 1 and 3 (mod 4), we shall model the value −1 p by a random variable X −1 , which is independent from the X q 's for q prime, and takes the values ±1 with equal probability 1/2.We also extend the definition of Rademacher random completely multiplicative functions to the negative integers using multiplicativity, namely by setting X −n = X −1 X n for n ∈ N. Combining all these elements together leads us to model the Legendre path f p by the following random Fourier series For a fixed t ∈ [0, 1], it follows from [15, Lem.1] that the series defining F X (t) converges almost surely as the limit of the partial sums Note that F X,N are C([0, 1])-valued random variables.We will show, in Proposition 2.1 below, that the sequence of processes (F X,N ) N ⩾1 converges in C([0, 1]) to the process F X as N → ∞.As a consequence, we deduce that F X is almost surely the Fourier series of a continuous function.Figure 2 shows samples of the random process F X depending on the value of X −1 .
Let Q be a large positive integer.We shall view each p → f p for Q ⩽ p ⩽ 2Q as a random variable on the finite probability space {Q ⩽ p ⩽ 2Q} endowed with the uniform probability measure.We denote this random variable by F Q .Our main result shows that the stochastic process (F Q ) Q converges in law, in the space of continuous functions C([0, 1]), to the random process F X .
Theorem 1.2.-As Q → ∞, the sequence of processes (F Q ) Q converges to the process F X in the sense of convergence in law in the Banach space C([0, 1]) endowed with the topology of uniform convergence.More precisely, for any continuous and bounded map  Sample of F X (t) with X−1 = 1 Figure 2. Samples, with 10,000 points, of F X (t), where the x axis is t.
The proof of this result is split into two main parts.We first show in Section 3 that the sequence of processes (F Q ) Q converges to the random process F X in the sense of finite distributions, as Q → ∞.By Prokhorov's theorem (see for example [3, Th. 5.1]) convergence in law in C([0, 1]) is then equivalent to showing that the sequence of processes (F Q ) Q is tight, which we establish in Section 4.
Remark 1.3.-The "graphs" of various exponential sums were investigated by several authors since the seventies.See for example the works of Lehmer [20] and Loxton [21] and [22].Our work is motivated by the recent papers of Kowalski and Sawin [17] on Kloosterman paths of prime moduli (paths constructed using partial sums of Kloosterman sums), and Ricotta-Royer [27], Ricotta-Royer-Shparlinski [28], and Milicevich-Zhang [23] on generalizations to Kloosterman paths of prime power moduli.However, unlike these works, where the corresponding equidistribution problem can be seen as "vertical", ours can be viewed as a "horizontal" problem, which is usually believed to be more challenging.
In [14] the first author investigated the distribution of character paths (paths constructed using partial sums of non-principal Dirichlet characters modulo a prime q), and proved the analogue of Theorem 1.2 for these paths.In this case, the sequence of corresponding processes converges in law to an analogous random Fourier series as q → ∞, where the Fourier coefficients are Steinhaus random multiplicative functions, and where the random variable Y is uniformly distributed on the unit circle.We should note that since the Legendre symbol is a real character, the Legendre path is a time graph, while the Kloosterman paths and character paths (for non-real Dirichlet characters) are instead polygonal curves in the complex plane.
In her thesis, the first author proved Theorem 1.2 assuming the Generalized Riemann Hypothesis GRH.Our unconditional proof is completely different and uses several new ideas to remove GRH from the different parts of the proof.More precisely, to prove that the sequence of processes (F Q ) Q converges in law to the random process F X in the sense of finite distributions, we compute in Section 3 the multivariate moments of (f p (t 1 ), . . ., f p (t k )) (for 0 ⩽ t 1 < • • • < t k ⩽ 1) unconditionally, by using a recent result of the second author [19] to considerably shorten the Pólya Fourier expansion (1.4) of f p for almost all p.We also need to bound the contribution of the possible exceptional discriminant.At the end of Section 3, we describe an alternative approach to establishing convergence in the sense of finite distributions, which is discussed in Section 6.2 of Kowalski's book [16], and relies instead on showing that the sequence of "Fourier coefficients" of the Legendre path converges in the sense of finite distributions to those of F X .Although this approach is somewhat simpler, we decided to include both since the original argument gives a quantitative asymptotic formula for the joint moments of the Legendre path f p at different points, which is interesting on its own, and might have further applications.Furthermore, in Section 4 we prove tightness of the sequence of processes (F Q ) Q by appealing to a variant of Kolmogorov's Tightness Criterion from [16], and then using Burgess's bound for short character sums along with the quadratic large sieve inequalities of Heath-Brown [13] and Montgomery and Vaughan [26], in order to bound several moments of |f p (t) − f p (s)| depending on the range of |t − s| ∈ [0, 1].We should note that our proof is very different from the work of the first author [14] on character paths associated to the family of non-principal characters modulo a large prime q, since this family behave very differently from that of Legendre symbols.Indeed, the crucial ingredients used in [14] are the orthogonality relations for characters modulo q, as well as Deligne's bound for hyper-Kloosterman sums, which is used by the first author to control certain twisted moments of Gauss sums.
