On a conjecture of Erd\"os and certain Dirichlet series

0.11 MB·English

by  Tapas Chatterjee

ON A CONJECTURE OF ERDO¨S AND CERTAIN DIRICHLET SERIES
TAPASCHATTERJEE1ANDM.RAMMURTY2
5
1
0
ABSTRACT. Letf :Z/qZ → Zbesuchthatf(a)= ±1for1 ≤a <q,andf(q) =0. ThenErdo¨s
2
conjecturedthatP f(n) 6=0.Forqeven,itiseasytoshowthattheconjectureistrue.Thecase
n≥1 n
n q ≡ 3(mod4)wassolvedbyMurtyandSaradha. Inthispaper,weshowthatthisconjectureis
a truefor82%oftheremainingintegersq≡1(mod4).
J

7
1. Introduction
1

InawrittencommunicationwithLivingston,Erdo¨smadethefollowingconjecture(see[5]):
]
T iff isaperiodicarithmeticfunctionwithperiodq and
N
±1 ifq ∤n,
.
h f(n)=
t (0 otherwise,
a
m then
∞
[ f(n)
L(1,f) = 6= 0
n
1
n=1
v X
wheretheL-functionL(s,f)associatedwithf isdefinedbytheseries
5
8
∞
1 f(n)
L(s,f) := . (1)
4 ns
0 n=1
X
. In1973,Baker,BirchandWirsing(seeTheorem1of[1]),usingBaker’stheoryoflinearformsin
1
0 logarithms,provedtheconjectureforq prime. In1982,Okada[10]establishedtheconjectureif
5 2ϕ(q)+1 > q. Hence, if q is a prime power or a product of two distinct odd primes, the con-
1
jecture is true. In 2002, R. Tijdeman [12] proved the conjecture is true for periodic completely
:
v multiplicative functions f. Saradha and Tijdeman [11] showedthat if f is periodicand multi-
i
X plicative with |f(pk)| < p−1for everyprime divisor p of q and everypositive integerk, then
r theconjectureistrue.
a
Itiseasytoseethat
∞
f(n)
L(1,f) =
n
n=1
X
q
exists if and only if f(n) = 0. If q is even and f takes values ±1 with f(q) = 0, then
n=1
q
f(n)6= 0. Hencetheconjectureholdsforevenq.
n=1
P
In 2007, Murty and Saradha [8] proved that if q is odd and f is an odd integer valued odd
P
periodic function then the conclusion of the conjecture holds. In 2010, they proved that the
2010MathematicsSubjectClassification. 11M06,11M20.
Keywordsandphrases. Erdo¨sconjecture,non-vanishingofDirichletseries,Okada’scriterion.
1ResearchofthefirstauthorwassupportedbyapostdoctoralfellowshipatQueen’sUniversity.
2ResearchofthesecondauthorwassupportedbyanNSERCDiscoverygrantandaSimonsfellowship.
1
2 TAPASCHATTERJEEANDM.RAMMURTY
Erdo¨s conjecture is true if q ≡ 3 ( mod 4) (see Theorem 7 of [7]). Thus the conjecture is open
onlyin caseswhereq ≡ 1(mod4). However,it seemsthat anovelideawill beneededtodeal
withthesecases. Inthispaper,weadoptanewdensity-theoreticapproachwhichisorthogonal
