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arXiv:1011.4492v1 [math.NT] 19 Nov 2010

On the second smallest prime non-residue

Kevin J. McGown1

Department of Mathematics, University of California, San Diego,

9500 Gilman Drive, La Jolla, CA 92093

Abstract

Let χ be a non-principal Dirichlet character modulo a prime p. Let q1< q2

denote the two smallest prime non-residues of χ. We give explicit upper

bounds on q2that improve upon all known results. We also provide a good

upper estimate on the product q1q2which has an upcoming application to

the study of norm-Euclidean Galois fields.

Keywords:

2010 MSC: Primary 11A15, 11N25; Secondary 11A05

Dirichlet character, non-residues, power residues

1. Introduction and Summary

Let χ be a non-principal Dirichlet character modulo a prime p. We call

a positive integer n a non-residue of χ if χ(n) / ∈ {0,1}, and denote by q1<

q2< ··· < qnthe n smallest prime non-residues of χ. The question of putting

an upper bound on q1is a classical problem which goes all the way back to

the study of the least quadratic non-residue.

The literature on this problem is extensive and we will not review it here

except to say that the work of Burgess in the 1960’s significantly advanced

existing knowledge on this matter. Burgess’ famous character sum estimate

(see [1]) implies that qn= O(p1/4+ε) for all n.2For the case of q1, one can

apply the “Vinogradov trick” (see [3, 4, 5]) to Burgess’ result, which gives

the stronger bound of q1= O(p

1

4√e+ε) (see [1]).

Email address: mcgownk@math.oregonstate.edu (Kevin J. McGown)

1Current address: Department of Mathematics, Oregon State University, 368 Kidder

Hall, Corvallis, OR 97331

2The O constant here depends upon ε and n; see [2] for more detail.

Preprint submitted to Elsevier November 22, 2010

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p0

107

108

109

1010

1011

1012

1013

C

11.0421

8.2760

7.2906

6.8121

6.5496

6.3964

6.3033

p0

1014

1015

1016

1017

1018

1019

1020

C

6.2452

6.2077

6.1829

6.1659

6.1536

6.1445

6.1374

Table 1: Values of C for various choices of p0

Making these results explicit with constants of a reasonable magnitude

turns out to be difficult, and often times it is results of this nature that one

requires in application. In this paper, we will restrict ourselves to the study

of q1and q2, and we will only be interested in bounds which are completely

explicit and independent of the order of χ.3

The best known explicit bound on q1was given by Norton (see [6]) by

applying Burgess’ method (see [1, 7]) with some modifications.

Theorem 1 (Norton). Suppose that χ is a non-principal Dirichlet charac-

ter modulo a prime p, and that q1is the smallest (prime) non-residue of χ.

Then q1< 4.7p1/4logp, and moreover, the constant can be improved to 3.9

when the order of χ and (p − 1)/2 have a common factor.

We prove the following theorem, which can be viewed as a generalization of

Norton’s result but with a slightly larger constant.

Theorem 2. Fix a real constant p0≥ 107. There exists an explicit constant

C (see Table 1) such that if χ is a non-principal Dirichlet character modulo

a prime p ≥ p0and u is a prime with u ≥ e2logp, then there exists n ∈ Z+

with (n,u) = 1, χ(n) ?= 1, and

n < C p1/4logp.

Provided that q1is not too small, the above theorem immediately gives an

explicit bound on q2.

3In Corollary 3 we do assume that χ has odd order, but we emphasize that none of our

constants depend upon the order of χ.

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Corollary 1. Fix a real constant p0≥ 107. Let χ be a non-principal Dirich-

let character modulo a prime p ≥ p0. Suppose that q1 < q2 are the two

smallest prime non-residues of χ. If q1> e2logp, then

q2< C p1/4logp,

where the constant C is the same constant as in the statement of Theorem 2

(see Table 1).

Using a lemma of Hudson and an explicit result of the author on con-

secutive non-residues, we can remove the restriction on q1for a small price.

Corollary 2. Let χ be a non-principal Dirichlet character modulo a prime

p ≥ 1019. Suppose that q1< q2are the two smallest prime non-residues of χ.

Then

q2< 53p1/4(logp)2.

The value q2has not been as extensively studied as q1, and it appears that

prior to now, the best explicit bound was essentially q2 ≤ cp2/5for some

absolute constant c (see [8, 9, 10, 11]). Corollary 2 constitutes an explicit

bound on q2 which even improves slightly on the best known O-bound of

p1/4+ε.