Remark 1.4.-Recall that the behavior of the Legendre path attached to p depends on the value of the Legendre symbol −1  p .Indeed, it follows from (1.6) that for any real number Z ⩾ 1 we have As before, we can view each p → f p for Q ⩽ p ⩽ 2Q and p ≡ 1 (mod 4) (respectively p ≡ 3 (mod 4)) as a random variable on the finite probability space {Q ⩽ p ⩽ 2Q and p ≡ 1 (mod 4)} (respectively {Q ⩽ p ⩽ 2Q and p ≡ 3 (mod 4)}) endowed with the uniform probability measure, and we denote this random variable by F Q,+ (respectively F Q,− ).We also do the same with the random Fourier series, where we fix X −1 = 1 or −1.To this end we define for t ∈ [0, 1] J.É.P. -M., 2024, tome 11 Then, it follows from the proof of Theorem 1.2 that the sequence of processes (F Q,± ) Q converges to the process F X,± .
The second author investigated in [19] the distribution of 1 √ p max 1⩽x⩽p |S p (x)|, as p varies over the primes satisfying Q ⩽ p ⩽ 2Q.In particular, he proved that the tail of the distribution is double exponentially decreasing in a large uniform range, which is believed to be best possible.Since the maximum of the Legendre path f p occurs at its vertices and Corollary 1.5.-Let µ be the probability measure on [0, ∞) associated to the random variable ||F X || ∞ .Then for any bounded continuous function h on [0, ∞), we have This result is the analogue of [4, Th. 1.4], which was proved (using a different and more direct approach) by Bober, Goldmakher, Granville and Koukoulopoulos for the family of non-principal characters modulo a large prime q.
2. Properties of the random Fourier series F X (t) 2.1.Almost sure continuity of F X .-Recall that F X (t) is defined as the limit of the finite sum F X,N (t) as N → ∞, and that for fixed t ∈ [0, 1], the series defining F X (t) is almost surely convergent by [15,Lem. 1].In order to prove that F X is almost surely continuous, we use the fact that F X,N are C([0, 1])-valued random variables, and then prove that the sequence of processes (F X,N ) N converges in law to the process F X .To this end we will show that (F X,N ) N converges to F X in the sense of finite distributions, and then show that the sequence (F X,N ) N is tight.
Proposition 2.1.-As N → ∞, the sequence of processes (F X,N ) N converges in the Banach space C([0, 1]).In particular, its limit F X is almost surely the Fourier series of a continuous function.
Since E(X n ) = 1 if n is a square, and equals 0 otherwise, we deduce that [16,Lem. 3] we deduce that the sequence of processes (F X,N ) N converges to the process F X in the sense of finite distributions.
To complete the proof, we need to show that the sequence of processes (F X,N ) N ⩾1 is tight.By Kolmogorov's tightness criterion (see for example [16,Prop. B.11.10]), we need to prove the existence of constants C ⩾ 0, α > 0 and δ > 0 such that for any N ⩾ 1, and real numbers 0 ⩽ s < t ⩽ 1 we have We will prove this with α = 4 and δ = 1/2.Let 0 ⩽ s < t ⩽ 1 be real numbers.For an integer n ̸ = 0 we define Then we have Let ε > 0 be a small real number.Note that |e(α and hence We now use the bound Inserting this estimate in (2.2) and choosing ε = 1/4 implies that for all positive integers N and real numbers 0 ⩽ s < t ⩽ 1, where C > 0 is an absolute constant.This completes the proof.□

The sequence of joint moments of the random process is determinate
In Section 3 we shall use the method of moments to prove that the sequence of processes (F Q ) Q converges to the process F X in the sense of convergence of finite distributions, as Q → ∞.More precisely, given an integer k ⩾ 1 and a k−tuple 0 ⩽ t 1 < • • • < t k ⩽ 1, we will prove that the joint moments of (F Q (t 1 ), . . ., F Q (t k )) converge to those of (F X (t 1 ), . . ., F X (t k )) as Q → ∞.However, we first need to verify that these moments are determinate, i.e. that they have only one representing measure.For n = (n 1 , . . ., n k ) ∈ (Z ⩾0 ) k we let n = k j=1 n j , and define Then we have J.É.P. -M., 2024, tome 11 since P(X −1 = 1) = P(X −1 = −1) = 1/2.Therefore, using the definition of F X,+ (t) and expanding the moment M X,+ (n) we obtain (2.7) where sin(2πb i,j t i ).
