Journal of Mathematical Analysis and Applications 267, 89–96 (2002)doi:10.1006/jmaa.2001.7752, available online at http://www.idealibrary.com on

On a Cochrane Sum and Its HybridMean Value Formula1

Zhang Wenpeng

Research Center for Basic Science, Xi’an Jiaotong University,Xi’an, Shaanxi, People’s Republic of China

Submitted by Bruce C. Berndt

Received February 21, 2001; published online January 24, 2002

The main purpose of this paper is to use a mean value theorem of DirichletL-functions to study the asymptotic property of a hybrid mean value of a Cochranesum and to give an interesting mean value formula. 2002 Elsevier Science (USA)

Key Words: A sum analogous to Dedekind sum; mean value; asymptotic formula.

1. INTRODUCTION

For a positive integer k and an arbitrary integer h, the classical Dedekindsum S�h� k� is defined by

S�h� k� =k∑

a=1

((a

k

))((ah

k

))�

where

��x�� ={x− �x� − 1

2 if x is not an integer;0 if x is an integer.

The various properties of S�h� k� were investigated by many authors. Forexample, Carlitz [3] obtained a reciprocity theorem of S�h� k�. Conrey et al.[4] studied the mean value distribution of S�h� k� and proved the importantasymptotic formula

k∑h=1

′�S�h� k��2m = fm�k�(k

12

)2m

+O((k

95 + k2m−1+ 1

m+1

)log3 k

)�(1)

1 This work is supported by the Natural Science Foundation and Province Natural ScienceFoundation of the People’s Republic of China.

89

0022-247X/02 $35.00 2002 Elsevier Science (USA)

All rights reserved.

90 zhang wenpeng

where∑

h′ denotes the summation over all h such that �k� h� = 1, and

∞∑n=1

fm�n�ns

= 2ζ2�2m�ζ�4m� · ζ�s + 4m− 1�

ζ2�s + 2m� ζ�s��

The author [5] improved the error term of (1) for m = 1. In October2000, during his visit to Xi’an, Professor Todd Cochrane introduced a sumanalogous to the Dedekind sum,

C�h� k� =k∑

a=1

′((

a

k

))((ah

k

))�

where a is defined by the equation aa ≡ 1 modk. Then he suggested that westudy the arithmetical properties and mean value distribution properties ofC�h� k�. We have not made any progress on these problems so far. But wefound that there are some relationships between C�h� k� and Kloostermansums,

K�m�n�k� =k∑

b=1

′e(mb+ nb

k

)�

where e�y� = e2πiy . In fact, for some special integers k, we can use theestimates for character sums and the mean value theorem of Dirichlet L-functions to get an interesting hybrid mean value formula involving C�h� k�and K�h� 1�k�. We shall prove the following main result:

Theorem. Let k be a square-full number (i.e., p�k if and only if p2�k).Then we have the asymptotic formula

k∑h=1

′K�h� 1�k�C�h� k� = −1π2 kφ�k� +O

(k exp

(3 lnkln lnk

))�

where exp�y� = ey , and φ�k� is the Euler function.

This theorem also holds if k is a prime. But for general k, it isan unsolved problem whether there exists an asymptotic formula for∑′k

h=1 K�h� 1�k�C�h� k�. We conjecture that the asymptotic formula

k∑h=1

′K�h� 1�k�C�h� k� ∼ −1π2 kφ�k�� as k → ∞�

holds for all integers k > 2.

a cochrane sum and its hybrid mean value formula 91

2. SOME LEMMAS

To prove the Theorem, we need the following lemmas.

Lemma 1. Let integer k ≥ 3 and �a� k� = 1. Then we have

C�a� k� = −1π2φ�k�

∑χmodk

χ�−1�=−1

�χ�a�( ∞∑n=1

G�χ� n�n

)2

�

where χ denotes a Dirichlet character modulo k with χ�−1� = −1, andG�χ� n� =∑k

b=1 χ�b�e� bnk � denotes the Gauss sum corresponding to χ.Proof. From the orthogonality relation for character sums modulo k we

have

C�a� k� =k∑

b=1

′((

b

k

))((ab

k

))(2)

