J
OURNAL DE
T
HÉORIE DES
N
OMBRES DE
B
ORDEAUX
W
ENPENG
Z
HANG
On the mean values of Dedekind sums
Journal de Théorie des Nombres de Bordeaux, tome
8, n
o2 (1996),
p. 429-442
<http://www.numdam.org/item?id=JTNB_1996__8_2_429_0>
© Université Bordeaux 1, 1996, tous droits réservés.
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
On the mean values of Dedekind Sums
par WENPENG ZHANG
RÉSUMÉ. On étudie dans ce papier le comportement asymtotique de valeurs moyennes de sommes de Dedekind. On donne en particulier une formule
asymptotique améliorant un résultat antérieur.
ABSTRACT. In this paper we study the asymptotic behavior of the mean
value of Dedekind sums, and give a sharper asymptotic formula.
1. Introduction
For a
positive
integer k
and anarbitrary integer h,
the Dedekind sumS(h, k)
is definedby
where
The various
properties
ofS(h, k)
wereinvestigated by
many authors.Per-haps
the most famousproperty
of Dedekind sums is thereciprocity
formula(see
(3~, [5], [6]):
for all
(h, k)
= >0, k
> 0. A three term version of(1)
was discoveredby
Rademacher[7].
Walum[8]
has shown that forprime
p >
3,
Mots-cles : Dedekind sums; Dirichlet L-functions; Mean value; Asymptotic formula. Manuscrit regu le 9 mai 1996
and
Recently,
J.B.Conrey
et al.[4]
studied the mean value distribution ofS(h, k),
andproved
thatwhere
~~
denotes the summation over all h such that(k, h)
=1,
andh
In the
spirit
of[4]
and~8],
we use the estimates for character sum and theproperties
of Dirichlet’s L-functions togive
asharper
asymptotic
formulafor
(4),
where m = 1 and k =pl,
apower of a
prime.
Thatis,
we shall prove thefollowing
two theorems.THEOREM 1. Let p be a
prime
and n be apositive integer.
Thenfor
k =p’~,
we have the
asymptotic
formula
where the constant
implied by
the0-symbol
does notdepend
on anyparam-eter,
andexp(y)
= ey .THEOREM 2. Let n be an
1,
p be aprime,
and k =p’~.
Then weCOROLLARY. Let p be a
prime.
Then we have theasyrnptotic
formulas
2. Some Lemmas
To prove the
theorems,
we need thefollowing
lemmas.LEMMA 1.
Let q
be aninteger > 3,
andlet X
be any Dirichlet charactermodulo q with
x(-1) _ -1.
Then we havewhere
L(s, X)
denotes DirichletL-function
corresponding
to X modulo q.Proof.
Let N > 3 be aninteger.
Then we haveNote that since
x( -1) = -1,
we also haveTaking
N --~ oo in theabove,
andnoting
thatand
the lemma follows.
LEMMA 2. Let k be an
integer >
3 and(h, k)
= 1. Thenwhere
0(k)
is the Eulerfunction.
k
Proof.
Note that7r:
= 0 ifX(-1)
= 1. For(c, k)
=1,
fromm
Lemma 1 and the
orthogonality
of characters modulo k we have theidentity
Hence
Thus
where
e(y)
=e21riy, i2
= -1,
(h, k)
= 1.Note that from the definitions of
S(h, k),
and from(9)
and(10)
weget
where we have used the identities
This
completes
theproof
of Lemma 2.Proof.
For any parameterN > q, applying
Abel’sidentity
we havewhere
d(n)
is the divisor function. We haveA(y, X) =
L
x(n)d(n).
Note that the identitiesThus from
(12)
weget
Note that for
(ab, q)
=1,
from theorthogonality
relation for character sumsmodulo q
we havewhere we have used the estimate
d(n)
« exp and((s)
is theRiemann zeta-function.
This proves lemma 3.
3. Proof of the theorems
In this
section,
we shallcomplete
theproof
of the theorems. First we prove theorem 1. For k =p’,
applying
Lemma 2 and Lemma 3 we havethat
Using
the method ofproving
Lemma 3 weget
~ 17~
Note that the estimate
holds.
Now, using
theCauchy inequality
and Lemma 3 weget
From
(16), (17)
and(18)
we obtainTHEOREM A. Let q be an 3. Then we have
where the summation is over all odd Dirichlet characters modulo q.
THEOREM B. Let q be a
square-full
q >
4 and aprime
p~q
irriplies
p2lq).
Then we havewhere the summation is over all odd
primitive
Dirichlet characters moduloq.
Using
Lemma 2 we cangive
asimple
proof
of Theorem A and Theorem B.In
fact,
note that for k >2,
from the definition ofS(h, k)
we haveFrom this
identity,
Lemma 2 with h = 1 and the Mbbius inversion formulawe
get
If q
be asquare-full number,
thenusing
the Mbbius inversion formulaand Theorem A we
get
This
completes
theproof
of Theorem B.ACKNOWLEDGEMENTS
The author expresses his
gratitude
to the referee for hishelpful
anddetailed comments.
REFERENCES
1. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New
York, 1976.
2. _____, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag,
New York, 1976.
3. L. Carlitz, The reciprocity theorem for Dedekind sums, Pacific J. Math. 3 (1953),
523-527.
4. J. B. Conrey, Eric Fransen, Robert Klein and Clayton Scott, Mean values of Dedekind sums, J. Number Theory 56
(1996),
214-226.5. L. J. Mordell, The reciprocity formula for Dedekind sums, Amer. J. Math. 73 (1951), 593-598.
6. H. Rademacher, On the transformation of log ~(03C4), J. Indian Math. Soc. 19 (1955), 25-30.
7. H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical
Mono-graphs, Math. Assoc. Amer., Washington D.C., 1972.
8. H. Walum, An exact formula for an average of L-series, Illinois J. Math. 26 (1982),
1-3.
9. Zhang Wenpeng, Lecture Notes in Contemporary Mathematics, Science Press,
10. Zhang Wenpeng, A note on a class mean square values of L-functions, Journal of
Northwest University 20
(1990),
9-12.11. D. E. Knuth, Notes on generalized Dedekind sums, Acta Arith. 33
(1977),
297-325.Wenpeng ZHANG
Department of Mathematics Northwest University Xi’an, Shaanxi P. R. China.