We observe that uniformly for t we have where d n is the n-th divisor function.Similarly, one has where (1 − cos(2πb i,j t i )).
We also have the following analogous bound to (2.8) We now prove the following lemma.
Lemma 2.2.-The sequence of moments (M X (n)) n∈(Z ⩾0 ) k have only one representing measure.
Proof.-It is sufficient to show that the moments satisfy the Carleman condition [29,Th. 15.11]: Let t ∈ [0, 1] be fixed.Combining the identities (2.6), (2.7), and (2.9) with the upper bounds (2.8) and (2.10) (with k = 1) we deduce that Combining these estimates implies the existence of positive constants C 1 , C 2 such that for all integers n ⩾ 2 This concludes the proof.□ 3. Proof of convergence in the sense of finite distributions Convergence in the sense of finite distributions of the process.-In this section we will prove the following theorem.
the sense of convergence of finite distributions.In other words, for every k ⩾ 1 and for every k−tuple In order to prove this result it suffices to show that the joint moments of (F Q (t 1 ), . . ., F Q (t n )) converge to those of (F X (t 1 ), . . ., F X (t n )) as Q → ∞, since it follows from Section 2.2 that the distribution of the latter random vector is completely determined by its joint moments.Therefore, Theorem 3.1 is a direct consequence of the following proposition.Proposition 3.2.-Let A > 1 be a fixed constant.Let k be a positive integer, n = (n 1 , . . ., n k ) ∈ (Z ⩾0 ) k and 0 ⩽ t 1 < • • • < t k ⩽ 1 be real numbers.Then we have where M X (n) is defined in (2.5), and the implicit constant in the error term is not effective.
Conditionally on GRH, the first author established this result in her thesis [14], with a much stronger error term Q −1/2+o (1) .In order to compute these moments unconditionally, the main ingredient is a recent work of the second author [19], which allows us to substantially reduce the length of the Pólya Fourier expansion of f p , for almost all primes p such that Q ⩽ p ⩽ 2Q (see Lemma 3.4 below).Another difficulty in proving Proposition 3.2 (which explains the weak error term of (3.1)) arises from the possible existence of Landau-Siegel zeros.Let x be a large real number.By the results of [6,Chap. 20], it follows that for all square-free integers |d| ⩽ exp( √ log x) with at most one exception q 1 , we have for some positive constant c, where χ d denotes the Kronecker symbol attached to d.Moreover, by Siegel's theorem this exceptional discriminant if it exists must satisfy for any fixed constant A > 0, where the implicit constant is not effective.To lighten our notation we shall denote by χ p := • p the Legendre symbol modulo a prime p, throughout the remaining part of the paper.The following lemma is standard (see [6]) but we include its proof for the sake of completeness.
for some positive constant c.Moreover, the same estimate holds for Q⩽p⩽2Q p≡3 mod 4 χ p (n).
Proof.-We only prove the estimate for the character sum over primes p ≡ 1 (mod 4) since the proof in the case p ≡ 3 (mod 4) is similar.The first estimate when n is a square follows simply from the prime number theorem in arithmetic progressions together with the trivial bound ω(n) ⩽ log n, where ω(n) is the number of prime factors of n.Now, suppose that n is not a square, and write n = dm 2 where d is square-free and d ̸ = |q 1 |.By the law of quadratic reciprocity we deduce that Q⩽p⩽2Q p≡1 mod 4 The second estimate follows from (3.2), together with the fact that Q⩽p⩽2Q p≡1 mod 4 In order to compute the joint moments of f p (t 1 ), . . ., f p (t k ) we need to considerably shorten the sum in the main term of (1.4), for all t ∈ {t 1 , t 2 , . . ., t k }.To this end we shall use the following result from a recent work of the second author [19] on the distribution of large values of quadratic character sums.
