on a generalized cochrane sum and its hybrid mean value formula

THE RAMANUJAN JOURNAL, 9, 373–380, 2005c© 2005 Springer Science + Business Media, Inc. Manufactured in the Netherlands.

On a Generalized Cochrane Sum and Its HybridMean Value Formula∗

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

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

Received July 26, 2002; Accepted August 21, 2003

Abstract. The main purpose of this paper is to use a mean value theorem of Dirichlet L-functions to study theasymptotic property of a hybrid mean value of a generalized Cochrane sum, and give an interesting mean valueformula.

Key words: generalized Cochrane sum, mean value formula, asymptotic property

2000 Mathematics Subject Classification: Primary—11F20; Secondary—11F99

1. Introduction

For any positive integer k and n and an arbitrary integer h, the general Dedekind sumS(h, n, k) is defined by

S(h, n, k) =k∑

a=1

Bn

(a

k

)Bn

(ah

k

),

where

Bn(x) ={

Bn(x − [x]), if x is not an integer;

0, if x is an integer.

Bn(x) is the Bernoulli polynomial, Bn(x) defined on the interval 0 < x ≤ 1 is the n-thBernoulli periodic function. In [3] and [4], the second author has given some mean valueproperties of S(h, n, k). In October 2000, during his visit to Xi’an, Professor Todd Cochrane

∗This work is supported by the N.S.F. and P.N.S.F of P.R. China.

374 HONGYAN AND WENPENG

introduced a sum analogous to the Dedekind sum,

C(h, k) =k∑′

a=1

((a

k

))((ah

k

)),

where a is defined by the equation aa ≡ 1 mod k. Then he suggested us to study thearithmetical properties and mean value distribution properties of C(h, k). In [5] and [6],the second author had studied the hybrid mean value problem of C(h, k) and Kloostermansums

K (m, n; q) =q∑′

b=1

e

(mb + nb

q

),

where e(y) = e2π iy , and obtained the following two conclusions:

k∑′

h=1

K (h, 1; k)C(h, k) = −1

2π2kφ(k) + O

(k exp

(3 ln k

ln ln k

))

holds for all square-full number k, where exp(y) = ey and φ(k) is the Euler function.For any positive integer k ≥ 3,

k∑′

h=1

K (h, 1; k)C(h, k) = −1

2π2kφ(k)

∏

p‖k

(1 − 1

p(p − 1)

)+ O

(k

32 +ε

),

where ε is any fixed positive number and∏

p‖k denotes the product over all prime divisorsp of k such that p | k and p2 �| k.

In this paper, we define a generalized Cochrane sum as follows:

C(h, q; m, n) =q∑′

a=1

Bm

(a

q

)Bn

(ah

q

).

Then we use the estimates for character sums and the mean value theorem of DirichletL-functions to study the hybrid mean value properties of C(h, q; m, n) and K (h, 1; q), andobtain an interesting hybrid mean value formula for it. That is, we shall prove the followingmain result:

Theorem. Let q be a square-full number (i.e. p | q if and only if p2 �| q). For any oddnumbers m and n, we have the asymptotic formula

q∑′

h=1

K (h, 1; q)C(h, q; m, n) = 2m!n!

(2π i)m+nqφ(q) + O

(q exp

(4 ln q

ln ln q

)),

where exp(y) = ey and φ(q) is the Euler function.

GENERALIZED COCHRANE SUM AND ITS HYBRID MEAN VALUE FORMULA 375

This theorem also holds if q is a prime. For general integer q ≥ 3, using the method of[6] we can also get an asymptotic formula for

q∑′

h=1

K (h, 1; q)C(h, q; m, n),

but the error term is O(q32 +ε).

2. Some lemmas

To prove the Theorem, we need the following Lemmas.

Lemma 1. Let integer q ≥ 3 and (h, q) = 1. Then for any odd numbers m and n, wehave

C(h, q; m, n) = 4m!n!

(2π i)m+nφ(q)

∑

χ mod qχ (−1) = −1

χ (h)

( ∞∑

r=1

G(χ, r )

rm

)( ∞∑

s=1

G(χ, s)

sn

),

where χ denotes a Dirichlet character modulo q with χ (−1) = −1, and G(χ, n) =∑qb=1 χ (b) e( bn

q ) denotes the Gauss sum corresponding to χ .

Proof: From the orthogonality relation for character sums modulo q we have

C(h, q; m, n) =q∑′

a=1

Bm

(a

q

)Bn

(ah

q

)

= 1

φ(q)

∑

χ mod q

{q∑

a=1

χ (a)Bm

(a

q

)}{q∑

b=1

χ (b)Bn

(bh

q

)}. (1)

Note that

Bn(x) = − n!

(2π i)n

+∞∑

r=−∞r �=0

e(xr )

rn,

and for any integer h with (h, q) = 1, G(χ, hn) = χ (h)G(χ, n). From these identities and(1) we have

C(h, q; m, n) = 1

φ(q)

∑

χ mod q

q∑

a=1

(− m!

