Dedekind sums with predi table signs
by
Kurt Girstmair (Innsbru k)
1. Introdu tion and main results. Letb and n be integers, b6= 0,
with(b;n)=1. The (inhomogeneous)Dedekindsum isde nedby
s(n;b)= jbj
X
k=1
((k=b))((kn=b))
wherethesymbol((:::))has theusualmeaning( f.,e.g., [8℄). We note the
relations
(1) s(n; b)=s(n;b) and s(n+b;b)=s(n;b):
Hen eweobtainallDedekindsumsifbisrestri tedtonaturalnumbersand
n to the range 0 n < b. The general de nition, however, willbe useful
later.
In general, it is not easy to guess what the sign of s(n;b) may be.
Radema her ([7℄, Satz 3) showed s(n;b) > 0 for 0 < n <
p
b 1. In this
notewe givea onsiderablegeneralizationofRadema her's result. Roughly
speaking,weshallshowthatthereareagreatmanyintervalsI in[0;b[su h
thats(n;b)takesa predi table and xed sign forea h n2I.
To this end we x the natural numberb forthe time being. Let d <b
beanothernatural number. De ne
(2)
d
=
b(d 1)(d 2)
2(bd 1)
;
d
=
(b d) 2
bd 1
and
(3)
d
=
d +
q
d +
2
d
;
the square root being positive. To ea h fra tion =d, 2 Z, ( ;d) =1, we
atta h an intervalof length2
d
=dwith midpoint b =d, namely,
I( ;d)=fx2R :jx b =dj<
d
=dg:
1991 Mathemati sSubje tClassi ation: Primary11F20.
Both \half-intervals"
I( ;d) =fx2I( ;d):x<b =dg; I( ;d) +
=fx2I( ;d):x>b =dg
are nonempty. Moreover, ea h number n 2 Z, (n;b) = 1, lying in I( ;d)
belongsto one of these half-intervals. Otherwise n=b =d, but then djb
be ause of ( ;d) = 1, so b=d divides (n;b), whi h is 1. This is impossible
sin eb=d>1. Our rst mainresult is
Theorem 1. As above, let d <b be natural numbers and an integer
with ( ;d)=1. Let n be an integer in I( ;d), (n;b) =1. Then s(n;b) <0
if n2I( ;d) ,and s(n;b)>0 if n2I( ;d) +
.
If d = 1, then
d
= 0 and
d
= b 1, so
d
=d =
d
= p
b 1 and
I(0;1) +
= ℄ 0;
p
b 1 [. Therefore, the ase d = 1 of Theorem 1 ontains
Radema her's above-mentioned result. In view of (1) and the well-known
identity
(4) s( n;b)= s(n;b);
it is lear that Radema her's theorem is equivalent to this spe ial ase of
Theorem1.
We look at the intervals I( ;d) more losely. It suÆ es, of ourse, to
onsider onlythose parts of them that are ontained in[0;b[ . Apart from
thehalf-intervalsI(0;1) +
andI(1;1) =℄ b p
b 1;b[ ,thesepartsarejust
the omplete intervals I( ;d), 2d <b,with 1 <d, ( ;d)=1. It will
beshownbelowthat, ifb4,then
(5)
p
b=d 3
1<
d
=d<
p
b=d 3
( f. Lemma2,(20), (21)). Thismeans thatthelength 2
d
=dof an interval
I( ;d) is of order of magnitude p
b if d 3
is small relative to b. In this
ase we say thatI( ;d) is \large". Obviously,largeintervals ontain many
integers n. There is no reason, however, to rule out \small" intervals. It
follows from (5) that I( ;d), ( ;d) = 1, ontains at least one integer if
d < (3=4)b 1=3
; and it turns out that at least some of the intervals I( ;d)
ontain an integer as long as d <
p
b. Conversely, I( ;d)\Zis empty for
d p
b (see theremarkfollowingthe proofof Theorem1). In view ofthis,
itis naturalto studythesubset
(6) R (b)=I(0;1) +
[I(1;1) [ [
2d<
p
b [
1 <d
( ;d)=1 I( ;d)
of [0;b[ . The setR (b) willbe alled theregion of predi table sign. It would
bedesirable to knowthenumber
(7) S(b)=jR (b)\Zj
Theorem 2. If b islarge enough,then
1:8b 2=3
<S(b)<4:75b 2=3
:
A ording to Theorem 2 the number of integers in the region of pre-
di table sign is substantially larger than the size of large intervals I( ;d).
