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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 ElsevierNovember 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.

2

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Proof. Using the explicit version of the P´ olya–Vinogradov inequality proven

in [14], we find

?

(n,q1)=1

n<x

χ(n)

=

??????

?????

?

n<x

χ(n) − χ(q1)

?

n<x/q1

χ(n)

??????

≤

?

?

n<x

χ(n)

?????+

??????

?

n<x/q1

χ(n)

??????

≤ 2

1

3log3m1/2logm + 6.5m1/2

?

.

If χ(n) = 1 for all n ≤ x with (n,q1) = 1, then

?

(n,q1)=1

n<x

χ(n) ≥ (1 − q−1

1)x − 1.

Thus for 1 < x < q2, we have

(1 − q−1

1)x − 1 ≤ 2

?

1

3log3m1/2logm + 6.5m1/2

?

.

Using the fact that q1≥ 2 and letting x approach q2from the left, we obtain

?

and the result follows. ?

q2≤ 4

1

3log3m1/2logm + 6.5m1/2

?

+ 2,

Proof of Corollary 3. If q1< e2logp, we use Lemma 8 to obtain q2< 2p1/2logp

and hence q1q2< 2e2p1/2(logp)2< 15p1/2(logp)2. If q1≥ e2logp, then we

apply Theorem 1 (using the fact that χ has odd order) and Corollary 1 to

find q1q2≤ C′p1/2(logp)2with C′= (3.9)(6.1536) < 24. ?

References

[1] D. A. Burgess, On character sums and primitive roots, Proc. London

Math. Soc. (3) 12 (1962) 179–192.

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[2] R. H. Hudson, A note on prime kth power nonresidues, Manuscripta

Math. 42 (1983) 285–288.

[3] I. M. Vinogradov, Sur la distribution des residus et des non-residus des

puissances, J. Phys. Math. Soc. Perm. 1 (1918) 94–96.

[4] I. M. Vinogradov, On a general theorem concerning the distribution of

the residues and non-residues of powers, Trans. Amer. Math. Soc. 29

(1927) 209–217.

[5] I. M. Vinogradov, On the bound of the least non-residue of nth powers,

Trans. Amer. Math. Soc. 29 (1927) 218–226.

[6] K. K. Norton, Numbers with small prime factors, and the least kth power

non-residue, Memoirs of the American Mathematical Society, No. 106,

American Mathematical Society, Providence, R.I., 1971.

[7] D. A. Burgess, A note on the distribution of residues and non-residues,

J. London Math. Soc. 38 (1963) 253–256.

[8] A. Brauer, ¨Uber den kleinsten quadratischen Nichtrest, Math. Z. 33

(1931) 161–176.

[9] A. Brauer, On the non-existence of the Euclidean algorithm in certain

quadratic number fields, Amer. J. Math. 62 (1940) 697–716.

[10] C. T. Whyburn, The second smallest quadratic non-residue, Duke Math.

J. 32 (1965) 519–528.

[11] R. H. Hudson, Prime k-th power non-residues, Acta Arith. 23 (1973)

89–106.

[12] K. J. McGown, Norm-Euclidean Galois fields (in preparation).

[13] K. J. McGown, On the constant in Burgess’ bound for the number of

consecutive residues or non-residues (submitted).

[14] G. Bachman, L. Rachakonda,

Williams and the P´ olya-Vinogradov inequality, Ramanujan J. 5 (2001)

65–71.

On a problem of Dobrowolski and

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