J
EFFREY
L. M
EYER
A reciprocity congruence for an analogue of the
Dedekind sum and quadratic reciprocity
Journal de Théorie des Nombres de Bordeaux, tome
12, n
o1 (2000),
p. 93-101
<http://www.numdam.org/item?id=JTNB_2000__12_1_93_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
A
Reciprocity Congruence
for
anAnalogue
of the
Dedekind
Sum and
quadratic
reciprocity
par JEFFREY L. MEYER
RÉSUMÉ. Une loi de
réciprocité est
établie pour des sommesappa-raissant dans les formules de transformations pour les
logarithmes
des fontions
theta,
sommesqui
sont lesanalogues
des sommes deDedekind dans la transformation du
logarithme
de la fonction eta. ABSTRACT In the transformation formulas for thelogarithms
of the classicaltheta-functions,
certain sums arise that areanalogous
to the Dedekind sums in the transformation of the
logarithm
ofthe eta-function. A new reciprocity law is established for one
of these
analogous
sums and thenapplied
to prove the law ofquadratic reciprocity.
1. INTRODUCTION
In Berndt
[1]
andGoldberg
[2]
the sumS4(d,c)
is defined for c > 0by
.,
The sum is one of several involved in the
multiplier systems
fortransfor-mations of the
logarithms
of the classical theta-functions. We define two of them here. Let q = and forIm(z)
> 0 defineWe prove a new
reciprocity
theorem for the sumS4(d, c).
As anapplication
of the
theorem,
we deduce the law ofquadratic
reciprocity.
the
principal
branch of thelogarithm
at all times. Berndt[1]
proved
thatGoldberg
[2]
gives
thefollowing
formula. If a is even andb,
c and d areodd, then
z q 4
With the
reciprocity
theorem for the Dedekind sums(d,
c)
andconnec-tions between
s(d, c)
and theLegendre-Jacobi
symbol
(~),
one can deducethe
quadratic reciprocity
law(See
Rademacher and Grosswald[3],pp.
34-35).
Berndt[1]
proved elegant reciprocity
theorems for several sums thatarise in the transformation formulas of the
logarithms
of the classical theta-functions. None of thesereciprocity
theorems allow for both of the argu-ments in the sum to be odd. For theapplication
of the sumsarising
inthe theta-function transformations to
quadratic
reciprocity, however,
weneed a new
reciprocity
theorem where c and d are both odd.And,
unlikethe Dedekind sum, the sum
S4(d, c)
does not possess areciprocity
theo-rem.
However,
areciprocity
relation modulo 8 for the sumS4(d, c)
doesexist and this is sufficient to deduce the
quadratic
reciprocity
theorem. For additionalproperties
ofS4(d, c)
see[7]
and[6].
To establish the connection between
S4(d, c)
and theLegendre-Jacobi
symbol,
we turn to Rademacher’s book[8]
(pp. 180-182)
for the needed results. We use the same doublesubscript
notation as Rademacher to state the theorem. ForIm(z)
> 0 and{0,1},
letThus
Note that we may allow
integers
other that 1 and 0 assubscripts
since it can be shown thatWe state one of Rademacher’s transformation formulas.
where
2. RECIPROCITY THEOREM FOR
S4 (d, C)
The next result is the foundation on which the new
reciprocity
theoremis based.
Theorem 2. Let c and d be
positive, coprime,
oddintegers.
ThenProof.
Choose a and b with b> d,
b even and ad - bc = 1 and setThen
Note that in each matrix the upper
right entry
is even and the determinantis 1. We use
(1)
with Vreplaced
with VW to find thatThen we
apply
(1)
with zreplaced
by
Wz to see thatand
finally
we use(1)
with Vreplaced by
W to deduce thatWe
replace
log
84(W z)
in(6)
with(7)
and then combine the result with(5)
to conclude thatWe have used the
following
lemma[5]
to conclude that there are no branchwhere k is
independent of z
andNext we
multiply
(8)
by
4/ (7ri),
rearrange, and use the fact thatS4(-c, d) _ -S4(c, d)
to conclude thatThe
symmetry
between c and d on the left-hand side of theequation
inTheorem 3 leads
immediately
to the next result.Corollary
1.If c
and d arecoprirrte,
odd, positive integers,
thenn n
The final
step
towards the desiredreciprocity
relation is thefollowing
congruence relation.
Lemma 2. Let d be an odd
prime
and c > d be anodd, positive integer
coprime
to d. ThenProof. Using
the definition ofS4(d, c)
and the fact thatwe see that
If m and n are
positive
andcoprime
integers,
recall that(see [4]
p.186,
forWe now
apply
(10)
to the first of the final two sums in(9)
andseparate
the terms in the second sum on theright according
to theparity
of j.
