Almost periodicity of some error terms in prime number theory

ACTAARITHMETICA
106.3(2003)
Almostperiodicityofsomeerrorterms
inprimenumbertheory
by
JerzyKaczorowski(Poznan) andOlivierRamare(Lille)
1.Introductionandstatementofresults.Theaimofthispaperis
toinvestigatedistributionofvaluesofalargeclassoffunctionsofarithmetic
signi cance assumingasuitablygeneralizedRiemannHypothesis.Probably
thesimplestexampleofamemberofthisclassisde ned bythefollowing
formula: 8
1v=2 v v 2v>e (e)+e log(1 e ) log2 if v>0;0< 2

v(1) (v)=0 1 1 ev=2 v ve>e (e )+e +v+ log +C if v<0;: 0 v2 1+e
where CistheEulerconstantandasusual(for x>1)
X X
e (x)= (n); (x)= (n)=n;
nx nx
1 1e e e (x)= ( (x+0)+ (x 0)); (x)= ( (x+0)+ (x 0)):0 02 2
Thisfunctionforpositive visonlyamildmodi cation ofthenormalized
remaindertermintheprimenumberformula,wherewetakethee ect of
thetrivialzerosofthezetafunctionintoaccount.Theproperde nition of
fornegativevaluesoftheargumentfollowsfromtheworkofthe rst0
namedauthor[8]andessentiallycomesfromthefunctionalequationofthe
Riemannzetafunction.Forreal ylet
logR
1 vN(R)= e dv:y f (v)>yg0R
1
Ourproblemistoseehowthisnumberandrelatedquantitiesbehave.
Thoughnorealnumberisknownforwhich (x) >lix,the rst named
authorsucceededin[9]inprovingundertheRiemannHypothesisthatthe
2000MathematicsSubjectClassi cation:Primary11N05;Secondary11M26,11K70,
42A75.
J.KaczorowskipartiallysupportedbytheKBNGrantnumber2PO3A02417.
[277]
278 J.KaczorowskiandO.Ramare
setofsuch xhasapositiveasymptoticlowerdensity,whichinoursetting
istranslatedintoliminf N (R)>0.Evenbetter,heshowedthatforR!1 1
someconstantc >1,everyintervaloftheshape[V;c V],V >1,containsa0 0
positiveproportionofsuchpointsandthesameholdswhen 1isreplaced
byanyrealnumbery.InfactthelimsupandliminfofN(R)asRgoestoy
in nityare >0and<1.
Similarproblemsrelatedtothedistributionofprimesinarithmeticpro-
gressionsareofinterest.Thereaderisreferredtothesurveypaper[11]and
theliteraturecitedthere.Thecommonfeatureoftheseresultsisthatthey
dependonakindofalmostperiodicity(cf.also[13]).Itisalsoclearthat
themethodusedintheproofsisofageneralcharacterandcansuccessfully
beappliedtomanysimilarproblems.Theprincipalaimofthispaperisto
considerthewholesubjectfromageneralpointofview.Itseemsthatthe
frameworkoftheSelbergclassisappropriatehere.
Let s=+itand f(s):=f(s).TheSelbergclassS(cf.[14])isde ned
bythefollowingaxioms:
(i)(Dirichletseries)Every F2SisaDirichletseries
1X
sF(s)= a(n)n ;
n=1
absolutelyconvergentfor >1.
(ii)(Analyticcontinuation) Thereexistsaninteger m0suchthat
m(s 1) F(s)isentireof nite order.
(iii)(Functionalequation) F2Ssatis es afunctionalequationoftype
(s)=!(1 s),where
rY
s(s)=Q ( s+)F(s)=(s)F(s);j j
j=1
say,with Q>0, >0,Re 0andj!j=1.j j
"(iv)(Ramanujanhypothesis) Forevery ">0, a(n)n :
P1 s(v)(Eulerproduct) F 2Ssatis es logF(s)= b(n)n ,wheren=1
k b(n)=0unless n=p with k1,and b(n)n forsome <1=2.
TheSelbergclasscontainsmost L-functionsusedinnumbertheory.
ThemostobviousexamplesaretheRiemannzetafunctionandtheshifts
L(s+i ),2 R,ofDirichletL-functionswithprimitivecharacter(modq),
q2.OtherexamplesincludeDedekindzetafunctionsofalgebraicnumber
