2012

c Cambridge Philosophical Society 2012 Math. Proc. Camb. Phil. Soc. (2012), 153, 147–166  147 doi:10.1017/S0305004112...

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c Cambridge Philosophical Society 2012 Math. Proc. Camb. Phil. Soc. (2012), 153, 147–166 

147

doi:10.1017/S030500411200014X First published online 28 February 2012

Large deviations of the limiting distribution in the Shanks–R´enyi prime number race B Y YOUNESS LAMZOURI† Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green Street, Urbana, IL, 61801, U.S.A. e-mail: [email protected] (Received 24 August 2011) Abstract Let q  3, 2  r  φ(q) and a1 , . . . , ar be distinct residue classes modulo q that are relatively prime to q. Assuming the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI), M. Rubinstein and P. Sarnak [11] showed that the vector-valued function E q;a1 ,...,ar (x) = (E(x; q, a1 ), . . . , E(x; q, ar )), where E(x; q, a) = √ (log x/ x)(φ(q)π(x; q, a)−π(x)), has a limiting distribution µq;a1 ,...,ar which is absolutely continuous on Rr . Furthermore, they proved that for r fixed, µq;a1 ,...,ar tends to a multidimensional Gaussian as q → ∞. In the present paper, we determine the exact rate of this convergence, and investigate the asymptotic behavior of the large deviations of µq;a1 ,...,ar .

1. Introduction A classical problem in analytic number theory is the so-called “Shanks and R´enyi prime number race” (see [3]) which is described in the following way. Let q  3 and 2  r  φ(q) be positive integers. For an ordered r -tuple of distinct reduced residues (a1 , a2 , . . . , ar ) modulo q we denote by Pq;a1 ,...,ar the set of real numbers x  2 such that π(x; q, a1 ) > π(x; q, a2 ) > . . . > π(x; q, ar ). Will the sets Pq;aσ (1) ,...,aσ (r ) contain arbitrarily large values, for any permutation σ of {1, 2, . . . , r }? A result of J. E. Littlewood [7] from 1914 shows that the answer is yes in the cases (q, a1 , a2 ) = (4, 1, 3) and (q, a1 , a2 ) = (3, 1, 2). Similar results to other moduli in the case r = 2 were subsequently derived by S. Knapowski and P. Tur´an [3] (under some hypotheses on the zeros of Dirichlet L-functions), and special cases of the prime number race with r  3 were considered by J. Kaczorowski [4, 5]. In 1994, M. Rubinstein and P. Sarnak [11] completely solved this problem, conditionally on the assumptions of the Generalized Riemann Hypothesis (GRH) and the Linear Independence Hypothesis (LI) (which is the assumption that the nonnegative imaginary parts of the zeros of all Dirichlet L-functions attached to primitive characters modulo q are linearly independent over Q). To describe their results, we first define some notation. For any real † Supported by a Postdoctoral Fellowship from the Natural Sciences and Engineering Research Council of Canada. Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

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number x  2 we introduce the vector-valued function E q;a1 ,...,ar (x) := (E(x; q, a1 ), . . . , E(x; q, ar )), where log x E(x; q, a) := √ (φ(q)π(x; q, a) − π(x)) . x The normalization is such that, if we assume GRH, E q;a1 ,...,ar (x) varies roughly boundedly as x varies. Rubinstein and Sarnak showed, assuming GRH, that the function E q;a1 ,...,ar (x) has a limiting distribution µq;a1 ,...,ar . More precisely, they proved  X  dx 1 = f (E q;a1 ,...,ar (x)) f (x1 , . . . , xr )dµq;a1 ,...,ar , (1·1) lim X →∞ log X 2 x Rr for all bounded, continuous functions f on Rr . Furthermore, assuming both GRH and LI, they showed that µq;a1 ,...,ar is absolutely continuous with respect to the Lebesgue measure on Rr if r < φ(q). (When r = φ(q), µq;a1 ,...,ar is shown to be absolutely continuous with  respect to the Lebesgue measure on the hyperplane rj=1 x j = 0.) As a consequence, under GRH and LI, the logarithmic density of the set Pq;a1 ,...,ar defined by  1 dt lim x→∞ log x t∈P t q;a1 ,...,ar [2,x] exists and is positive.