toearliermethods. Hereisthemainconsequenceofourmethod:
Theorem1.1. LetS(X) = |{q ≡ 1(mod4), q ≤ X|Erdo¨sconjectureistrueforq}|.
Then
S(X)
liminf ≥ 0.82.
X→∞ X/4
In other words, the Erdo¨s conjecture is true for at least 82 % of the integers q ≡ 1 (mod 4).
OurmethoddoesnotextendtoshowthattheErdo¨sconjectureistruefor100 %ofthemoduli
q ≡ 1 (mod 4). We examine this question briefly at the end of the paper. It seems to us that
moreideasareneededtoresolvetheconjecturefully.
These questions have a long history beginning with Baker, Birch and Wirsing [1]. Their
work was generalized by Gun, Murty and Rath [4] to the setting of algebraic number fields.
Theforthcomingpaper[2]givesnewproofsofsomeofthebackgroundresultsofthisarea. We
alsoreferthereaderto[12]foranexpandedsurveyoftheearlyhistory.
2. NotationsandPreliminaries
Fromnowonwards,wedenotethefieldofrationalsbyQ,thefieldofalgebraicnumbersby
Q,Euler’stotientfunctionbyϕandEuler’sconstantbyγ. Wesayafunctionf isErdo¨sianmod
q iff isaperiodicfunctionwithperiodq and
±1 ifq ∤ n,
f(n)=
(0 otherwise.
Alsowewillwritef(X) . g(X)tomean
f(X)
limsup ≤ 1.
g(X)
X→∞
Similarly,wewritef(x)& g(x)tomean
f(X)
liminf ≥ 1.
X→∞ g(X)
2.1. Okada’scriterion.
Proposition2.1. Lettheq-thcyclotomicpolynomialΦ beirreducibleoverthefieldQ(f(1),··· ,f(q)).
q
LetM(q)bethesetofpositive integerswhicharecomposedofprimefactorsofq.
ThenL(1,f) = 0ifandonlyif
f(am)
= 0
m
m∈M(q)
X
foreveryawith1≤ a < q,(a,q) = 1,and
q
f(r)ǫ(r,p) = 0
r=1
X
(r,q)>1
ONACONJECTUREOFERDO¨S 3
foreveryprimedivisor pofq,where
v (r) ifv (r)< v (q),
p p p
ǫ(r,p) =
(vp(q)+ p−11 otherwise
andforanyintegerr,v (r)istheexponentofpdividingr.
p
This Proposition is a modification, due to Saradha and Tijdeman [11], of a 1986 result of
Okada[9]. NotethatOkadadeducedthesufficientcondition2ϕ(q)+1 > q statedintheintro-
ductionfromhisoriginalversionofthiscriterion.
2.2. Wirsing’sTheorem. ThefollowingpropositionisduetoWirsing[13].
Proposition2.2. Letf beanon-negativemultiplicativearithmeticfunction,satisfying
|f(p)|≤ Gforallprimesp,
p−1f(p)logp ∼ τ logX,
p≤X
X
withsomeconstantsG> 0,τ > 0and
p−k|f(pk)| < ∞;
p k≥2
XX
if0 < τ ≤ 1,then,inaddition,thecondition
|f(pk)| = O(X/logX)
p k≥2
X X
pk≤X
isassumedtohold.Then
X e−γτ f(p) f(p2)
f(n)= (1+o(1)) (1+ + +···).
logX Γ(τ) p p2
n≤X p≤X
X Y
2.3. Mertens Theorem. We also need a classical theorem of Mertens in a later section. We