elds, andHecke L-functionsformedwithprimitivecharacters.Moreover,
theArtin L-functions L(s;%;K=Q)associatedwithirreduciblerepresenta-
tionsoftheGaloisgroupGal(K=Q)belongtoSprovidedastandardcon-
jectureholds.TheL-functionsL(s)associatedwithholomorphicnewformsfAlmostperiodicityoferrorterms 279
f(z)oncongruencesubgroupsofSL(2;Z)belongtoSoncesuitablynormal-
ized.Thesameistrueforthenon-holomorphicones,providedcertainconjec-
tureshold.TheRankin{Selbergconvolutionoftwonormalized L-functions
associatedwithholomorphicnewformsisinS,andthesameistrueforthe
symmetricsquare L-functionassociatedwithaholomorphicnewformon
SL(2;Z).Finally,weremarkthatmanyotherimportant L-functionswould
belongtoS,providedcertainwellknownconjectures,suchastheLanglands
conjecture,hold.
Wereferto[12]forbasicfactsconcerningS.Theminimalinteger min
(ii)iscalledthepolarorderof Fanddenotedby m .WeremarkthattheF
function(s)in(iii)isde ned uniquelyuptoamultiplicativeconstant.We
callitthe-factorofFanddenoteby .ItisknownthatF2ShastrivialF
zerosatpoints
+kj(2) ; 1jr; k0:
j
Becauseofapossiblepoleats=1,thetrivialzeroats=0,ifitexists,has
multiplicity
(3) #f1jr: =0g m :j F
Allotherzerosarecallednon-trivialandlieintheverticalstrip01.
Weexpectthatallnon-trivialzeroslieonthecriticalline=1=2.Inother
wordsweexpectthattheGeneralizedRiemannHypothesis(GRH)holdsin
theSelbergclass.
Let F2Sandlet %= +i denotethegenericnon-trivialzeroof F.
Moreover,let !denotethegenerictrivialzeroof F.Foracomplexnumber
zfromtheupperhalf-plane(Imz >0)wewrite
X
%zk(z;F):= e
Im%>0
andforRez >0let
X
!zk(z; ):= e :F
!
Itiseasytoverifythatbothseriesconvergeintheindicatedhalf-planes,the
convergencebeinguniformoncompactsubsets.Hencek(;F)isholomorphic
forImz>0,and k(; )forRez >0.F
Moreover,wede ne
z
K(z;F):= k(s;F)ds;
i1
wheretheintegrationistakenalongtheverticalhalf-line.ForImz >0we280 J.KaczorowskiandO.Ramare
P
%zhave K(z;F)= e =%.Forreal x=0letIm%>0
8
x>> k(t; )dt if x>0; F<
1K(x; ):=F x>> t> ek( t; )dt if x<0;> F:
1
f(x;F):= lim(K(x+iy;F)+K(x+iy;F));
+y!0
x=2F(x;F):=e f(x;F):
Wealsowrite
X X (n)Fe (n):=b(n)logn; (x;F):= (n); (x;F):= ;F F
n
nx nx
sothataccordingto(v)wehave,for >1,
10 XF (n)F(s)= :
sF n
n=1
Theorem1.(a)(Analyticcontinuationofk(z; ))Thefollowingfor-F
mulagivesmeromorphiccontinuationofk(z; )tothewholecomplexplane:F
r z= X j je
k(z; )= m :F Fz= j1 e
j=1
(b)(Analyticcontinuationof k(z;F)) k(z;F)hasmeromorphiccontin-
uationtotheRiemannsurface Mof logz.For z2Mwehave
1 z zk(z;F)= (k(z; ) e k( z; )+(1 e)m )logz+N(z;F);F F 1F2i
where N issingle-valuedandmeromorphicon Chavingsimplepolesat1
mostatthepoints z=0or z=lognforapositiveinteger n.Wehave
1 1 (n)F
Res N(z;F)= (n); Res N(z;F)= :z=logn 1 F z= logn 1
2i 2i n
ia(c)(Functionalequation) Writepoints z2 M intheform z= re ,
c ia c iar >0, a2 Randlet M3z7!z 2Mbede nedby(re ) =re .Then
forall z2Mwehave
X
z zck(z;F)+k(z ;F)=m e k(z; ) e ;F F
=0
wherethesummationistakenoverallnon-trivial,realzerosof F(ifany).
6Almostperiodicityoferrorterms 281
(d)(Boundaryvalues)For x>0, x=lognwehave
xX ex x(4) f(x;F)= (e ;F)+m e K(x; ) +c ;F F F

Im%=0
whereasfor x<0, x= lognwehave
xX ejxje(5) f(x;F)= (e ;F)+m x K(x; ) +d ;F F F