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Upper bound estimate of character sums over Lehmer’s numbers

Abstract

Let p be an odd prime. For each integer a with 1ap1, it is clear that there exists one and only one a ¯ with 0 a ¯ p1 such that a a ¯ 1modp. Let denote the set of all integers 1ap1, in which a and a ¯ are of opposite parity. The main purpose of this paper is using the analytic method and the properties of Kloosterman sums to study the estimate problem of the mean value a A χ(a), and give a sharp upper bound estimate for it, where χ denotes any non-principal even character modp.

MSC:11L40.

1 Introduction

Let q3 be an odd number. For each integer a with 1aq1 and (a,q)=1, it is clear that there exists one and only one b with 0bq1 such that ab1modq. Let A(q)=A denote the set of cases, in which a and b are of opposite parity. For q=p, an odd prime, professor Lehmer [1] asked to study |A| or at least to say something nontrivial about it, where p is a prime, and |A| denote the number of all elements in . We call such a number a Lehmer’s number. It is known that |A|2 or 0mod4 when p±1mod4. For general odd number q3, Zhang ([2] and [3]) studied the asymptotic properties of |A|, and obtained a sharp asymptotic formula for it. That is, he proved the asymptotic formula

|A|= 1 2 ϕ(q)+O ( q 1 2 d 2 ( q ) ln 2 q ) ,

where ϕ(q) is the Euler function, and d(q) is the Dirichlet divisor function.

Let M(a,p) denote the number of all integers 1b, cp1 such that bcamodp and (2,b+c)=1, and E(a,p)=M(a,p) p 1 2 . Then Zhang [4] also studied the mean value properties of E(a,p), and proved that

a = 1 p 1 E 2 (a,p)= 3 4 p 2 +O ( p exp ( 3 ln p ln ln p ) ) ,

where exp(y)= e y . Some related works can also be found in [5, 6] and [7].

In this paper, we consider the estimate problem of the character sums

a A χ(a)= a = 1 p 1 2 a + a ¯ χ(a),
(1.1)

and give a sharper upper bound estimate for it, where χ denotes any non-principal even character modp.

About character sums (1.1), it is clear that its value is zero, if χ is an odd character modp. If χ is a non-principal even character modp, then how large is the upper bound estimate of (1.1)? About this problem, it seems that none had studied it yet, at least we have not seen any related results. The problem is interesting, because it can help us to understand the deep properties of the character sums over some special sets, for example, Lehmer’s numbers.

The main purpose of this paper is using the analytic method and the properties of Kloosterman sums to study this problem, and prove the following result.

Theorem Let p>3 be an odd prime. Then for any non-principal character χmodp, we have the estimate

a A χ(a) p 1 2 ln 2 p.

For general odd number q3, whether there exists a similar upper bound estimate for (1.1) is an interesting problem, we will further study it.

2 Several lemmas

In this section, we will give several lemmas, which are necessary in the proof of our theorem. Hereinafter, we will use many properties of character sums, all of which can be found in [8], so they will not be repeated here. First we have the following.

Lemma 1 Let q>2 be an odd number. Suppose that χ is an odd character modq, then we have the identity

( 1 2 χ ( 2 ) ) a = 1 q aχ(a)=χ(2)q a = 1 q 1 2 χ(a).

Proof See reference [9]. □

Lemma 2 Let q>2 be an odd number. Then for any even character χ 1 modq, we have the identity

a A χ(a)= 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 ( 1 2 χ χ 1 ( 2 ) ) ( 1 2 χ ( 2 ) ) ( 1 q a = 1 q a χ χ 1 ( a ) ) ( 1 q a = 1 q a χ ( a ) ) ,

where χ mod q χ ( 1 ) = 1 denotes the summation over all odd characters χmodq.

Proof From the orthogonality relation for character sums modq and the definition of we have

a A χ ( a ) = 1 2 a = 1 q 1 ( 1 ( 1 ) a + a ¯ ) χ 1 ( a ) = 1 2 a = 1 q 1 χ 1 ( a ) 1 2 a = 1 q 1 ( 1 ) a + a ¯ χ 1 ( a ) = 1 2 ϕ ( q ) χ mod q ( a = 1 q 1 ( 1 ) a χ 1 χ ( a ) ) ( b = 1 q 1 ( 1 ) b χ ( b ) ) ,
(2.1)

where a = 1 q 1 denotes the summation over all 1aq such that (a,q)=1.

If χ(1)=1, then

b = 1 q 1 ( 1 ) b χ(b)=0.
(2.2)

If χ(1)=1, then

b = 1 q 1 ( 1 ) b χ(b)=2χ(2) b = 1 q 1 2 χ(b).
(2.3)

Note that if χ(1)=1 and χ 1 (1)=1, then χ χ 1 (1)=1. So combining (2.1), (2.2), (2.3) and Lemma 1, we have

a A χ ( a ) = 1 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 ( 2 χ χ 1 ( 2 ) a = 1 ( q 1 ) / 2 χ 1 χ ( a ) ) ( 2 χ ( 2 ) b = 1 ( q 1 ) / 2 χ ( b ) ) = 2 ϕ ( q ) χ mod q χ ( 1 ) = 1 ( 1 2 χ χ 1 ( 2 ) ) ( 1 2 χ ( 2 ) ) ( 1 q a = 1 q a χ χ 1 ( a ) ) ( 1 q a = 1 q a χ ( a ) ) .

This proves Lemma 2. □

Lemma 3 Let p be an odd prime, let χ be any character modp. Then for any integers m and n, we have the estimate

a = 1 p 1 χ(a)e ( m a + n a ¯ p ) ( m , n , p ) 1 2 p 1 2 ,

where e(y)= e 2 π i y , (m,n,p) denotes the greatest common divisor of m, n and p.