recordthetheoremhere(seeforexample,p. 130of[6]):
Proposition2.3. lim logX (1− 1) = e−γ.
p
X→∞ p≤X
Q
3. ExceptionstotheconjectureofErdo¨s
WesaythattheErdo¨sconjectureisfalse(modq),ifthereisanErdo¨sianfunctionf forwhich
L(1,f) = 0. Thefollowingpropositionplaysafundamentalroleinourapproach.
Proposition3.1. IftheErdo¨sconjectureisfalse(modq)withqodd,then
1
1 ≤ .
ϕ(d)
d|q
X
d≥3
4 TAPASCHATTERJEEANDM.RAMMURTY
Proof. Bythehypothesis,thereisanErdo¨sianfunctionf modqforwhich,wehaveL(1,f) = 0.
ApplyingOkada’scriterion,weget
f(b)
= 0. (2)
b
b∈M(q)
X
Letd = (b,q),sothatb = db with(b ,q/d) = 1. Then(2)canbewrittenas
1 1
1 f(db )
1
−f(1) = .
d b
1
Xd|q b1∈XM(q)
d≥3 (b1,q/d)=1
Takingabsolutevalueofbothsides,weget
1 1
1 ≤ . (3)
d b
1
Xd|q b1∈XM(d)
d≥3
Noticethattheinnersumcanbewrittenas
−1
1 1 1 1 d
= 1+ + +··· = 1− = .
b p p2 p ϕ(d)
b1∈XM(d) 1 Yp|d (cid:18) (cid:19) Yp|d (cid:18) (cid:19)
Hencefrom(3),weget
1
1 ≤ .
ϕ(d)
d|q
X
d≥3
(cid:3)
Twoimmediatecorollariesoftheabovepropositionarethefollowing:
Corollary 3.2. Ifq isaprimepoweroraproductoftwodistinctoddprimes,thentheErdo¨s
conjectureistrue(modq).
Proof. Thisisapleasantelementaryexercise. (cid:3)
Hence we have recovered the two basic cases of the conjecture which were given in the
introduction,ofcourse,alsoasaconsequenceofOkada’sCriterion.
Letd(n)bethedivisorfunction,i.e. d(n)isthenumberofdivisorsofn.
Corollary3.3. Ifthesmallestprimefactorofqisatleastd(q),thentheErdo¨sconjectureistrue
forq.
Proof. Letlbethesmallestprimefactorofq. Fromtheaboveproposition,iftheErdo¨sconjecture
isfalse(modq),thenwehave
1
1 ≤
ϕ(d)
d|q
X
d≥3
1 d(q)−2
< 1 = ,
ϕ(l) l−1
d|q
X
d≥3
ONACONJECTUREOFERDO¨S 5
thestrict inequality in thepenultimate stepcoming from thefact that q has at least twoprime
divisors. Thus,l < d(q). Henceifl ≥ d(q),thentheErdo¨sconjectureistrue(modq).
(cid:3)
Notethat, theCorollary 3.3 was notknownpreviously. Itimplies that theconjectureis true
for any squarefree number q with k prime factors, provided the smallest prime factor of q is
greater than 2k. Proposition 3.1 opens the door for a new approach to the study of Erdo¨s’s
conjecture. Letusconsiderthefollowing
S (X) = |{q ≡ 1(mod4), q ≤ X|Erdo¨sconjectureisfalse(modq)}|
1
Then,wehave
1 1
S (X) ≤ ≤ 1
1
ϕ(d) ϕ(d)
q≤X d|q 3≤d≤X q≤X
X X X X
q≡1(mod4)d≥3 dodd q≡1(mod4)
d|q
1 X 1 X 1
≤ +O(1) ≤ +O
ϕ(d) 4d ϕ(d)4d  ϕ(d)
3≤d≤X (cid:18) (cid:19) 3≤d≤X 3≤d≤X
X X X
dodd dodd
 
