SUMS AND TWO-TERM EXPONENTIAL SUMS

SUMS AND TWO-TERM EXPONENTIAL SUMS

WANG TINGTING and ZHANG WENPENG

In this paper, we use the elementary and analytic methods to study the mean value properties of the Cochrane sums weighted by two-term exponential sums, and give two exact computational formulae for them.

AMS 2010 Subject Classification: 11L40, 11F20.

Key words: Cochrane sums, two-term exponential sums, hybrid mean value, com- putational formula.

1. INTRODUCTION

Let q be a natural number and h an integer prime to q. The Cochrane sums C(h, q) is defined by

C(h, q) =

q

X

a=1 0

a q

ah q

,

where

((x)) =

( x−[x]−¹₂ ifxis not an integer, 0 ifxis an integer, a is defined by aa ≡ 1 modq and

q

P

a=1

0 denotes the summation over all 1 ≤ a≤q such that (a, q) = 1.

About the arithmetical properties of the Cochrane sums and related sums, some authors have studied it, and obtained many interesting results, see for example [1], [3], [4] and [5].

In this paper, we consider the computational problem of the mean value (1)

q

X

m=1 0

q

X

n=1

0|C(m, n, k, h;q)|²·C(mn, q),

MATH. REPORTS14(64),4 (2012), 355–362

where the two-term exponential sums C(m, n, k, h;q) is defined as C(m, n, k, h;q) =

q

X

a=1

e

ma^k+na^h q

, e(y) =e^2πiy. Some results related to C(m, n, k, h;q) can be found in [6]–[7].

But for mean value (1), it seems that none has studied it yet, at least we have not seen any related results before. This sum is interesting, because it has close relations with the class numberh_p of the quadratic fieldQ(√

−p), so we can give a new expression forhp. In this paper, we use the elementary and analytic methods to study this problem, and give two exact computational formulae for (1). That is, we shall prove the following two conclusions:

Theorem 1. Let p be an odd prime with p≡1 mod 4, then we have the computational formulae

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p) =

=

( 0 if p≡1 mod 8,

−

3 p

·^p

3 2

π²

τ²(χ₄)L²(1, χ₄) +τ²(χ₄)L²(1, χ₄) if p≡5 mod 8, where χ₄ is an odd character modp such that χ²₄ is the Legendre’s symbol, L(1, χ4) denotes the Dirichlet L-function corresponding to χ4, and τ(χ) =

p−1

P

a=1

χ(a)e ^a_p

denotes the classical Gauss sums.

Theorem 2. Let p >3 be an odd prime withp≡3 mod 4, then we have

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p) =

( −p·h²_p if p≡7 mod 12, p·h²_p if p≡11 mod 12.

Corollary. For any prime p >3 with p≡3 mod 4, we have h²_p =

3 p

·1 p ·

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p), where ^∗_p

denotes the Legendre’s symbol.

For some special positive integerskandh, we can also give an identity for

p−1

X

m=1 p−1

X

n=1

|C(m, n, k, h;p)|²·C(mn, p).

But for general positive integer q, whether there exists a computational for- mula for (1) is an open problem, where k≥3 and h≥1 are two integers.

2. SEVERAL LEMMAS

In this section, we shall give several lemmas, which are necessary in the proof of our theorems. First we have the following:

Lemma 1. Let p >3 be a prime, then for any integer nwith (n, p) = 1, we have the identity

p

X

a=1

a²+n p

=−1, where ^∗_p

denotes the Legendre’s symbol.

Proof. This is a well known result, here we give a proof for the sake of completeness. Since ^∗_p

≡ χ2 is a primitive character modp, so from the properties of Gauss sums τ(χ) =

p−1

P

a=1

χ(a)e ^a_p

we know that

p

X

a=1

a²+n p

= 1

τ(χ₂)

p

X

a=1 p−1

X

b=1

b p

e

b(a²+n) p

(2)

= 1

τ(χ2)

p−1

X

b=1

b p

e

nb p

^p X

a=1

e ba²

p

.

From Theorem 7.5.4 of [2] we know that for any integer u with (u, p) = 1, we have

(3)

p

X

a=1

e ua²

p

= u

p

τ(χ₂) = u

p ^p−1

X

a=1

a p

e

a p

. Then from (2) and (3) we have

p

X

a=1

a²+n p

= 1

τ(χ₂)

p−1

X

b=1

b p

e

nb p

b p

τ(χ₂) =

p−1

X

b=1

e nb

p

=−1.