Both onstantsinTheorem2areratherpessimisti |thetrueorderofmag-
nitudeofS(b)seemstobe3:1b 2=3
. FurtherdetailsonthegrowthofS(b)
an befoundinSe tions 3 and4.
The diagramsbelow may give an idea of thebehaviourof the valuesof
s(n;b) inside and outside R (b). They display the ase b = 1009, a prime,
where S(b)=266. The small ir les represent pairs(n;12s(n;b)). Observe
that
j12s(n;b)j<b
holds for arbitrary numbers n, b with (n;b) = 1 ( f. (14)). In the rst
diagram the valuesn = 1 and n =b 1 have been omitted|just to save
spa e,sin ethese aretheonlyones withj12s(n;b)j losetob;foranyother
n, j12s(n;b)j<b=2. Thediagramssuggestthat R (b) ontainsallintegersn
forwhi hjs(n;b)jis\large"butnot only these; onversely,the omplement
[0;b[nR (b) seemsto onsist only of numbersnwithjs(n;b)jsmall. Indeed,
our omputationsshowthatj12s(n;b)jseldomex eeds p
bifnisnotinR (b),
whereasthere aremanynumbers ninR (b) withj12s(n;b)j<
p
b.
2. The proof of Theorem 1. Theorem 1 is based on a relation for
Dedekindsums(Lemma1) thatgeneralizestheusualthree-termrelationof
Radema her [6℄. Thislemma is a onsequen e of thetransformationlawof
thelogarithmofDedekind's-fun tion. Relationsofthismore generaltype
were given by Dieter [4℄ and frequently used by Bruggeman ( f., e.g., [3℄,
formula (3.1); [2℄, part 2.3). Nevertheless it seems that these relations are
not ommonly known ( f. the redis overy in [5℄). For the onvenien e of
the readerwe in lude a short proof,sin e it may be toilsome to adapt the
resultsof [4℄to the situation onsideredhere.
Let d, b be natural numbers and n, integers with (n;b) = ( ;d) = 1.
We write
(8) n b =d=q=d;
whereq isan integer. Supposeq 6=0 and put
"=sign(q) (2f1g):
Moreover, letj and k be integerssu hthat
(9) j+dk=1
and put
a
a
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a
a
b=1009: thevaluesof12s(n;b)forn2R (b),n6=1;b 1
Lemma 1. In the above situation,
12s(n;b)=12s( ;d)+"12s(r;q)+ b
2
+d 2
+q 2
bdq
3":
Proof. The transformation lawof thelogarithm of Dedekind's-fun -
tion says
(10) (AB)=(A)+(B) 3sign(A)sign(B)sign(AB)
( f. [8℄, pp. 49 ). Here A, B denote matri es inSL(2;Z) and and sign
arede ned inthefollowingway: If
A=
Æ
;
thensign(A)=sign( ) (2f0;1g), and
(A)=
=Æ if =0;
In oursituationwe put
A=
u v
b n
; B=
k
d j
;
whereu;v areintegerssu h that
(11) un vb=1:
Onapplying(10) one readilyobtains(observe (4))
12s(n;b)=12s( j;d)+"12s(r;q) (12)
+(u vd+bk jn)=q+(u+n)=b+(j )=d 3":
Be ause of (9), j 1 mod d, and a well-known identity( f. [8℄, p. 26)
says
s( j;d)=s( ;d):
Therefore,therightsideof(12)hasthedesiredshapeifonlythesumofthe
three fra tions equals (b 2
+d 2
+q 2
)=(bdq). But this follows from a short
al ulation whi h takes theidentities(9)and (11) into a ount.