Thus(9)
becomesLet
where = x -
(~~
denotes the fractionalpart
of x. The final sum in(11)
can besimplified
since d c and[d 2/2] = (d2 -
1)/2
for d odd. Note thatWe conclude from
(11)
and(13)
thatSince d is odd and
thus d2 -
1 (mod 8),
we deduce the congruenceWe claim that N is even. Let
where
LPRI (m)
denotes the leastpositive
residue of m modulo thepositive
integer
l. Observe thatif and
only
if there exists apositive
integer k
such thator, in other
words,
if andonly
ifLPRd2(k(cd
+1))
>d2/2.
Thus N = n. The final claim in theproof
of the congruence is thefollowing:
The number n defined above is even.The
proof
of this claim is a carefulapplication
of the method used in thestandard
proof
of Gauss’s Lemma(see
[4],
p.133 ,
forexample).
Let r1,r2,... rn be the n residues of
k(cd
+1),
1 k(d2 -
1)/2,
falling
in the upper half of the leastpositive
residuesystem
(mod
d2)
and let sl, s2, ... , sl be those in the lower half. Then n + I =(d2 -
1)/2.
Nextwe consider
r~
=d2 -
ri for i =
1, 2, ... ,
n. Each of ther~
are distinct and we can say further that nor!
for any i andj.
Note that there are
[(d2 - 1)/(2d)~ _
(d - 1)/2
positive multiples
of dthat are less than
(d2 -
1)/2.
Now for k =md,
1 m(d -
1)/2,
From
(18),
we conclude thatThus all of the residues
produced
when 1~ is amultiple
of d are among theSj and are in fact the
positive
multiples
of d that are less thand2 /2.
We remove them from the listI s 17. 0 .
reindex this set andput
’I 1
So now we consider the n
rz’s,
and the l’sj’s,
with n-t- l’
=d(d -
1)/2
residuesaltogether. By
the distinctness of ther~,
we conclude that ther2
and
the Sj
are somerearrangement
of the numbers1, 2, ... ,
(d2 -1)/2
withthe
multiples
of d removed. Thusyielding
the congruenceNow we rewrite
(19)
withthe ri
andthe Sj
in theiroriginal
form,
with somepossible
rearrangement,
toget
Recall the exclusion of the
multiples
of d on each side. Because d isprime,
we may cancel the common factors from each side of the congruence(20)
and conclude that
But
by
the binomialtheorem,
From
(21)
and(22),
we deduce that n is even and theproof
of the claim iscomplete.
Andthus,
from(14),
we haveWe now assemble Theorem
3,
Corollary
1,
and Lemma 2 to reach ourreciprocity
result.Theorem 3. Let d be an odd
prime
and c > d be anodd, positive integer
coprime
to d. Then3. THE LAW OF
QUADRATIC
RECIPROCITYAs an
application
of Theorem4,
we offer a newproof
of the law ofquadratic reciprocity.
ad-6c 1. Let V
a
b
and HW-b
-a .
Note that a isnecessarily
ad-bc 1. Let V
e dJ
and Wde. Note
that a isnecessarily
odd. From
(3), (4)
andrecalling
that a, c, and d are odd and b is even, we see thatand
From Theorem
1,
(23)
and(24),
we haveand
We also
have,
from(1)
and(2),
and
Next,
wemultiply
(25)
and(26)
andpartially simplify
the result to deduce thatFrom
(29),
(30)
and Lemma1,
we deduce thatNext,
wesimplify
(31)
to find thatNote that since d is odd and
ad-bc = 1, we
have that a - d+dbc(mod 8).
From this fact and the
application
of Theorem 3 to(32),
we conclude that, , , -,
, , , ,
A
straight-forward
calculation shows that,_ .. - - -- - - - ,
Using
(34)
in(33),
we deduce thatas desired.
REFERENCES
[1] B.C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math. 303/304 (1978), 332-365.
[2] L.A. Goldberg, Transformations of theta-functions and analogues of dedekind sums. Ph.D.
thesis, Thesis, University of Illinois, Urbana, 1981.
[3] H. Rademacher E. Grosswald, Dedekind sums. Carus Math. Monogr., vol. 16, Mathematical Association of America, Washington, D.C., 1972.
[4] H. Montgomery I. Niven, H. Zuckerman, An introduction to the theory of numbers. 5th ed.,
John Wiley and Sons, New York, 1991.
[5] J. Lewittes, Analytic continuation of Eisenstein series. Trans. Amer. Math. Soc. 171 (1972), 469-490.
[6] J.L. Meyer, Analogues of dedekind sums. Ph.D. thesis, University of Illinois, Urbana, 1997. [7] _ Properties of certain integer-valued analogues of Dedekind sums. Acta Arithmetica
82 (1997), 229-242.
[8] H. Rademacher, Topics in analytic number theory. Springer-Verlag, New York, 1973.
Jeffrey L. MEYER
15 Carnegie Hall
Department of Mathematics
Syracuse University Syracuse, NY 13244, USA