= 1φ�k�

∑χmodk

{k∑

b=1

χ�b�((

b

k

))}{ k∑c=1

χ�c�((

ca

k

))}�

Note that

��x�� = − 1π

∞∑n=1

sin�2πnx�n

� sin x = 12i�exi − e−xi��

and for any integer a with �a� k� = 1, G�χ� an� = �χ�a�G�χ� n� and

k∑c=1

χ�c�((

ca

k

))= 0� if χ�−1� = 1�

From these identities and (2) we have

C�a� k� = 1π2φ�k�

∑χmodk

χ�−1�=−1

{ ∞∑n=1

∑kb=1 χ�b� sin�2πnb/k�

n

}

×{ ∞∑n=1

∑kc=1 χ�c� sin�2πnca/k�

n

}

= −14π2φ�k�

∑χmodk

χ�−1�=−1

{ ∞∑n=1

G�χ� n� −G�χ�−n�n

}

×{ ∞∑n=1

G�χ� an� −G�χ�−an�n

}

92 zhang wenpeng

= −1π2φ�k�

∑χmodk

χ�−1�=−1

{ ∞∑n=1

G�χ� n�n

}{ ∞∑n=1

G�χ� an�n

}

= −1π2φ�k�

∑χmodk

χ�−1�=−1

�χ�a�( ∞∑n=1

G�χ� n�n

)2

�

This proves Lemma 1.

Lemma 2. Let q be a square-full number. Then for any nonprimitive char-acter χ modulo q, we have the identity

τ�χ� = G�χ� 1� =q∑

a=1

χ�a�e(a

q

)= 0�(3)

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 bea Dirichlet character modulo q. Then we have the identities

∑χmod q

∗χ�r� = ∑d��q� r−1�

µ

(q

d

)φ�d�

and

J�q� =∑d�q

µ�d�φ(q

d

)�

where∑

χmod q∗ denotes the summation over all primitive characters modulo

q, µ�n� is the Mobius function, and J�q� denotes the number of primitivecharacters modulo q.

Proof. From the properties of characters we know that for any characterχ modulo q, there exists one and only one d�q and primitive character χ∗

d

modulo d such that χ = χ∗dχ

0q, where χ0

q denotes the principal charactermodulo q. So we have∑

χmod q

χ�r� =∑d�q

∑χmod d

∗χ�r�χ0q�r� =

∑d�q

∑χmod d

∗χ�r��

Combining this formula and Mobius inversion, and noting the identity

∑χmod q

χ�r� ={φ�q�� if r ≡ 1�mod q�;0� otherwise,

we have ∑χmod q

∗χ�r� =∑d�q

µ�d� ∑χmod q/d

χ�r� = ∑d��q� r−1�

µ

(q

d

)φ�d��

a cochrane sum and its hybrid mean value formula 93

Taking r = 1, we immediately get

J�q� =∑d�q

µ�d�φ(q

d

)�

This proves Lemma 3.

Lemma 4. Let k be any integer with k > 2. Then we have the asymptoticformula

∑χmodk

χ�−1�=−1

∗L2�1� χ� = 12J�k� +O

(exp

(3 lnkln lnk

))�

Proof. Let τ�n� be the divisor function. Then for any parameter N ≥ kand nonprincipal character χ modulo k, applying Abel’s identity, we have

L2�1� χ� =∞∑n=1

χ�n�τ�n�n

= ∑1≤n≤N

χ�n�τ�n�n

+∫ ∞

N

A�y� χ�y2 dy�(4)

where A�y� χ� =∑N<n≤y χ�n�τ�n�. Note the partition identities

A�y� χ� = 2∑n≤√

y

χ�n� ∑m≤y/n

χ�m� − 2∑

n≤√N

χ�n� ∑m≤N/n

χ�m�

−( ∑n≤√

y

χ�n�)2

+( ∑n≤√

N

χ�n�)2

�

Applying the Polya–Vinogradov inequality,∣∣∣∣b∑

n=aχ�n�

∣∣∣∣� √k lnk�

we have ∑χmodk

χ�−1�=−1

∗�A�y� χ�� �√yk · k · lnk�(5)

Thus from (5) we get

∑χmodk

χ�−1�=−1

∗∫ ∞

N

A�y� χ�y2 dy ≤

∫ ∞

N

1y2

∑

χmodkχ�−1�=−1

∗�A�y� χ��

dy(6)