Proof of Theorem 1. We onsider the ase n > b =d rst. Let q be
de ned by(8), soq >0and "=1. By thelemma, s(n;b)>0holdsif, and
onlyif,
(13) 12s( ;d)+12s(r;q)+(b 2
+d 2
+q 2
)=(bdq) 3>0:
Next we applytheestimate
(14) j12s(x;y)j(jyj 1)(jyj 2)=jyj;
whi hholdsforarbitrary oprimeintegersx;y,y6=0( f. [7℄). Thereby,the
left sideof (13) is>0 ifonly
(d 1)(d 2)=d (q 1)(q 2)=q+(b 2
+d 2
+q 2
)=(bdq) 3>0:
Thisis thesame assayingthatf(d;q)<0, wheref(d;q) isthepolynomial
de nedby
(15) f(d;q)=bq(d 1)(d 2)+bd(q 1)(q 2) (b 2
+d 2
+q 2
)+3bdq:
We onsiderf(d;q) asapolynomialinq onlyand note
f(d;q)=(bd 1)=q 2
2
d
q
d
( f. (2)). Hen e f(d;q) isnegative if,and onlyif, q liesbetweenthe zeros
d
q
d +
2
d
of f(d;q). Sin e q is positive,this meansnothing butq <
d
( f. (3)) and
+
In the ase n < b =d we have q < 0 and " = 1. One shows, in
the same way, that s(n;b) < 0 if f(d;jqj) < 0, whi h means jqj <
d and
n2I( ;d) .
Remark. We drawthereader'sattention tothefa tthatthede nition
(15)off(d;q)is symmetri ind andq. Thisallows rephrasingtheassertion
\n 2 I( ;d)" in another way. Indeed, let q be de ned by (8). Then \n 2
I( ;d)" is the same as saying jqj <
d
or f(d;jqj) < 0. This, however, is
equivalent to f(jqj;d) < 0 or d <
jqj
. Now the (still unproved) estimate
(5), appliedto
jqj
,gives
jqj
<
p
b=jqj ; son2I( ;d) an holdonlyif
d<
p
b=jqj :
In parti ular,I( ;d)\Zis empty ifd p
b|aswe said inSe tion1.
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a a aa
a a
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a a
a
a
a a
a a a a
a a a
a a
a a
b=1009: thevaluesof12s(n;b)forn62R (b),1n<b
3. The sums S
1
(b) and S
2
(b). Obviously,thesets I( ;d)\Zmustbe
thede nitions(6) and (7)show
S(b)= X
1d<
p
b S
b;d
with
(16) S
b;d
= X
0 <d
( ;d)=1
jI( ;d)\Zj:
The proofof Theorem2 isbased on theseparatetreatment of thesums
(17) S
1 (b)=
X
1d<b 1=3
S
b;d
; S
2 (b)=
X
b 1=3
d<
p
b S
b;d :
Infa t, someoftheestimates usedforthe rstsumdonotwork inthe ase
of these ond and onversely. We shallshow
Proposition 1.(a) For ea h suÆ iently large natural number b,
18
2
b 2=3
9
8 p
b<S
1 (b)<
27
2
b 2=3
:
(b) If b is large enough, then
S
2
(b)<2b 2=3
+b 0:51
:
Both parts(a) and(b) togetheryield
1:8b 2=3
<S
1
(b)S(b)=S
1 (b)+S
2
(b)<2:74b 2=3
+2:01b 2=3
=4:75b 2=3
and hen e Theorem2. Next we list some estimates neededfor theproof of
Proposition1.
Lemma 2. Let d <b be natural numbers and
d ,
d ,
d
as in (2) and
(3). Then
( d 3)=2<j
d
j<d=2;
(18)
p
b=d p
d=b<
p
d
<
p
b=d ; (19)
d
=d>
p
b=d 3
1=
p
bd 1=2;
(20)
d
=d<
p
b=d 3
: (21)
Finally,if 12d<b,
(22)
d
<
1+ 4
d
b
d 2
:
Proof. Observethat
2j
d j=
b(d 1)(d 2)
b(d 1)(d 2)
>
d 2
3d
:
Moreover, ifd =1,j
d
j<d=2is true. Ford2,
2j
d j=
(d 1)(d 2)
d 1=b
d 2<d:
Thisproves(18). Inorder to prove(19), note
d
=
(b d) 2
bd 1
>
(b d) 2
bd
;
hen e p
d
>(b d)=
p
bd= p
b=d p
d=b . Further,
d
= b d
d
b d
b 1=d
b d
d
<b=d:
Assertion (20)is immediatefrom the de nition(3) of
d
, theupperbound
for j
d
j and the lower bound for p
d
whi h are displayed in (18), (19),
respe tively. In orderto show(21) we use
q
d +
2
d
p
d +j
d j;
whi h an be veri ed bysquaring. This gives
d
p
d
<
p
b=d , by (19).