Proof See [10] and [11]. □

Lemma 4 Let p be an odd prime, let χ be any even character modp. Then for any integer c with (c,p)=1, we have the estimate

| χ mod p χ ( 1 ) = 1 χ(c)τ(χ)τ(χ χ 1 )L(1, χ ¯ χ 1 ¯ )L(1, χ ¯ )| p 3 2 ln 2 p.

Proof For any non-principal character χmodp, applying Abel’s identity (see Theorem 4.2 of [8]), we have

L(1, χ ¯ χ 1 ¯ )L(1, χ ¯ )= n = 1 p 3 χ ¯ ( n ) d | n χ ¯ 1 ( d ) n + p 3 A ( y , χ ¯ ) y 2 dy,
(2.4)

where A(y, χ ¯ )= p 3 < n y χ ¯ (n) d | n χ ¯ 1 (d).

From [12], we know that for any real number y> p 3 , we have the estimate

χ mod p χ ( 1 ) = 1 |A(y, χ ¯ ) | 2 y ϕ 2 (p).
(2.5)

From (2.4), (2.5), Lemma 3, the orthogonality relation for character sums modp, the definition of Gauss sums, and noting that |τ(χ)|= p , we have

χ mod p χ ( 1 ) = 1 χ ( c ) τ ( χ ) τ ( χ χ 1 ) L ( 1 , χ ¯ χ 1 ¯ ) L ( 1 , χ ¯ ) = a = 1 p 1 b = 1 p 1 χ 1 ( a ) e ( a + b p ) n = 1 p 3 d | n χ ¯ 1 ( d ) n χ mod p χ ( 1 ) = 1 χ ( a b c n ¯ ) + O ( χ mod p χ ( 1 ) = 1 | τ ( χ ) | | τ ( χ χ 1 ) | p 3 | A ( y , χ ¯ ) | y 2 d y ) = p 1 2 n = 1 p 3 d | n χ ¯ 1 ( d ) n a = 1 p 1 b = 1 p 1 a b c n mod p χ 1 ( a ) e ( a + b p ) p 1 2 n = 1 p 3 d | n χ ¯ 1 ( d ) n a = 1 p 1 b = 1 p 1 a b c n mod p χ 1 ( a ) e ( a + b p ) + O ( p χ mod p χ ( 1 ) = 1 p 3 | A ( y , χ ¯ ) | y 2 d y ) = p 1 2 n = 1 p 3 d | n χ ¯ 1 ( d ) n a = 1 p 1 χ 1 ( a ) e ( a + n c ¯ a ¯ p ) p 1 2 n = 1 p 3 d | n χ ¯ 1 ( d ) n a = 1 p 1 χ 1 ( a ) e ( a n c ¯ a ¯ p ) + O ( p ) = O ( p 3 2 n = 1 p 3 d ( n ) n ) = O ( p 3 2 ln 2 p ) .

This proves Lemma 4. □

3 Proof of the theorems

In this section, we will complete the proof of our theorem. First, if χ(1)=1, then from Theorems 12.11 and 12.20 of [8], we have

1 p b = 1 p bχ(b)= i π τ(χ)L(1, χ ¯ ).
(3.1)

From (3.1), Lemma 2 and Lemma 4, we have

a A χ ( a ) = 2 ϕ ( p ) χ mod p χ ( 1 ) = 1 ( 1 2 χ χ 1 ( 2 ) ) ( 1 2 χ ( 2 ) ) × ( 1 p a = 1 p a χ χ 1 ( a ) ) ( 1 p a = 1 p a χ ( a ) ) = 2 π 2 ( p 1 ) χ mod p χ ( 1 ) = 1 ( 1 2 χ χ 1 ( 2 ) ) ( 1 2 χ ( 2 ) ) τ ( χ ) τ ( χ χ 1 ) L ( 1 , χ χ 1 ¯ ) L ( 1 , χ ¯ ) = 2 π 2 ( p 1 ) χ mod p χ ( 1 ) = 1 τ ( χ ) τ ( χ χ 1 ) L ( 1 , χ χ 1 ¯ ) L ( 1 , χ ¯ ) 4 π 2 ( p 1 ) χ mod p χ ( 1 ) = 1 χ ( 2 ) τ ( χ ) τ ( χ χ 1 ) L ( 1 , χ χ 1 ¯ ) L ( 1 , χ ¯ ) 4 π 2 ( p 1 ) χ 1 ( 2 ) χ mod p χ ( 1 ) = 1 χ ( 2 ) τ ( χ ) τ ( χ χ 1 ) L ( 1 , χ χ 1 ¯ ) L ( 1 , χ ¯ ) + 8 π 2 ( p 1 ) χ 1 ( 2 ) χ mod p χ ( 1 ) = 1 χ ( 4 ) τ ( χ ) τ ( χ χ 1 ) L ( 1 , χ χ 1 ¯ ) L ( 1 , χ ¯ ) = O ( 1 p p 3 2 ln 2 p ) = O ( p 1 2 ln 2 p ) .

This completes the proof of our theorem.

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Acknowledgements

The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N.S.F. (11071194, 11001218) of P.R. China and G.I.C.F. (YZZ12062) of NWU.

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Correspondence to Wenpeng Zhang.

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Authors’ contributions

DH studied the estimate problem of the mean value a A χ(a), and gave a sharp upper bound estimate for it. WZ participated in the research and summary of the study.

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Han, D., Zhang, W. Upper bound estimate of character sums over Lehmer’s numbers. J Inequal Appl 2013, 392 (2013). https://doi.org/10.1186/1029-242X-2013-392

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Keywords

  • Lehmer’s numbers
  • character sums
  • Kloosterman sums
  • upper bound estimate