 r 2 Here and throughout we shall use the notations ||x|| = j=1 x j and |x|∞ = max1ir |xi | for the Euclidean norm and the maximum norm on Rr respectively. In [11], Rubinstein and Sarnak also studied the behavior of the tail µq;a1 ,...,ar (||x|| > V ) when q is fixed. They showed, under GRH, that for all distinct reduced residues a1 , . . . , ar modulo q we have √ exp (− exp(c2 (q)V ))  q µq;a1 ,...,ar (||x|| > V )  q exp(−c1 (q) V ), (1·2) for some c1 (q), c2 (q) > 0 which depend on q. In this paper we investigate large deviations of the distribution µq;a1 ,...,ar uniformly as q → ∞, under the additional assumption of LI. For a non-trivial character χ modulo q, we denote by {γχ } the sequence of imaginary parts of the non-trivial zeros of L(s, χ). Let χ0 denote the principal character modulo q and define S = χχ0 mod q {γχ }. Moreover, let {U (γχ )}γχ ∈S be a sequence of independent random variables uniformly distributed on the unit circle. Rubinstein and Sarnak established, under GRH and LI, that the distribution µq;a1 ,...,ar is the same as the probability measure corresponding to the random vector X q;a1 ,...,ar = (X (q, a1 ), . . . , X (q, ar )), where X (q, a) = −Cq (a) +

(1·3)

  2Re(χ(a)U (γχ ))  , 1 2 + γ χχ0 γχ >0 χ 4

χ mod q

χ0 is the principal character modulo q and Cq (a) := −1 +



1.

b2 ≡a mod q 1bq Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

The limiting distribution in the Shanks–R´enyi prime number race

149

Note that for (a, q) = 1 the function Cq (a) takes only two values: Cq (a) = −1 if a is a nonsquare modulo q, and Cq (a) = Cq (1) if a is a square modulo q. Furthermore,  an elementary argument shows that Cq (a) < d(q)   q  for any  > 0, where d(q) = m|q 1 is the usual divisor function. Let Covq;a1 ,...,ar be the covariance matrix of X q;a1 ,...,ar . A straightforward computation shows that the entries of Covq;a1 ,...,ar are  Covq;a1 ,...,ar ( j, k) = where Var(q) :=

  χχ0 γχ χ mod q

and Bq (a, b) :=

1 4

if j = k if j  k,

Var(q) Bq (a j , ak )

  1 = 2 + γχ2 χχ0 γ >0

 χ χχ0 γχ >0 χ mod q

χ mod q

b a 1 4

χ



1 4

1 + γχ2

a b

+ γχ2

for (a, b) ∈ A(q), where A(q) is the set of ordered pairs of distinct reduced residues modulo q. Assuming GRH, Rubinstein and Sarnak showed that Var(q) ∼ φ(q) log q and Bq (a, b) = o(φ(q) log q) as q → ∞,

(1·4)

uniformly for all (a, b) ∈ A(q). Combining these estimates with an explicit formula for the Fourier transform of µq;a1 ,...,ar (see (3·1) below) they established, under GRH and LI, that

2     x1 + · · · + xr2 −r/2 dx + oq (1), (1·5) µq;a1 ,...,ar ||x|| > λ Var(q) = (2π) exp − 2 ||x||>λ for any fixed λ > 0. We refine their result using the approach developed in [6]. More precisely,  we prove that the asymptotic formula (1·5) holds uniformly in the range 0 < λ  log log q with an optimal error term Or (1/ log2 q). T HEOREM 1. Assume GRH and LI. Fix an integer r  2. Let q  be large and a1 , . . . , ar be distinct reduced residues modulo q. Then, in the range 0 < λ  log log q we have



2     x + · · · + xr2 1 . dx + Or exp − 1 µq;a1 ,...,ar ||x|| > λ Var(q) = (2π)−r/2 2 log2 q ||x||>λ Moreover, there exists an r -tuple of distinct reduced residue classes (a1 , . . . , ar ) modulo q, such that in the range 1/4 < λ < 3/4 we have

2     1 x1 + · · · + xr2 µq;a1 ,...,ar ||x|| > λ Var(q) − (2π)−r/2 dx  r . exp − 2 log2 q ||x||>λ Since Var(q) ∼ φ(q) log q, it follows from Theorem 1 that

V2 (1 + o(1)) µq;a1 ,...,ar (||x|| > V ) = exp − 2φ(q) log q

(1·6)