Finally,(19)implies
d
< j
d j+
q
b=d+ 2
d :
We show
(23)
q
b=d+ 2
d
(1+4=d)b=d 2
+j
d j;
whi h yields(22). However,
1+ 4
d
b
d 2
+j
d j
2
2
d +2j
d j
1+ 4
d
b
d 2
2
d
+(d 3)
1+ 4
d
b
d 2
;
by(19). One he ks that(d 3)(1+4=d)dwheneverd12 andobtains
(23).
Proof of Proposition 1(a). Sin e I( ;d) is an open interval of length
2
d
=d, itis learthat
2
d
=d 1jI( ;d)\Zj2
d
=d+1:
Therefore, (16)gives
(24) '(d)(2
d
=d 1)S
b;d
'(d)(2
d
=d+1);
where'(:::) denotesEuler'sfun tion. Now theassertionfollowsfrom (17),
(20), (21), and thefollowingthree formulas thathold forlargeb:
X
1=3 '(d)
d 3=2
= 12
2
b 1=6
+C+O(b 1=6
logb);
with 0:56<C<0,
(25)
X
1d<b 1=3
'(d)
p
d
=O(
p
b);
and
X
1d<b 1=3
'(d)= 3
2
b 2=3
+O(b 1=3
logb):
Ofthese, (25)is quiteelementarysin eits leftsideis
X
1d<b 1=3
p
d =O((b 1=3
) 3=2
):
The remaining two formulas are appli ations of standard results ( f. [1℄,
p.62, Theorem3.7,and p.71, Exer ise7).
The upperboundforS
b;d
given in(24)isnotgoodenoughfortheproof
of Proposition1(b). Instead, we shalluse
Lemma 3. Let 1d<b. Then
S
b;d
2
d
+(d;b):
Proof. By (24), theassertion is true ford =1. Suppose, hen eforth,
d2. S
b;d
isthe numberof all integers n, 0n<b,su hthat n2I( ;d)
holdsforsome ,0 <d,( ;d)=1. Butsaying\n2I( ;d)" isthesame
assaying
(26) jnd b j<
d :
Forevery k2Zde ne(k)
b
2Zbythe onditions
(k)
b
k mod b; b=2(k)
b
<b=2
(so(k)
b
isa ertainrepresentativeofthe ongruen e lassofkmodb). Now
(21), togetherwith theinequalities1<
p
b=d<b=db=2, yields
d
<b=2.
Consequently,(26) an hold onlyifj(nd)
b j<
d
. Butthen
(27) S
b;d
jfn :0n<b; j(nd)
b j<
d gj:
We writeÆ=(d;b),d=d 0
Æ,andb=b 0
Æ. Theidentity
(nd)
b
=(nd 0
)
b 0
Æ
readily shows
(28) jfn:0n<b; j(nd)
b j<
d gj
=Æjfn:0n<b 0
; j(nd 0
)
b 0
j<
d
=Ægj:
Sin e (d 0
;b 0
) = 1, the map n 7! (nd 0
)
b
0 is inje tive on fn : 0 n < b 0
g.
Thus,
jfn:0n<b 0
; j(nd 0
)
b 0j
<
d
=Ægj2
d
=Æ+1:
By (27)and (28), S 2 +Æ,asdesired.
Proof of Proposition 1(b). Lemma3 yields
S
2
(b)2 X
b 1=3
d<
p
b
d +
X
b 1=3
d<
p
b (d;b):
The se ondsum is dominatedby
X
1d<
p
b
(d;b) X
djb
djfn: 1n<
p
b; djngj X
djb d
p
b=d= p
b X
djb 1;
whi hisb 0:51
assoonasbislargeenough. Asto the rstsum,weassume
b 1=3
12 and use(22) togetherwith theformulas
X
db 1=3
1=d 2
=b 1=3
+O(b 2=3
);
X
db 1=3
1=d 3
=O(b 2=3
)
( f. [1℄, pp. 55 ). This on ludes theproof.
4. Additional observations. The proof of Theorem 2 might suggest
thatthesumS
2
(b)doesnota tuallyplaya roleforthegrowthof S(b)|for
instan e, we did not even use S
2
(b) > 0. Numeri al examples, however,
indi ate that S
1
(b) and S
2
(b) have the same order of magnitude, namely
b 2=3
( f. Table 1,whi h displayssome ases inwhi h bisa prime).