1 X
≤ +O(logX)
ϕ(d)4d
3≤d≤X
X
dodd
wherewehaveusedthewell-knownfactthat(seeforexample,p. 67of[6])
1
= O(logX).
ϕ(d)
d≤X
X
Hence,weget
X 1
S (X) .
1
4 dϕ(d)
3≤d
X
dodd
X 1 1
. 1+ + +··· −1
4  pϕ(p) p2ϕ(p2) 
podd(cid:18) (cid:19)
Y
 
X 1 1
. 1+ + +··· −1
4  p(p−1) p3(p−1) 
podd(cid:18) (cid:19)
Y
 
X 1 1 1
. 1+ 1+ + +··· −1
4  p(p−1) p2 p4 
podd(cid:18) (cid:18) (cid:19)(cid:19)
Y
 
X p
. 1+ −1 .
4  (p−1)(p2 −1) 
podd(cid:18) (cid:19)
Y
 
Theproductiseasilycomputednumerically andwehave: S (X) . 0.33(X/4).Asanimmedi-
1
atecorollarywegetthefollowing:
6 TAPASCHATTERJEEANDM.RAMMURTY
Corollary3.4. |{q ≡ 1(mod4), q ≤ X|Erdo¨sconjectureistrueforq}|
& 0.67X.
4
3.1. Refinementusingthesecondmoment. Byconsideringhighermoments,wecanimprove
the lower bound in theabove corollary. Webegin with the secondmoment. We include these
estimatessincetheyareofindependentinterestandself-contained.
Proposition3.5. |{q ≡ 1(mod4), q ≤ X|Erdo¨sconjectureistrueforq}|
& 0.78X.
4
Proof. Letusfirstconsiderthefollowinginequality:
2
1
S1(X) ≤  
ϕ(d)
q≤X d|q
X X 
q≡1(mod4)d≥3 
  1
≤
ϕ(d )ϕ(d )
1 2
qX≤X d1|Xq,d2|q
q≡1(mod4)3≤d1,d2<q
1
≤ 1
ϕ(d )ϕ(d )
1 2
3≤d1X,d2≤X qX≤X
d1,d2 odd q≡1(mod4)
d1|q,d2|q
1
≤ 1
ϕ(d )ϕ(d )
1 2
3≤d1X,d2≤X qX≤X
d1,d2 odd q≡1(mod4)
[d1,d2]|q
1 X
≤ +O(1) .
ϕ(d )ϕ(d ) 4[d ,d ]
3≤d1X,d2≤X 1 2 (cid:18) 1 2 (cid:19)
d1,d2 odd
Hence,wehave
X 1
S (X) ≤ +O(log2X).
1
4 ϕ(d )ϕ(d )[d ,d ]
1 2 1 2
3≤d1X,d2≤X
d1,d2 odd
Byasimplenumericalcalculation,wededucethat
X
S (X) . 0.22 .
1
4
Hencetheconjectureholdsforatleast78%ofthepositiveintegerscongruentto1mod4. (cid:3)
Similarlyonecancomputehigherfractionalmomentstogetanoptimalresult. Foranyr > 1,
wehave
r
1
S1(X) ≤   .
ϕ(d)
q≤X d|q
X X 
q≡1(mod4)d≥3 
 
ONACONJECTUREOFERDO¨S 7
We studythis as a function of r. Using Maple we computed that the minimal value occurs at
r ∼ 3.85(see[3])andweget
X
S (X) . 0.18 .
1
4
Thus,weget|{q ≡ 1(mod4), q ≤ X|Erdo¨sconjectureistrueforq}|
& 0.82X,i.e.
4
S(X)
liminf ≥ 0.82.
X→∞ X/4
Hence, we have shown the theoremstated in the Introduction, i.e. the conjecture holds for at
least82%ofthenumberscongruentto1mod4.
3.2. An alternative approach. In this subsection, we discuss an alternative approach to this
problem. Itleadstoa slightlyweakerresult. Howeverthismethodis ofindependentinterest,
so we record it here. We begin with a further refinement of Proposition 3.1 by considering
fractional momentsthere. FromProposition3.1, wegetiftheErdo¨sconjectureis false forodd
q,then
1
1 ≤ .
ϕ(d)
d|q
X
d≥3
Adding1bothsidesoftheaboveinequality,weget
1
2 ≤ ,
ϕ(d)
d|q
X
whichcanberewrittenas
1 1
1 ≤ .
2 ϕ(d)
d|q
X
Henceforanyα > 0,Proposition3.1canberewrittenas:
Proposition3.6. IfErdo¨sconjectureisfalseforoddq,then
α
1 1
1 ≤ .
2α  ϕ(d)
d|q
X
 
Asbefore,S (X) = |{q ≡ 1(mod4), q ≤ X|Erdo¨sconjectureisfalseforq}|. Thenfromthe
1
aboveproposition,weget
α
1 1
S (X) ≤ .
1 2α  ϕ(d)
q≤X d|q
X X
q≡1(mod4)
 