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in the range (φ(q) log q)1/2  V  (1 + o(1))(φ(q) log q log log q)1/2 . Our next result shows that a similar behavior holds in the much larger range (φ(q) log q)1/2  V  φ(q) log q. T HEOREM 2. Assume GRH and LI. Fix an integer r  2 and a real number A  1. Let q be large. Then for all distinct reduced residues a1 , . . . , ar modulo q, we have



V2 V2  µq;a1 ,...,ar (||x|| > V )  exp −c1 (r, A) , exp −c2 (r, A) φ(q) log q φ(q) log q uniformly in the range (φ(q) log q)1/2  V  Aφ(q) log q, where c2 (r, A) > c1 (r, A) are positive numbers that depend only on r and A. Using an analogous approach, we prove that (1·6) does not hold when V /(φ(q) log q) → ∞ as q → ∞, which shows that a transition occurs at V  φ(q) log q. T HEOREM 3. Assume GRH and LI. Fix an integer r  2 and let q be large. If V /(φ(q) log q) → ∞ and V /(φ(q) log2 q) → 0 as q → ∞, then for all distinct reduced residues a1 , . . . , ar modulo q, we have

V2 V exp c6 (r )  µq;a1 ,...,ar (||x|| > V ) exp −c4 (r ) φ(q) log q φ(q) log q and

µq;a1 ,...,ar (||x|| > V )  exp −c3 (r )



V2 V exp c5 (r ) , φ(q) log q φ(q) log q

where c4 (r ) > c3 (r ), and c6 (r ) > c5 (r ) are positive numbers which depend only on r . In the range V /(φ(q) log2 q) → ∞ one can prove, using the same ideas, that there are positive constants c8 (r ) > c7 (r ) such that

  V V  µq;a1 ,...,ar (||x|| > V )  exp − exp c7 (r ) . exp − exp c8 (r ) φ(q) φ(q) (1·7) In particular, these bounds show that the asymptotic behavior of the tail µq;a1 ,...,ar (||x|| > V ) changes again at V  φ(q) log2 q. Assuming the RH and using the LI for the Riemann zeta function, H. L. Montgomery [9] had previously obtained a similar result for µ1 , the limiting distribution of the error term in the prime number theorem π(x) − Li(x). His result states that √ √ √   √   exp − c10 V exp 2π V  µ1 (|x| > V )  exp − c9 V exp 2π V , for some absolute constants c10 > c9 > 0. A more precise estimate was subsequently derived by W. Monach [8], namely √  √  (1·8) µ1 (|x| > V ) = exp − (e−A0 + o(1)) 2π V exp 2π V , where A0 is an absolute constant defined in Theorem 4 below. In our case, it appears that µq;a1 ,...,ar (|x|∞ > V ) is more natural to study, if one wants to gain a better understanding of the decay rate of large deviations of µq;a1 ,...,ar in the range V /(φ(q) log2 q) → ∞. We achieved this using the saddle-point method. We also note that in contrast to our previous results, r can vary uniformly in [2, φ(q) − 1] as q → ∞. Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

The limiting distribution in the Shanks–R´enyi prime number race

151

T HEOREM 4. Assume GRH and LI. Let q be large, and 2  r  φ(q) − 1 be an integer. If V /(φ(q) log2 q) → ∞ as q → ∞, then for all distinct reduced residue classes a1 , . . . , ar modulo q, the tail µq;a1 ,...,ar (|x|∞ > V ) equals    2 1/4 φ(q) log 2(φ(q) − 1)V q 2π V exp 1+O exp −e−L(q) , L(q)2 + π φ(q) − 1 V where

⎛ ⎞  log p φ(q) ⎝ ⎠ + A0 − log π L(q) = log q − φ(q) − 1 p − 1 p|q

and



1

A0 := where I0 (t) =

∞

0

log I0 (t) dt + t2

n=0 (t/2)





1

log I0 (t) − t dt + 1 = 1.2977474, t2

/n! is the modified Bessel function of order 0. √ Lastly, we should also mention that in the range V  φ(q) log q one may allow r to vary uniformly, as in Theorem 4, if one is willing to replace µq;a1 ,...,ar (||x|| > V ) by µq;a1 ,...,ar (|x|∞ > V ) in the statements of Theorems 2 and 3. 2n

