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Almost periodicity of some error terms in prime number theory

De
21 pages
ACTA ARITHMETICA 106.3 (2003) Almost periodicity of some error terms in prime number theory by Jerzy Kaczorowski (Poznan) and Olivier Ramare (Lille) 1. Introduction and statement of results. The aim of this paper is to investigate distribution of values of a large class of functions of arithmetic significance assuming a suitably generalized Riemann Hypothesis. Probably the simplest example of a member of this class is defined by the following formula: ?0(v) = ? ?? ?? e?v/2 ( ? ?0(ev) + ev ? 12 log(1? e?2v)? log 2pi ) if v > 0, e?v/2 ( ?˜0(e?v) + ev + v + 1 2 log 1? ev 1 + ev + C ) if v < 0, (1) where C is the Euler constant and as usual (for x > 1) ?(x) = ∑ n≤x ?(n), ?˜(x) = ∑ n≤x ?(n)/n, ?0(x) = 12 (?(x+ 0) + ?(x? 0)), ?˜0(x) = 12(?˜(x+ 0) + ?˜(x? 0)). This function for positive v is only a mild modification of the normalized remainder term in the prime number formula, where we take the effect of the trivial zeros of the zeta function into account.

  • riemann zeta

  • generalized riemann

  • indicated half-planes

  • selberg class

  • arithmetic pro

  • function ?

  • class contains most


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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