α
Letf (q) = 1 andχbethenon-trivialDirichletcharactermod4. Thentheabove
α d|q ϕ(d)
inequalitybeco(cid:16)mes (cid:17)
P
1
S1(X) ≤ 2α+1  fα(q)+ χ(q)fα(q). (4)
q≤X q≤X
X X 
q odd q odd
 
 
8 TAPASCHATTERJEEANDM.RAMMURTY
Again,notethatf (q)isamultiplicativearithmeticfunction. Onecancheckthatitalsosatisfies
α
all the other hypothesesof Wirsing’s theorem(Proposition 2.2) with G = 2α and τ = 1. So in
lightofWirsing’stheorem,weget
e−γ f (p) f (p2)
α α
f (q) ∼ X 1+ + +···
α logX p p2
q≤X p≤X(cid:18) (cid:19)
X Y
qodd p6=2
and
e−γ χ(p)f (p) χ(p2)f (p2)
α α
χ(q)f (q)∼ X 1+ + +··· .
α logX p p2
q≤X p≤X(cid:18) (cid:19)
X Y
qodd p6=2
Again,fromMertenstheoremweknowthat
e−γ
(1−1/p) ∼ .
logX
p≤X
Y
Hencewehave
X f (p) f (p2)
α α
f (q) ∼ (1−1/p) 1+ + +···
α 2 p p2
q≤X p≤X (cid:18) (cid:19)
X Y
q odd p6=2
X
∼ p (say)
1
2
and
X χ(p)f (p) χ(p2)f (p2)
α α
χ(q)f (q) ∼ (1−1/p) 1+ + +···
α 2 p p2
q≤X p≤X (cid:18) (cid:19)
X Y
q odd p6=2
X
∼ p (say).
2
2
Nowusingtheabovetwoinequality(4)becomes,
X
S (X) . (p +p ).
1 2α+2 1 2
Finally usingMaple (see[3]), we findthat thequantity onthe righthand sideis minimized at
α ∼ 8.11andweget
X
S (X) . 0.20 .
1
4
Hence,weget
S(X)
liminf ≥ 0.80.
X→∞ X/4
Remarks. Onecannothopetoobtain100%bythesemethods. Infact,onecanshowthatthere
isapositivedensity(albeitsmall)ofqforwhichtheinequalityofProposition3.1holds. Indeed,
since
1 1
≥ 1+
ϕ(d) p−1
d|q p|q (cid:18) (cid:19)
X Y
we can make the product( and hence the sum)arbitrarily large by ensuringthat q is divisible
byall theprimesinan initial segment. Wecan evenensurethattheseprimesare congruentto
ONACONJECTUREOFERDO¨S 9
1(mod4). Wethentakenumberswhicharedivisiblebythisq andcongruentto1(mod4)and
deducethatforallthesenumbers,theinequalityinthepropositionholds. Sincetheproducton
therightdivergesslowlytoinfinityaswegothroughsuchnumbersq,weobtaininthisway,a
smalldensityofnumbersforwhichtheinequalityholds.
Acknowledgements. WethankMichaelRothforhishelponusingtheMaplelanguageaswell
asSanoliGun,PurusottamRathandEkataSahafortheircommentsonanearlierversionofthis
paper. Wealsothanktherefereeforhelpfulcommentsthatimprovedthequalityofthepaper.
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(T.Chatterjee)DEPARTMENTOFMATHEMATICS,INDIANINSTITUTEOFTECHNOLOGYROPAR,PUNJAB-140001,
INDIA.
(M. Ram Murty) DEPARTMENT OF MATHEMATICS AND STATISTICS, QUEEN’S UNIVERSITY, KINGSTON, ON-
TARIO,CANADA,K7L3N6.
E-mailaddress,TapasChatterjee:[email protected]
E-mailaddress,M.RamMurty:[email protected]