2

2. The intermediate range: proof of Theorems 2 and 3  In this section we investigate the 2behavior of the tail µq;a1 ,...,ar (||x|| > V ) when V  φ(q) log q as long as V /(φ(q) log q) → 0 as q → ∞. Recall that µq;a1 ,...,ar is also the probability measure corresponding to the random vector X q;a1 ,...,ar (defined in (1·3)). Our idea starts with the observation that the random variables   2Re(χ(a)U (γχ ))  Y (q, a) := X (q, a) − E(X (q, a)) = , (2·1) 1 2 + γ χχ0 γχ >0 χ 4 χ mod q

are identically distributed for all reduced residues a modulo q. Indeed, for all (a, q) = 1 the random variables {U˜ (γχ )}γχ ∈S , where U˜ (γχ ) = χ(a)U (γχ ), are independent and uniformly distributed on the unit circle. Hence, if (a, q) = 1 then Y (q, a) has the same distribution as   2 cos(2πθ(γχ ))  , (2·2) Y (q) := 1 2 + γ χχ0 γχ >0 χ 4 χ mod q

where {θ(γχ )}γχ ∈S are independent random variables uniformly distributed on [0, 1]. Our first lemma shows, in our range of V , that large deviations of µq;a1 ,...,ar are closely related to those of Y (q). L EMMA 2·1. The random √ variable Y (q) is symmetric. Moreover, if q is sufficiently large, r  2 is fixed and V  φ(q), then for all distinct reduced residues a1 , . . . , ar modulo q we have

V P(Y (q) > 2V )  µq;a1 ,...,ar (||x|| > V )  2r P Y (q) > √ . 2 r Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

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∞ m Proof. Note that E(exp(it cos(2πθ(γχ )))) = J0 (t), where J0 (t) := m=0 (−1) 2m 2 (t/2) /m! is the Bessel function of order 0. Therefore, since the Fourier transform (characteristic function) of Y (q) is ⎛ ⎞   2t ⎠, E(eitY (q) ) = J0 ⎝  1 2 + γ χχ0 γχ >0 χ 4 χ mod q

and J0 is an even function then Y (q) is symmetric. Now, µq;a1 ,...,ar (||x|| > V ) = P(||X q;a1 ,...,ar || > V )  P(X (q, a1 ) > V ) = P(Y (q) > V +Cq (a1 )). The lower bound follows from the fact that Cq (a1 ) < d(q) = q o(1) . On the other hand, if √ |X (q, a j )|  V / r for all 1  j  r , then ||X q;a1 ,...,ar ||  V . Hence

 r r  V V P |X (q, a j )| > √ P |Y (q)| > √ − |Cq (a j )| .  µq;a1 ,...,ar (||x|| > V )  r r j=1 j=1 √ Using that V  φ(q), |Cq (a)| = q o(1) for all (a, q) = 1, and Y (q) is symmetric we obtain from the last inequality



V V µq;a1 ,...,ar (||x|| > V )  r P |Y (q)| > √ = 2r P Y (q) > √ , 2 r 2 r if q is sufficiently large. This establishes the lemma. Let Sq (T ) :=







χχ0 0T 4

1 1 4

+ γχ2

,

1 . + γχ2

To investigate large deviations of Y (q), we shall establish the following result, which is very similar to Montgomery and Odlyzko [10, theorem 1]. T HEOREM 2·2. Let V be a positive real number. If Sq (T )  V /4 then

V2 P(Y (q)  V )  exp − . 16K q (T ) Moreover, if Sq (T )  8V then

120V 2 . exp − K q (T )

(2·3)



−50

P(Y (q)  V )  2

(2·4)

In order to prove this result, we shall use the properties of the Laplace transform (or the moments generating function) of Y (q), which is defined by L(λ) := E(eλY (q) ). Since Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

The limiting distribution in the Shanks–R´enyi prime number race E(exp(t cos(2πθ(γχ )))) = I0 (t), we have the following product formula for L(λ) ⎛ ⎞   2λ ⎠ L(λ) = I0 ⎝  for λ  0. 1 2 + γ χχ0 γχ >0 χ 4

153

(2·5)

χ mod q

Note that the infinite product in (2·5) convergent for λ  0, since I0 (t)    is absolutely 1 2 exp(t /4) by Lemma 2·3 below and χχ0 γχ >0 ( 4 + γχ2 )−1 < ∞.  2n 2 Recall that I0 (λ) = ∞ n=0 (λ/2) /n! , and so I0 (λ)  1 for λ  0. First, we establish some classical estimates for I0 (λ). L EMMA 2·3. For all λ  0 we have

  2 I0 (λ)  min eλ , eλ /4 .

Moreover, we have

 I0 (λ) 

eλ /16 eλ/8 2

if 0  λ  4 if λ  4.