To shorten our notation we define Then, we note that where Therefore, in view of (2.6) it suffices to establish the following asymptotic formulas We shall only prove this estimate for M Q,+ (n), since the proof for M Q,− (n) is similar.
Using (1.4) with Z := Q 21/40 we get Furthermore, it follows from Lemma 3.4 with the choice B = 6An + 1 that Combining the estimates (3.6) and (3.7) we deduce that Expanding the main term we obtain (3.8) where B Y,n,t,+ (a) is defined by We shall first estimate the contribution of the squares a = m 2 which gives rise to the main term of M Q,+ (n).Indeed, the contribution of these terms to the right hand side of (3.8) equals for some positive constant c 0 , by the prime number theorem for arithmetic progressions.Let ε > 0 be small.By (2.8) we have we deduce that this contribution equals (3.9) . Next, we bound the contribution of the terms a which are not of the form m 2 or |q 1 |m 2 .Since Y n ⩽ exp √ log Q by our assumptions on Y and Q, it follows from Lemma 3.3 that the contribution of these terms to the main term on the right hand side of (3.8) is (3.10) ≪ √ a, and where c > 0 is the constant in Lemma 3.3.Finally, we bound the contribution of the terms a = |q 1 |m 2 .Using the bound B Y,n,t,+ (a) ≪ n a 1/4 , and bounding the inner character sum in the right hand side of (3.8) trivially, we deduce that the contribution of these terms is . Inserting the estimates (3.9), (3.10), and (3.11) in (3.8) completes the proof.□

3.2.
Establishing convergence in the sense of finite distributions for the sequence of "Fourier coefficients" of the path.-In this section, we describe a different approach to establishing convergence in the sense of finite distributions.Let C 0 ([0, 1]) be the subspace of C([0, 1]) consisting of functions vanishing at 0, and let C 0 (Z) be the Banach space of complex-valued functions on Z converging to 0 at infinity with the sup norm.Then note that f p ∈ C 0 ([0, 1]) for all primes p.We will follow the notation in Chapter 6 of Kowalski's book [16].Let FT p = (FT p (h)) h∈Z be defined by FT p (0) = 0 and for h ̸ = 0 FT p (h) := Here convergence in the sense of finite distributions of a sequence (X n ) of C 0 (Z)-valued random variables to X means that for any H ⩾ 1, the vectors (X n,h ) |h|⩽H converge in law to (X h ) |h|⩽H , in the sense of convergence in law in C 2H+1 .More specifically, in our case one needs to compute the moments (3.12) where {j h } 0<|h|⩽H are non-negative integers, since FT p (0) = 0 for all p.
Let p be a large prime number.It follows from formula (6.5) of [16,Chap. 6] that for fixed h and we deduce that where ε p is defined in (1.6).By using the Taylor expansions of sin(πh/(p − 1)) and e(−h/(2(p − 1))) we deduce that Using this estimate and taking H and j = (j h ) 0<|h|⩽H fixed we obtain J.É.P. -M., 2024, tome 11 Therefore, using Lemma 3.3 and the bound (3.3) together with the fact that H and j are fixed, we get where c is the constant from Lemma 3.3.This shows that the Fourier coefficients of the Legendre path converge to those of F X in the sense of finite distributions, as desired.

Tightness of the sequence of processes F Q
To complete the proof of Theorem 1.2 we need to show that the sequence of processes (F Q ) Q is tight.To this end we shall use the following variant of Kolmogorov's tightness criterion.-Let (Y n ) n⩾1 be a sequence of C([0, 1])-valued random variables.Suppose that there exist positive real numbers α 1 , α 2 , α 3 , β 1 , β 2 , δ, C, such that β 2 < β 1 and for any real numbers 0 ⩽ s < t ⩽ 1 and any n ⩾ 1, we have Then (Y n ) n⩾1 is tight.Let 0 ⩽ s < t ⩽ 1 be real numbers.To use this criterion for the sequence   -This final range is the hardest to deal with.Indeed, a new feature occurs here (and more generally in the range Q −1/2+ε ⩽ |t − s| ⩽ 1), caused by the fact that the random variables X n are not independent, and F X,p are not sub-Gaussian.In this case we shall use the quadratic large sieve to prove the following key proposition.