(2π i)m

) +∞∑

r=−∞r �=0

χ (a)e(

raq

)

rm

×

q∑

b=1

(− n!

(2π i)n

) +∞∑

s=−∞s �=0

χ (b)e(

sbhq

)

sn


= m!n!

(2π i)m+n

1

φ(q)

∑

χ mod q

+∞∑

r=−∞r �=0

1

rm

q∑

a=1

χ (a)e

(ra

q

)

×

+∞∑

s=−∞s �=0

1

sn

q∑

b=1

χ (b)e

(sbh

q

)

= m!n!

(2π i)m+n

1

φ(q)

∑

χ mod q

+∞∑

r=−∞r �=0

G(χ, r )

rm

+∞∑

s=−∞s �=0

G(χ, sh)

sn

= m!n!

(2π i)m+n

1

φ(q)

∑

χ mod qχ (−1)=−1

{2

∞∑

r=1

G(χ, r )

rm

}{2

∞∑

s=1

G(χ, sh)

sn

}

= 4m!n!

(2π i)m+n

1

φ(q)

∑

χ mod qχ (−1)=−1

χ (h)

{ ∞∑

r=1

G(χ, r )

rm

}{ ∞∑

s=1

G(χ, s)

sn

}.

This proves Lemma 1.

Lemma 2. Let q be a square-full number. Then for any non-primitive character χ moduloq, we have the identity

τ (χ ) = G(χ, 1) =q∑

a=1

χ (a)e

(a

q

)= 0.

Proof: (See Theorem 7.2 of [7]).

Lemma 3. Let q and r be integers with q ≥ 2 and (r, q) = 1, and let χ be a Dirichletcharacter modulo q. Then we have the identities

∑

χ mod q

∗χ (r ) =

∑

d|(q,r−1)

µ

(q

d

)φ(d)

and

J (q) =∑

d|qµ(d)φ

(q

d

),

where∑∗

χ mod q denotes the summation over all primitive characters modulo q, µ(n) is theMobius function, and J (q) denotes the number of primitive characters modulo q.

Proof: (See Lemma 3 of [5]).


Lemma 4. Let q, t and s be any positive integers with q > 2. Then we have the asymptoticformula

∑

χ mod qχ (−1)=−1

∗L(s, χ )L(t, χ ) = 1

2J (q) + O

(exp

(3 ln q

ln ln q

)).

Proof: Without loss of generality we can assume t ≥ s. Let r (n) = ∑d|n ds−t . Then for

any parameter N ≥ q and non-principal character χ modulo q, applying Abel’s identity,we have

L(s, χ )L(t, χ ) =∞∑

n=1

χ (n)r (n)

ns=

∑

1≤n≤N

χ (n)r (n)

ns+ s

∫ ∞

N

A(y, χ )

ys+1dy, (2)

where A(y, χ ) = ∑N<n≤y χ (n)r (n). Note the partition identities

A(y, χ ) =∑

n≤√y

χ (n)ns−t∑

m≤y/n

χ (m) +∑

m≤√y

χ (m)∑

n≤y/m

χ (n)ns−t

−∑

n≤√N

χ (n)ns−t∑

m≤N/n

χ (m) −∑

m≤√N

χ (m)∑

n≤N/m

χ (n)ns−t

−(

∑

n≤√y

χ (n)ns−t

)(∑

n≤√y

χ (n)

)+

(∑

n≤√N

χ (n)ns−t

)(∑

n≤√N

χ (n)

).

Applying the Polya - Vinogradov inequality,∣∣∣∣∣

b∑

n=a

χ (n)

∣∣∣∣∣ √q ln q,

we may prove the estimate:∑

χ mod qχ (−1)=−1

∗ |A(y, χ )| √q · q · ln q · √

y. (3)

In fact, if s = t , then we have∑

χ mod qχ (−1)=−1

∗ |A(y, χ )| √qy · q · ln q.

If t > s, note that

∑

n≤y/m

χ (n)

nt−s= L(t − s, χ ) − (t − s)

∫ ∞

y/m

∑N<n≤y χ (n)

yt−s+1dy

ln q + √q

(y

m

)s−t

.


we also have

∑

χ mod qχ (−1)=−1

∗ |A(y, χ )| √q · q · ln q · √

y.

So (3) holds with t ≥ s. Thus from (3) we get

∑

χ mod qχ (−1)=−1

∗s∫ ∞

N

A(y, χ )

ys+1dy ≤

∫ ∞

N

s

ys+1

(∑

χ mod qχ (−1)=−1

∗ |A(y, χ )|)

dy

∫ ∞

N

1

ys+1(√

y√

q3 ln q) dy q32 ln q

(√

N )2s−1. (4)

Note that for (a, q) = 1, from Lemma 3 we have

∑

χ mod qχ (−1)=−1

∗χ (a) = 1

2

∑

χ mod q

∗(1 − χ (−1))χ (a)

= 1

2

∑

χ mod q

∗χ (a) − 1

2

∑

χ mod q

∗χ (−a)

= 1

2

∑

u|(q,a−1)

µ

(q

u

)φ(u) − 1

2

∑

u|(q,a+1)

µ

(q

u

)φ(u).

