An identity involving the mean value of two term character sums

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R E S E A R C H

Open Access

An identity involving the mean value of

two-term character sums

Xiaoyan Guo

*

_{and Jingzhe Wang}

*_{Correspondence:}

[email protected] Department of Mathematics, Northwest University, Xi’an, Shaanxi, P.R. China

Abstract

The main purpose of this paper is, using the properties of Gauss sums and the estimate for character sums, to study the mean value problem of the two-term character sums and give an interesting identity for it.

MSC: 11M20

Keywords: two-term character sums; mean value; identity; analytic method; Gauss sums

1 Introduction

Letq≥ be an integer,χ be a non-principal charactermodq. For any integernwith (n,q) = , the two-term character sumsN(n,r,s,χ;q) are deﬁned as

N(n,r,s,χ;q) =

q

a=

χar+nas,

wherer>sare two ﬁxed positive integers. These sums play a very important role in the study of analytic number theory, so they caused many number theorists’ interest and favor. Some works related toN(n,r,s,χ;q) can be found in [–]. If fact, the sumsN(n,r,s,χ;q) is a special case of the general character sums of the polynomials

N+M

a=N+

χf(a), ()

whereMandNare any positive integers, andf(x) is a polynomial. Ifq=pis an odd prime, then Weil (see []) obtained the following important conclusion: Letχ be a qth-order character modp. Iff(x) is not a perfectqth powermodp, then we have the estimate

N+M

x=N+

χf(x)p_ln_p_, _()

where ‘’ constant depends only on the degree off(x).

Now we are concerned about whether there exists a computational formula for the mean value

∗

χmodq q

a=

χar+nas

k

, ()

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wheres>r≥ are two integers. In this paper, we shall use the analytic method and the properties of Gauss sums to study this problem, and give an exact computational formula for (). That is, we shall prove the following theorem.

Theorem  Let q> be an odd integer.Then,for any real number k and integer n with

(n,q) = ,we have the identity

∗

χmodq

q

a=

χa+na

k

=J(q)·qk· pq

 – 

p– + 

pk₍_p_{– )}

,

where ∗_χ_mod_qdenotes the summation over all primitive charactersmodq,pα_{q denotes}

that pα_|_{q and p}α+_q_,_{and J}₍_q_{) =}

d|qμ(d)φ( q

d)denotes the number of all primitive char-actersmodq.

Theorem  Let q> be an odd integer.Then,for any real number k and integer n with

(n,q) = ,we have the identity

∗

χmodq

q

a=

χa+na

k

=J(q)·qk· pq,p–

 – 

p– + 

pk₍_p_{– )}

×

pq,|p–

 – 

p– + 

pk₍_p_{– )}

.

From these two theorems we may immediately deduce the following corollaries.

Corollary  Let q be an odd square-full number(that is,for any prime p, if p|q,then

p_|_q_). _Then_, _{for any real number k and integer n with} ₍_n_,_q_{) = ,} _{we have the}

iden-tity

∗

χmodq

q

a=

χa+na

k

= ∗

χmodq q

a=

χa+na

k

=J(q)·qk.

Corollary  Let q be an odd square-free number(that is,for any prime p,p_†_q_)._Then_,

for any real number k and integer n with(n,q) = ,we have the identity

∗

χmodq

q

a=

χa+na

k

=qk· p|q

p–  + 

pk

.

Corollary  Let q be an odd square-full number.Then,for any integer n with(n,q) = ,we

have the identity

∗

χmodq

q

a=

χa+na

–

= ∗

χmodq q

a=

χa+na

–

=J(q)

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For general charactersχmodq,whether there exists an identity for

χmodq

q

a=

χar+nas

k

is an interesting open problem,where r>s are two positive integers.

2 Several lemmas

In this section, we shall give several lemmas, which are necessary in the proof of our the-orems. Hereinafter, we shall use many properties of character sums and Gauss sums, all of which can be found in references [] and [], so they will not be repeated here. First we have the following lemmas.

Lemma  Let q> be an integer,J(q)denotes the number of all primitive characters modq.

Then we have the identity

J(q) =

d|a

μ(d)·φ

q d

,

whereμ(n)is the Möbius function,φ(n)is the Euler function.

Proof This is a well-known result, here we give a simple proof. It is clear that for any non-principal characterχmodq, there exists one and only oned|qand a primitive character χ∗ moddsuch thatχ=χ∗·χ, whereχdenotes the principal charactermodq. So, from these properties, we have

φ(q) =

χmodq

 =

d|q J(d).

From this identity and the Möbius inversion formula, we may immediately deduce

J(q) =

d|a

μ(d)·φ

q d

.

This proves Lemma .