�∫ ∞

N

1y2

(√y√k3 lnk

)dy � k

32 lnk√N

�

94 zhang wenpeng

Note that for �a� k� = 1, from Lemma 3 we have

∑χmodk

χ�−1�=−1

∗χ�a� = 12

∑χmodk

∗�1 − χ�−1��χ�a�

= 12

∑χmodk

∗χ�a� − 12

∑χmodk

∗χ�−a�

= 12

∑u��k� a−1�

µ

(k

u

)φ�u� − 1

2∑

u��k� a+1�µ

(k

u

)φ�u��

So that

∑χmodk

χ�−1�=−1

∗ ∑1≤n≤N

χ�n�τ�n�n

= 12∑

1≤n≤N�n�k�=1

∑u��k�n−1�

µ

(k

u

)φ�u�τ�n�

n(7)

− 12∑

1≤n≤N�n�k�=1

∑u��k�n+1�

µ

(k

u

)φ�u�τ�n�

n

= 12J�k�+O

(∑u�k

φ�u� ∑1≤'≤N/u

τ�'u+1�'u+1

)

+O

(∑u�k

φ�u� ∑2≤'≤N/u

τ�'u−1�'u−1

)

= 12J�k�+O

(∑u�k

φ�u�u

× ∑1≤'≤N/u

1'

exp( �1+ε�ln2lnN

lnlnN

))

= 12J�k�+O

(exp

(lnN

lnlnN

))�

where we have used the estimate τ�n� � exp� �1+ε� ln 2 ln nln ln n �. Taking N = k3,

and combining (4), (6), and (7), we obtain the asymptotic formula

∑χmodk

χ�−1�=−1

∗L2�1� χ� = 12J�k� +O

(exp

(3 lnkln lnk

))�

This proves Lemma 4.

a cochrane sum and its hybrid mean value formula 95

3. PROOF OF THE THEOREM

In this section we complete the proof of the Theorem. Let k be a square-full number. Then applying Lemma 1, we have

k∑h=1

′K�h� 1�k�C�h� k�(8)

= −1π2φ�k�

∑χmodk

χ�−1�=−1

′(

k∑h=1

�χ�h�K�h� 1�k�)( ∞∑

n=1

G�χ� n�n

)2

�

For any primitive character χ modulo k, from the properties of Gauss sumswe have

G�χ� n� = �χ�n�τ�χ�� τ�χ�τ��χ� = −k� if χ�−1� = −1�

andk∑

h=1

′�χ�h�K�h� 1�k� =k∑

b=1

′k∑

h=1

�χ�h�e(hb+ b

k

)(9)

=k∑

b=1

′χ�b�e(b

k

) k∑h=1

′�χ�h�e(h

k

)= τ2��χ��

From (8), (9), Lemma 2, Lemma 3, and Lemma 4 we immediately obtaink∑

h=1

′K�h� 1�k�C�h� k� = −1π2φ�k�

∑χmodk

χ�−1�=−1

τ2��χ�( ∞∑n=1

G�χ� n�n

)2

(10)

= −1π2φ�k�

∑χmodk

χ�−1�=−1

∗τ2��χ�τ2�χ�( ∞∑n=1

�χ�n�n

)2

= −k2

π2φ�k�∑

χmodkχ�−1�=−1

∗L2�1��χ�

= −k2

π2φ�k�∑

χmodkχ�−1�=−1

∗L2�1� χ�

= −k2

π2φ�k�J�k� +O

(k exp

(3 lnkln lnk

))�

If k is a square-full number, then J�k� = φ2�k�/k. So from (10) we havek∑

h=1

′K�h� 1�k�C�h� k� = −1π2 kφ�k� +O

(k exp

(3 lnkln lnk

))�

This completes the proof of the Theorem.

96 zhang wenpeng

ACKNOWLEDGMENTS

The author expresses his 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. L. Carlitz, The reciprocity theorem of Dedkind Sums, Pacific J. Math. 3 (1953), 513–522.4. J. B. Conrey, Eric Fransen, Robert Klein, and Clayton Scott, Mean values of Dedekind

sums, J. Number Theory 56 (1996), 214–226.5. Wenpeng Zhang, A note on the mean square value of the Dedekind sums, Acta Math.

Hungar. 86 (2000), 275–289.6. Wenpeng Zhang, On the mean values of Dedekind Sums, J. Theor. Nombres Bordeaux 8

(1996), 429–442.7. L. K. Hua, “Introduction to Number Theory,” pp. 175–176, Science Press, Beijing, 1979.