This proves Lemma 1.

Lemma 2. Let p be an odd prime with p≡5 mod 8, χ₄ denotes the odd character mod p such thatχ²₄ = ^∗_p

. Then we have the identity

p

X

a=1 p

X

b=1

χ4 (a³−b³)(a−b)

=− 3

p

·(p−1)·√

p·τ(χ4) τ(χ₄).

Proof. In fact χ4(n) = e _ind

g(n) 4

, where g is a primitive root mod p, andindg(n) denotes the index ofnin primitive rootgmod p(related contents can be found in [8]). Note that ²_p

= −1, from the properties of complete residue system mod p we have

p

X

a=1 p

X

b=1

χ4 (a³−b³)(a−b)

= (4)

=

p

X

a=1

χ4(a⁴) +

p

X

a=1 p−1

X

b=1

χ4 (a³b³−b³)(ab−b)

=

p−1

X

a=1

χ₄(a⁴) +

p−1

X

b=1

χ₄(b⁴)

! _p X

a=1

χ₄ (a³−1)(a−1)

!

= (p−1) 1 +

p

X

a=1

χ4 (a³−1)(a−1)

!

= (p−1) 1 +

p−1

X

a=0

χ₄ a²(a²+ 3a+ 3)

!

= (p−1) 1 +

p−1

X

a=1

χ₄(a⁴)·χ₄ 1 + 3a+ 3a²

!

= (p−1) 1 +

p−1

X

a=1

χ₄ 3a²+ 3a+ 1

!

= (p−1) 1 + 2

p ^p−1

X

a=1

χ₄ 12a²+ 12a+ 4

!

= (p−1) 1 + 2

p ^p−1

X

a=1

χ₄ 3(2a+ 1)²+ 1

!

= (p−1) 2

p ^p

X

a=1

χ4 3(2a+ 1)²+ 1

=−(p−1)

p

X

a=1

χ4(3a²+ 1).

From the properties of Gauss sums we also have

p

X

a=1

χ₄ 3a²+ 1

= 1

τ(χ₄)

p

X

a=1 p−1

X

b=1

χ₄(b)e

b(3a²+ 1) p

(5)

= 1

τ(χ₄)

p−1

X

b=1

χ4(b) e b

p ^p

X

a=1

e 3ba²

p

=

√p τ(χ4)

3 p

^p−1 X

b=1

χ₄(b) b

p

e b

p

=

√p τ(χ₄)

3 p

^p−1 X

b=1

χ₄(b)e b

p

= 3

p

·√

p·τ(χ₄) τ(χ₄), where we have used the fact that χ²₄(b) = ^b_p

. Combining (4) and (5) we have

p

X

a=1 p

X

b=1

χ₄ (a³−b³)(a−b)

=− 3

p

·(p−1)·√

p·τ(χ₄) τ(χ4). This proves Lemma 2.

Lemma 3. Let a, q are two integers with q ≥ 3 and (a, q) = 1. Then we have

C(a, q) = −1 π²φ(q)

X

χmodq χ(−1)=−1

χ(a)

∞

X

n=1

G(χ, n) n

!2

,

where χ runs through the Dirichlet characters mod q withχ(−1) =−1, and G(χ, n) =

q

X

a=1

χ(a) e an

q

denotes the Gauss sums corresponding to χ.

Proof. See [5, Lemma 1].

3. PROOF OF THE THEOREMS

In this section, we shall use the lemmas proved in section two to complete the proof of our theorems. First we prove Theorem 1. If p is an odd prime then all non-principal characters χmodp are primitive. Hence by Lemma 3 and the propertyG(χ, n) =χ(n)τ(χ) of the Gauss sums associated to primitive characters we have

(6) C(a, p) = −1

π²(p−1) X

χmodp χ(−1)=−1

χ(a)·τ²(χ)·L²(1, χ),

where τ(χ) =

p−1

P

b=1

χ(b)e ^b_p .