Lemma  Let p be an odd prime,i≥be an integer and q=pi_._Then_,_{for any real number} k and integer n with(n,q) = ,we have the identities

∗

χmodq

q

a=

χa+na

k

= ∗

χmodq

qk=φ(q)·qk–

and

∗

χmodp p

a=

χa+na

k

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Proof For any primitive characterχmodq, from the properties of Gauss sums, we know that

q

a=

χa+na=  τ(χ)

q

a=

q

b= χ(b)e

b(a₊_na₎

q

= 

τ(χ)

q

a=

q

b= χ(ba)e

ba(a₊_na₎

q

= 

τ(χ)

q

b= χ(b)e

nb

q q

a= χ(a)e

ba

q

=τ(χ) τ(χ)

q

b= χ(b)e

nb

q

=χ(n)τ(χ)·τ(χ ₎

τ(χ) . ()

Note thatq=pi_and_i_≥_{, so if}_χ_{is a primitive character}_mod_q_{, then}_χ_{is also a primitive} character modq, so that|τ(χ)|=|τ(χ)|=|τ(χ)|=√q. From these identities, Lemma  and formula (), we may immediately deduce that

∗

χmodq

q

a=

χa+na

k

= ∗

χmodq

qk=φ(q)·qk–.

This proves the ﬁrst formula of Lemma .

Now we prove the second formula. Ifχ is the Legendre symbolmodp, then we have |τ(χ₎_|_{= . So, from Lemma  and the method of proving (), we have}

∗

χmodp p

a=

χa+na

k

=  + (p– )pk.

This completes the proof of Lemma .

Lemma  Let p> be an odd prime,α≥be an integer and q=pα_._Then_,_{for any real}

number k and n with(n,q) = ,we have the identities

∗

χmodq

q

a=

χa+na

k

=φ(q)·qk–

and

∗

χmodp p

a=

χa+na

k

=

 + (p– )·pk _{if p}_≡__mod_;

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Proof From the properties of primitive characters modq and the method of proving Lemma , we have

q

a=

χa+na=  τ(χ) q a= q b= χ(b)e

b(a₊_na₎

q =  τ(χ) q b= χ(b)e

nb q q a= χ(a)e

ba

q

=χ(n)τ(χ

₎_·_τ_(χ₎

τ(χ) . ()

Sinceχis a primitive charactermodqand (,q) = , soχ_and_χ_{are also two primitive} characters modq. So, from Lemma , () and the properties of Gauss sums, we can deduce the ﬁrst identity of Lemma .

Note that if (,p– ) = ,χis the Legendre symbolmodp, then for any non-principal character χmodp, we have |τ(χ)|=√p and|τ(χ

)|= . So, from Lemma  and the method of proving (), we have

∗

χmodp p a=

χa+na

k

=  + (p– )pk. ()

If |p– , then there exist two -order charactersmodp, so from the method of proving (), we have the identity

∗

χmodp p a=

χa+na

k

=  + (p– )pk. ()

Combining () and (), we can deduce the second identity of Lemma .

Lemma  Let q> and q> be two integers with(q,q) = ,χmodqandχmodq.

Then,for any integer n with(n,qq) = ,we have

qq

a= χχ

a+na

= q a= χ a+na

· q b= χ b+nb

.

Proof From the properties of complete residue systemqq, we have

qq

a= χχ

a+na

= q a= q b=

χχ(aq+bq)+n(aq+bq)

= q a= χ

(aq+bq)+n(aq+bq)

q

b= χ

(aq+bq)+n(aq+bq)

= q a= χ

(aq)+n(aq)

q

b= χ

(bq)+n(bq)

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= q a= χ a+na

q

b= χ

b+nb

= q a= χ a+na

· q b= χ b+nb

.

This proves Lemma .

3 Proof of the theorems

In this section, we shall complete the proof of our theorems. First we prove Theorem . For any odd numberq> , it is clear that there exist two integersMandNsuch thatq=M·N, whereMis a square-free number, andNis a square-full number. Now, for any primitive character χmodq, there exist two primitive charactersχmodMandχmodN such thatχ=χχ. Note thatJ(N) = φ

₍_N₎

N , so from these properties, Lemma , Lemma  and

Lemma , we have

∗

χmodq q a=

χa+na k = ∗

χmodM M a=

χa+na k · ∗

χmodN N a=

χa+na

k

=

p|M

 + (p– )·pk pα_N

φpα·pα(k–)

=J(q)·qk· pq

 – 

p– + 

pk₍_p_{– )}

,

wherepα_q_{denotes that}_pα_|_q_and_pα+_q_{. This proves Theorem .}

Now we prove Theorem . From Lemma , Lemma  and the method of proving The-orem , we have

∗

χmodq q a=

χa+na k = ∗

χmodM M a=

χa+na k · ∗

χmodN N a=

χa+na

k

=

p|M

 + (p– )·pk pα_N

φpα_·_pα(k–)

=J(q)·qk· pq,p–

 – 

p– + 

pk₍_p_{– )}

×

pq,|p–

 – 

p– + 

pk₍_p_{– )}

.

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Competing interests

The authors did not provide this information.

Authors’ contributions

The authors did not provide this information.

Acknowledgements

The authors would like to thank the referee for their very helpful and detailed comments, which have signiﬁcantly improved the presentation of this paper. This work is supported by the National Natural Science Foundation of P.R. China (No. 11071194).

Received: 6 July 2013 Accepted: 17 October 2013 Published:12 Nov 2013

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10.1186/1029-242X-2013-533