So that

∑

χ mod qχ (−1)=−1

∗ ∑

1≤n≤N

χ (n)r (n)

ns

= 1

2

∑

1≤n≤N(n,q)=1

∑

u|(q,n−1)

µ

(q

u

)φ(u)

r (n)

ns− 1

2

∑

1≤n≤N(n,q)=1

∑

u|(q,n+1)

µ

(q

u

)φ(u)

r (n)

ns

= 1

2J (q) + O

(∑

u|qφ(u)

∑

1≤�≤N/u

τ (�u + 1)

(�u + 1)s

)+ O

(∑

u|qφ(u)

∑

2≤�≤N/u

τ (�u − 1)

(�u − 1)s

)

= 1

2J (q) + O

(∑

u|q

φ(u)

us

∑

1≤�≤N/u

1

�sexp

((1 + ε) ln 2 ln N

ln ln N

))

= 1

2J (q) + O

(exp

(ln N

ln ln N

)), (5)


where τ (n) is the divisor function. We have used the estimate

r (n) ≤ τ (n) exp

((1 + ε) ln 2 ln n

ln ln n

).

Taking N = q3, and combining (2), (4) and (5), we obtain the asymptotic formula

∑

χ mod qχ (−1)=−1

∗L(s, χ )L(t, χ ) = 1

2J (q) + O

(exp

(3 ln q

ln ln q

)).

This proves Lemma 4.

3. Proof of the theorem

In this section we complete the proof of Theorem. Let q be a square-full number. Thenapplying Lemma 1 we have

q∑′

h=1

K (h, 1; q)C(h, q; m, n)

= 4m!n!

(2π i)m+nφ(q)

∑

χ mod qχ (−1)=−1

(q∑′

h=1

χ (h)K (h, 1; q)

)( ∞∑

r=1

G(χ, r )

rm

)( ∞∑

s=1

G(χ, s)

sn

). (6)

For any primitive character χ modulo q, from the properties of Gauss sums we have

G(χ, n) = χ (n)τ (χ ), τ (χ )τ (χ ) = −q, ifχ (−1) = −1

and

q∑′

h=1

χ (h)K (h, 1; q) =q∑′

b=1

q∑

h=1

χ (h)e

(hb + b

q

)

=q∑′

b=1

χ (b)e

(b

q

) q∑′

h=1

χ (h)e

(h

q

)= τ 2(χ ). (7)

From (6), (7), Lemmas 2 and 3 we immediately obtain

q∑′

h=1

K (h, 1; q)C(h, q; m, n)

= 4m!n!

(2π i)m+nφ(q)

∑

χ mod qχ (−1)=−1

τ 2(χ )

( ∞∑

r=1

G(χ, r )

rm

)( ∞∑

s=1

G(χ, s)

sn

)


= 4m!n!

(2π i)m+nφ(q)

∑

χ mod qχ (−1)=−1

∗τ 2(χ )τ 2(χ )

( ∞∑

r=1

χ (r )

rm

)( ∞∑

s=1

χ (s)

sn

)

= 4m!n!

(2π i)m+n

q2

φ(q)

∑

χ mod qχ (−1)=−1

∗L(m, χ )L(n, χ )

= 4m!n!

(2π i)m+n

q2

φ(q)

∑

χ mod qχ (−1)=−1

∗L(m, χ )L(n, χ ).

If q is a square-full number, then J (q) = φ2(q)q . So from (8) and Lemma 4 we have

q∑′

h=1

K (h, 1; q)C(h, q; m, n) = 2m!n!

(2π i)m+nqφ(q) + O

(q exp

(4 ln q

ln ln q

)).

This completes the proof of Theorem.

Acknowledgments

The author express their gratitude to the referee for very helpful and detailed comments.

References

1. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.2. Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York,

1976.3. Wenpeng Zhang, “On the general dedekind sums and one kind identities of dirichlet L-functions,” Acta Math-

ematica Sinica 44 (2001), 269–272.4. Min Xie and Wenpeng Zhang, “On the 2k-th mean value formula of general dedekind sums,” Acta Mathematica

Sinica 44 (2001), 85–94.5. Wenpeng Zhang, “On a cochrane sum and its hybrid mean value formula,” Journal of Mathematical Analysis

and Application 267 (2002), 89–96.6. Wenpeng Zhang, “On a cochrane sum and its hybrid mean value formula (II),” Journal of Mathematical Analysis

and Application 276 (2002), 446–457.7. L.K. Hua, Introduction to Number Theory, Science Press, Beijing, 1979, pp. 175–176.

on a generalized cochrane sum and its hybrid mean value formula

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