Proposition 4.5.-Let β = 1/1000 and 0 ⩽ s < t ⩽ 1 be real numbers such that We shall require the following important large sieve inequalities for quadratic characters.The first is a direct consequence of a classical result due to Heath-Brown [13].
Lemma 4.6.-Let Q, Y ⩾ 2. Then for arbitrary complex numbers a n and for any ε > 0 we have Q⩽p⩽2Q n⩽Y As a corollary we deduce the following lemma.
Lemma 4.7.-Let ε > 0 be a fixed small real number and (a n ) n⩾1 be an arbitrary bounded sequence of complex numbers.Let Q be large, 1 ⩽ Y ⩽ Q and δ > 0 be real numbers.The number of primes p such that Q ⩽ p ⩽ 2Q and Proof.-By Lemma 4.6 we have by writing n 2 = |n 1 n 2 | and using that (a n ) n⩾1 is bounded.Finally, using the trivial bound d(n 2 ) ≪ ε n ε/2 we deduce that the number of primes Q ⩽ p ⩽ 2Q such that (4.3) holds is The next lemma is a large sieve inequality for prime discriminants, which is a special case of [25, Lem.9] (see also [2,Lem. 1]).Lemma 4.8 ([25, Lem.9]).-Let ε > 0 be a fixed small number.Let x, N be real numbers such that x ⩾ 2 and 2 ⩽ N ⩽ x 1/2−ε .Then for arbitrary complex numbers a 1 , . . ., a N we have p⩽x n⩽N Proof of Proposition 4.5.-Taking Z = Q in Pólya's Fourier expansion (1.4) gives for all primes p such that Q ⩽ p ⩽ 2Q and all t ∈ [0, 1] Therefore, our goal is to show that Hence, throughout the proof we will work under the assumption that (4.6) otherwise there is nothing to prove.Let ε > 0 be a small fixed real number.Let 0 < δ < 1 and 1 ⩽ Y ⩽ Q be parameters to be chosen, and define A(δ, Y ) to be the set of primes p ∈ Then it follows from Lemma 4.7 (with the choice a n = e(−ns) − e(−nt)) that We now write where To complete the proof we will choose Y in such a way to insure that I 2 (δ, Y ) ⩽ I/2.This will imply I ⩽ 2I 1 (δ, Y ), and hence it will only remain to bound the fourth moment on the right hand side of (4.9), which we shall handle using the quadratic large sieve.The choice of the parameter Y as well as which large sieve inequality to use (Heath-Brown's large sieve (Lemma 4.6), or the Montgomery-Vaughan large sieve (Lemma 4.8)) will depend on the size of |t − s|.We will consider two cases: 1⩽|n1|,|n2|⩽Y 2 n1n2=□ g(n 1 )g(n 2 ) n 1 n 2 (4.12) where g is defined by (2.1).We now choose Y = Q 1/2−ε |t − s| (β−1)/2 .Then by (4.10) we obtain if ε is suitably small and Q is large enough, by our assumption on |t − s|.Therefore, combining the estimates (4.9) and (4.11) we deduce that Choosing ε to be suitably small and inserting this estimate in (4.9) gives Finally, using (4.10) and our assumption on |t − s| we derive if ε is suitably small and Q is large enough.Combining these estimates we deduce that which completes the proof.□ Remark 4.9.-By optimizing this method and allowing the exponent β > 0 to be very close to 0, we can cover the wider range Q −1/2+ε ⩽ |t − s| ⩽ 1.

Figure 1 .
Figure 1.Legendre paths for p = 991 and p = 997, where the x-axis is pt.

Lemma 3 .
3. -Let Q be large and n ⩽ exp √ log Q be a positive integer.Then, we have Q⩽p⩽2Q p≡1 mod 4

Lemma 3 . 4 .
-Let B > 0 be a fixed constant.For all real numbers N 1 , N 2 verifying exp

1 0f
p (t)e(−ht)dt.By[16, Prop.B.11.8], one can replace Theorem 3.1 (convergence in the sense of finite distributions of the process) in the proof of Theorem 1.2, by the convergence in the sense of finite distributions of the sequence of the Fourier coefficients

4. 1 .
The "trivial" range: 0 < |t − s| ⩽ 1/Q.-Recall the definition of f p : the concatenation of points in the time graph of 1 √ p n⩽pt χ p (n).This implies the trivial bound