If oneassumesthatthebehaviouroftheregionR (b)relativeto integers
is\random",oneexpe tsthatits(usual)measure%(b)is losetoS(b). This
is the ase in the examples listed below. Here we note that the intervals
I( ;d) are mutually disjoint|so this is true not only for the integers in
these intervals. Therefore,
%(b)= X
1d<
p
b
'(d)2
d
=d:
However, we abstain fromprovingthesaid disjointness,some detailsbeing
fairlytoilsome. Withtheaidof Lemma 2it isnotdiÆ ultto show
%(b)<
36
2
b 2=3
:
The rightsideseems to be amore realisti upperboundforS(b)thanthat
of Theorem2.
Table1
b S(b) S
1 (b) S
2
(b) %(b) (36=
2
)b 2=3
10 5
+ 3 6338 4378 1960 6308.8 7858.6
10 6
+ 3 30210 20716 9494 30123.4 36475.7
10 7
+19 143010 97536 45474 142693.3 169305.1
10 8
+ 7 672954 457150 215804 671954.9 785843.6
10 9
+ 7 3153674 2136180 1017494 3150637.2 3647562.6
One may ask whether the intervals I( ;d) of Theorem 1 are \largest
possible" or, onversely, whether they an be extended to larger intervals
with the same behaviour of the sign. Indeed, an extension is possible in
individual ases but not ingeneral. Radema her ([7℄, Satz1) showed that
s(n;b)=0 ifn= p
b 1 ,sotheintervalsI( ;1) annot be enlargedifb 1
happens to bea square. Inaddition, we investigatedmanynumbers d>1
andfoundnumerousexamplesof sign hangesof s(n;b)assoon asnpasses
one of theboundariesof I( ;d).
Thebehaviourofs(n;b)insidetheintervalsI( ;d)isexplained,partially
at least, bythe followingobservation, whi h isbased on Lemma 1,too ( f.
also [3℄, part 2.4): Ifd issmall relative to band nis lose to the midpoint
b =d of I( ;d), the point (n;12s(n;b)) is lose to the point (x;y) of the
hyperbola
(x b =d)y=b=d 2
with x =n. In parti ular, thesign of s(n;b) agreeswith that of y. When
n moves away from b =d, thepoint (n;12s(n;b)) may gradually leave its
ompanion (n;y)|however, it must not ross the asymptote y =0 of the
hyperbola as long asn remains inside I( ;d). The reader may inspe t the
ases d=2;3in the rst diagramabove.
Finally,weobserve thatthe estimate(21) implies
jn=b =dj<1=(2d 2
)
for any n2 I( ;d) if onlyb 12 (re all that d must be <
p
b if I( ;d) is
nonempty). Therefore, thefra tion =d isa onvergent of n=b a ording to
Legendre's riterion. Inorder to test whethera given numbern, (n;b)=1,
is in the region R (b) one may pro eed as follows: Compute the ontinued
fra tion of n=b and he k whether some onvergent =d, d <
p
b, satis es
(26). If this is the ase, n is in I( ;d) and hen e inR (b), otherwisen lies
outside R (b).
Referen es
[1℄ T.M.Apostol,Introdu tiontoAnalyti NumberTheory,Springer,NewYork,1976.
[2℄ R. Bruggeman, On the distribution of Dedekind sums, in: Contemp. Math. 166,
Amer.Math.So .,1994,197{210.
[3℄ |,DedekindsumsforHe kegroups,A taArith.71(1995),11{46.
[4℄ U.Dieter,Beziehungenzwis henDedekinds henSummen,Abh.Math. Sem.Univ.
Hamburg21(1957),109{125.
[5℄ J. E. Pommersheim, Tori varieties, latti e points, and Dedekind sums, Math.
Ann.295(1993),1{24.
[6℄ H.Radema her,Generalizationofthere ipro ityformulaforDedekindsums,Duke
Math.J.21(1954),391{397.
[8℄ H. Radema her and E.Grosswald,Dedekind Sums,Carus Math. Monographs
16, Math.Asso .Amer.,1972.
InstitutfurMathematik
UniversitatInnsbru k
Te hnikerstr.25/7
A-6020Innsbru k,Austria
E-mail:Kurt.Girstmairuibk.a .at
Re eivedon18.7.1997
andinrevisedformon16.9.1997 (3229)