(2·6)

(2·7)

 ∞ 2n 2 2n 2 Proof. First, note that I0 (λ) = ∞ n=0 (λ/2) /n!  n=0 (λ/2) /n! = exp(λ /4). Now, x 2 using that e  1 + 4x for 0  x  1, we deduce that I0 (λ)  1 + λ /4  exp(λ2 /16) for 0  λ  4. On the other hand, we have the following integral representation of I0 (λ)  1 I0 (λ) = eλ cos(2πθ) dθ, (2·8) 0 λ

which shows that I0 (λ)  e for all λ  0. Moreover, since cos(2πθ)  1/2 if θ ∈ [0, 1/6]  [5/6, 1] then I0 (λ)  eλ/2 /3  eλ/8 if λ  4, which completes the proof. To prove Theorem 2·2 we shall use the following probability inequalities. L EMMA 2·4. Let X be a random variable on a probability space ( , µ) such that E(et X ) < ∞, for all t  0. Then, for all λ > 0 we have P(X  V )  e−λV E(eλX ).

(2·9)

Moreover, if λ and V are positive real numbers such that E(eλX ) = 2eλV , then P (X  V ) 

E(eλX )2 . 4E(e2λX )

(2·10)

Proof. The first inequality is standard and known as Chernoff’s bound. It is a simple consequence of Markov’s inequality. Indeed, we have P(X  V ) = P(eλX  eλV )  e−λV E(eλX ). λV Now we establish (2·10). Let Z = eλX . Then P(X  V  ) = P(Z  e ) = P(Z  E(Z )/2), by our assumption on λ and V . Moreover, since Z E(Z )/2 Z dµ  E(Z )/2, then applying the Cauchy-Schwarz inequality, we obtain





2   E(Z ) 2 E(Z ) ,  Z dµ  E Z 2 P Z  2 2 Z E(Z )/2

which completes the proof. Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

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Proof of Theorem 2·2. We follow the proof of [10, theorem 1]. We shall first establish (2·3). Using (2·5) and (2·9) we obtain ⎛ ⎞   2λ ⎠ P(Y (q)  V )  e−λV L(λ)  e−λV I0 ⎝  , 1 2 + γ χχ0 γχ >0 χ 4 χ mod q

for any λ > 0. Now, we use the first bound of (2·6) for I0 (2λ( 41 + γχ2 )−1/2 ) when γχ  T and the second bound of (2·6) when γχ > T , which yields

λV + λ2 K q (T ) P(Y (q)  V )  exp(−λV + 2λSq (T ) + λ2 K q (T ))  exp − 2 since Sq (T )  V /4. Choosing λ = V /(4K q (T )) gives the desired bound. Now suppose that Sq (T )  8V , and let λ0 be a solution of the equation L(λ) = 2eλV . Such a λ0 must exist, since both sides of this equation are continuous functions of λ, the left-hand side  is smaller than the right when λ = 0, and the reverse is true when λ  −1 max(V , 2 1/4 + T 2 ), since by Lemma 2·3 we have ⎛ ⎞

  2λ ⎠ λ  exp Sq (T )  e2λV > 2eλV . I0 ⎝  (2·11) L(λ)  4 1 + γ2 χχ 0T 0

χ mod q

χ

4

χ

which yields that λ0  8V /K q (T ). Inserting this bound in (2·12) completes the proof. In order to apply Theorem 2·2 to our setting, we have to understand the asymptotic behavior of the sums Sq (T ) and K q (T ). For a non-trivial character χ modulo q, we let qχ∗ be the conductor of χ, and χ ∗ be the unique primitive character modulo qχ∗ which induces χ. We begin by recording some standard estimates which will be useful in our subsequent work. L EMMA 2·5. Assume GRH. Let χ be a non-trivial character modulo q. Then  1 1 = log qχ∗ + O(log log q). 1 2 2 + γχ γ >0 4 χ

Moreover, we have

⎞  log p ⎠, log qχ∗ = φ(q) ⎝log q − p−1 p|q χ mod q 



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The limiting distribution in the Shanks–R´enyi prime number race and

155

 log p  log log q. p−1 p|q

Proof. The first estimate follows from [2, lemma 3·5], and the second is proved in [2, proposition 3·3]. Finally, we have  log p  log p 1   + 1  log log q, p − 1 p − 1 log q 2 p|q p|q p(log q)

which follows from the trivial bound



p|q

1  log q/ log 2.