By this identity and the definition ofC(m, n, k, h;q) we have

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p) = −π⁻² p−1

X

χmodp χ(−1)=−1

τ²(χ)·L²(1, χ)·

(7)

·

p

X

a=1 p

X

b=1 p−1

X

m=1 p−1

X

n=1

χ(mn)e

m(a³−b³) +n(a−b) p

= −π⁻² p−1

X

χmodp χ(−1)=−1

L²(1, χ)τ²(χ)τ²(χ)

p

X

a=1 p

X

b=1

χ a³−b³

χ(a−b)

= −p²·π⁻² p−1

X

χmodp χ(−1)=−1

L²(1, χ)

p

X

a=1 p

X

b=1

χ a³−b³

χ(a−b) and

p

X

a=1 p

X

b=1

χ a³−b³

χ(a−b) (8)

=

p−1

X

b=1

χ⁴(b)

! 1 +

p−1

X

a=1

χ a²+a+ 1

χ²(a−1)

!

=

p−1

X

b=1

χ⁴(b)

! _p X

a=1

χ a²+a+ 1

χ²(a−1)

! ,

where we have used the identity τ(χ)τ(χ) =τ(χ)τ(χ)χ(−1) =−pin (7).

If p = 8k+ 1, then for any character χmodp with χ(−1) = −1, χ⁴ is not a principal character mod p, so we have the identity

p−1

X

b=1

χ⁴(b) = 0.

(9)

If p= 8k+ 5, then there exist two and only two characters χ₄ and χ₄ modp with χ4(−1) =χ₄(−1) =−1 such thatχ⁴₄ is a principal character mod p. If χ=χ4 orχ₄, then

p−1

X

b=1

χ⁴(b) =p−1.

(10)

For any other character χmodpwith χ(−1) =−1, we have

p−1

X

b=1

χ⁴(b) = 0.

(11)

Ifp >3 andp≡3 mod 4, then for any character χmodp withχ(−1) =

−1,χ⁴ is also not a principal character modp, exceptχ= ^∗_p

, the Legendre’s symbol. In this case, from the properties of the complete residue system mod p and Lemma 1 we have the identities

(12)

p−1

X

b=1

b p

4

=p−1 and

p

X

a=1

a²+a+ 1 p

a−1 p

2

=

p

X

a=1

a²+a+ 1 p

− 3

p (13)

=

p

X

a=1

4a²+ 4a+ 4 p

− 3

p

=

p

X

a=1

(2a+ 1)²+ 3 p

− 3

p

=

p

X

a=1

a²+ 3 p

− 3

p

=−1− 3

p

. Now if p= 8k+ 1, then from (7), (8) and (9) we have

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p) = 0.

(14)

Ifp= 8k+ 5, then from (7), (10), (11) and Lemma 2 we have

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p) (15)

=− 3

p

·p³² π²

τ²(χ4)L²(1, χ₄) +τ²(χ₄)L²(1, χ4) . Now, our Theorem 1 follows from (14) and (15).

Ifp >3 andp≡3 mod 4, then note thatL(1, χ₂) =πh_p/√

p , from (7), (8), (12) and (13) we have the identity

p−1

X

m=1 p−1

X

n=1

|C(m, n,3,1;p)|²·C(mn, p)

= −p²

π²(p−1)·L²(1, χ2)·(p−1)·

1−1− 3

p

= 3

p

p·h²_p

=

( −p·h²_p ifp= 12k+ 7, p·h²_p ifp= 12k+ 11.

This completes the proof of Theorem 2.

Acknowledgements.This work is supported by the N.S.F. (11071194) of P.R. China.

REFERENCES

[1] J.B. Conrey, Eric Fransen, Robert Klein and Clayton Scott, Mean values of Dedekind sums. J. Number Theory56(1996), 214–226.

[2] L.K. Hua,Introduction to Number Theory. Science Press, Beijing, 1979.

[3] Z. Wenpeng, On the mean values of Dedekind Sums. J. Th´eor. Nombres Bordeaux 8 (1996), 429–442.

[4] Z. Wenpeng,A sum analogous to Dedekind sums and its hybrid mean value formula. Acta Arith.107(1003), 1–8.

[5] Z. Wenpeng, On a Cochrane sum and its hybrid mean value formula. J. Math. Anal.

Appl.267(2002), 89–96.

[6] T. Cochrane and Z. Zheng, Upper bounds on a two-term exponential sum. Sci. China (Series A)44(2001), 1003–1015.

[7] T. Cochrane and C. Pinner,A further refinement of Mordell’s bound on exponential sums.

Acta Arith.116(2005), 35–41.

[8] Tom M. Apostol, Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976.

Received 7 March 2011 Northwest University

Department of Mathematics Xi’an Shaanxi, P.R. China tingtingwang126@126.com

wpzhang@nwu.edu.cn