For the application the author has in mind to norm-Euclidean Galois

fields (see [12]), the following corollary is more useful.

Corollary 3. Let χ be a non-principal Dirichlet character modulo a prime

p ≥ 1018having odd order. Suppose that q1< q2are the two smallest prime

non-residues of χ. Then

q1q2< 24p1/2(logp)2.

2. Outline of the Proof

We will establish our results using a generalization of Burgess’ method.

The approach will be similar to a previous paper of the author (see [13]),

but it will be sufficiently different as these results do not follow from the

aforementioned ones or vice versa. The main idea behind Burgess’ method

is to combine upper and lower bounds for the following sum:

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Definition 1. If h,r ∈ Z+and χ is a Dirichlet character modulo p, then we

define

p−1

?

We will use the following lemma, proven in [13], which is a slight improve-

ment on Lemma 2 of [1].

S(χ,h,r) :=

x=0

?????

h

?

m=1

χ(x + m)

?????

2r

.

Lemma 1. Suppose χ is any non-principal Dirichlet character to the prime

modulus p. If r,h ∈ Z+, then

S(χ,h,r) <1

4(4r)rphr+ (2r − 1)p1/2h2r.

Apart from the use of Lemma 1, the proofs of Theorem 2 and Corollary 1

are completely self-contained; in particular, they do not rely on Theorem 1.

However, the derivation of Corollary 2 will use Theorem 1.2 of [13], and the

derivation of Corollary 3 will use Theorem 1 and an explicit version of the

P´ olya–Vinogradov inequality given in [14].

The meat of the proof of our results is to give a lower bound on S(χ,h,r),

under some extra conditions on the involved parameters. In §3 we prove the

following:

Proposition 1. Let h,r,u ∈ Z+with u prime and h ≤ u. Suppose that

χ is a Dirichlet character modulo a prime p ≥ 5 such that χ(n) = 1 for

all n ∈ [1,H] satisfying (n,u) = 1. Assume 2h < H ≤ (2hp)1/2and set

X := H/(2h) > 1. Then

S(χ,h,r) ≥

6

π2(1 − u−1)h(h − 2)2rX2f(X,u).

For each fixed u we have f(X,u) → 1 as X → ∞; the function f(X,u) is

explicitly defined in Lemma 5.

Combining Lemma 1 and Proposition 1 with a careful choice of the pa-

rameters h and r gives our main result from which Theorem 2 follows:

Theorem 3. Suppose that χ is a non-principal Dirichlet character modulo

a prime p ≥ 107, and that u is a prime with u ≥ e2logp. Suppose χ(n) = 1

for all n ∈ [1,H] with (n,u) = 1. If

H ≤ (2e2logp − 2)1/2p1/2,

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then

H ≤ Kg(p)p1/4logp,

where

K =πe

√2≈ 6.0385

and

g(p) =

?

?

?

?

?

?

1 +

4

3logp

?

?

2e2 ,89

?

1 −

1

e2logp

?

f

Kp1/4

? .

The function g(p) is positive and decreasing for p ≥ 107, with g(p) → 1 as

p → ∞. The function f(X,u) is defined in Lemma 5.

The proofs of Theorems 2 and 3 are carried out in §4. Finally in §5 we

derive Corollaries 1, 2, and 3.

3. Proof of Proposition 1

The idea is to locate a large number of disjoint intervals on which χ is

“almost constant.” For the remainder of this section p will denote a prime

with p ≥ 5, and h,H will denote positive integers. The following are the

intervals that will be of interest to us:

Definition 2. For integers with 0 ≤ t < q, we define the intervals

?pt

J(q,t) =

qq

I(q,t) =

q,H + pt

−H + pt

q

?

,

I(q,t)⋆=

?

?pt

q,H + pt

?

q

− h

?

,

?

,−pt

,

J(q,t)⋆=

−H + pt

q

,−pt

q− h

?

.

We note that the intervals I(q,t)⋆, J(q,t)⋆might be empty. In fact,

they are non-empty exactly when h < H/q, which will always be the case

whenever we employ them.

Lemma 2. Let X > 1 be a real number and suppose XH < p. Then the

intervals I(q,t) where 0 ≤ t < q ≤ X with (t,q) = 1 are disjoint, and

similarly for J(q,t).

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