From this lemma, one can deduce the more precise asymptotic Var(q) = φ(q) log q + O(φ(q) log log q). Our next result gives the classical estimate for  Nq (T ) :=



1.

χχ0 0 λ Var(q)) in the range 0 < λ  log log q, from which Theorem 1 will be deduced. Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

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T HEOREM 3·2. Assume GRH and LI. Fix an integer r  2. Let q be large and  a1 , . . . , ar be distinct reduced residue classes modulo q. Then in the range 0 < λ  log log q we have

2     x + · · · + xr2 dx exp − 1 µq;a1 ,...,ar ||x|| > λ Var(q) = (2π)−r/2 2 ||x||>λ

 (log log q)r 1 2 , Bq (a j , ak ) F j,k (λ) + Or + 2Var(q)2 1 jλ

2   x + · · · + xr2 x 2j − 1 xk2 − 1 exp − 1 dx. 2

Proof of Theorem 1. First, it follows from [6, corollary 5·4] that max |Bq (a, b)|  φ(q).

(a,b)∈A(q)

On the other hand, [6, proposition 5·1] yields |Bq (a, −a)|  φ(q), for all (a, q) = 1. Hence, we deduce max

(a,b)∈A(q)

1 |Bq (a, b)|  . Var(q) log q

(3·2)

We also remark that this last estimate follows implicitly from the work of D. Fiorilli and G. Martin [2]. The first part of Theorem 1 follows from Theorem 3·2 upon using (3·2) and noting that F j,k (λ)  r 1. Let 1  j < k  r . If 1/4 < λ < 3/4, then

2   2   2 x1 + · · · + xr2 −r/2 dx  δr , (2π) x j − 1 xk − 1 exp − 2 ||x||λ for some positive number δr which depends only on r . Moreover, we have

2   2   2 x1 + · · · + xr2 −r/2 dx = 0, (2π) x j − 1 xk − 1 exp − 2 x∈Rr since the antiderivative of (x 2 − 1)e−x λ < 3/4, that

2

/2

is −xe−x

2

/2

. Hence we deduce, in the range 1/4 <

F j,k (λ)  −δr , for all 1  j < k  r.

(3·3)

On the other hand, it follows from (3·2) that there exist distinct reduced residues a1 , a2 modulo q, such that 1 Bq (a1 , a2 )2 .  Var(q)2 log2 q

(3·4)

Let a3 , . . . , ar be distinct reduced residues modulo q that are different from a1 and a2 . Downloaded from https://www.cambridge.org/core. YBP Library Services, on 31 Aug 2018 at 20:12:24, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S030500411200014X

The limiting distribution in the Shanks–R´enyi prime number race Appealing to Theorem 3·2 along with (3·3) we get     µq;a1 ,...,ar ||x|| > λ Var(q) − (2π)−r/2



x 2 + · · · + xr2 exp − 1 2 ||x||>λ

 |F1,2 (λ)|



159

dx

Bq (a1 , a2 )2 (log log q)r − κ r 2Var(q)2 (log q)3

for some κr > 0 (which depends only on r ), if q is sufficiently large. Combining this inequality with (3·3) and (3·4) completes the proof. The first step in the proof of Theorem 3·2 consists in using the explicit formula (3·1) to approximate the Fourier transform µˆ q;a1 ,...,ar (t1 , . . . , tr ) by a multivariate Gaussian in the range ||t||  log q/Var(q). To lighten the notation, we set µ = µq;a1 ,...,ar throughout the remaining part of this section. L EMMA 3·3.  Assume GRH and LI. Fix an integer r  2 and a real number A = A(r ) > 0. If ||t||  A log q, then



2  1 t1 tr t1 + · · · + tr2 1− µˆ √ Bq (a j , ak )t j tk ,..., √ = exp − 2 Var(q) 1 j0 Var(q)1/4 Var(q) Var(q) + γχ2 4 0 χ mod q

χ

(3·5)  in the range ||t||  A log q. Furthermore, the main term on the right-hand side of (3·5) equals −

  1 Var(q) χχ γ >0 0

χ mod q

χ

1 4

 1 χ(a j )χ(ak )t j tk 2 + γχ 1 j,kr

 1 t 2 + · · · + tr2 − =− 1 Bq (a j , ak )t j tk . 2 Var(q) 1 j