A History of the Prime Number Theorem
Author(s): L. J. Goldstein
Reviewed work(s):
Source: The American Mathematical Monthly, Vol. 80, No. 6 (Jun. - Jul., 1973), pp. 599-615
Published by: Mathematical Association of America
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A HISTORY OF THE PRIME NUMBER THEOREM
of Maryland
L. J. GOLDSTEIN, University
The sequenceof primenumbers,whichbegins
2, 3,5,7, 11,13,17,19,23,29,31,37,
both professionalsand amateurs
has held untold fascinationfor mathematicians,
alike. The basic theoremwhichwe shalldiscussin thislectureis knownas theprime
and allows one to predict,at leastin grossterms,the way in which
numbertheorem
Let x be a positiverealnumber,and let7r(x)= thenumber
theprimesare distributed.
of primes <x. Then the primenumbertheoremassertsthat
(X)
lim
(1)
X_
0x/logx
= 1,
wherelog x denotesthenaturallog of x. In otherwords,theprimenumbertheorem
assertsthat
(2)
)0log
?
(
(logx)
(x-
oo),
whereo(x/logx) standsfora function
f(x) withthe property
lim
f(x)
l +, x/logx
= 0
Actually,forreasonswhichwill become clear later,it is muchbetterto replace(1)
and (2) by the followingequivalentassertion:
?O(lox) x)
r(x)== Jb doyy-+
7t(x)
log
log
(3)
to integrate
To provethat(2) and (3) are equivalent,it suffices
[x dy
12
log y
once bypartsto get
(4)
f
logy
logx
log2 +
log2y
Ph.D. underG. Shimura.
Aftera Gibbsinstructorship
LarryGoldsteinreceivedhis Princeton
His research
at Yale, hejoinedtheUniv.of M arylandas AssociateProfessor
and nowisProfessor.
Functions.
He is theauthorofAnalyNumberTheoryandAutomorphic
is inAnalytic
and Algebraic
tic Number Theory(Prentice-Hall 1971), and AbstractAlgebra,A First Course (Prentice-Hall, to
appear). Editor.
599
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600
[June-July
L. J. GOLDSTEIN
However,for x > 4,
fX
J2
(5)
dy
J2
y l2y
-
<
1/x
-
0
x
dy
'u'
+
I
dy
y
J2_log2
x-
1
log2
l
1
(1)
x
Vlogx '
wherewe have used the fact that 1/log2xis monotonedecreasingfor x > 1. It is
clearthat(4) and (5) showthat(2) and (3) are equivalentto one another.The advantage of the version(3) is thatthe function
Li(x)
=
fx
5
'
called the logarithmic
integral,providesa muchcloser numericalapproximationto
ir(x)thandoes x /logx. This is a ratherdeep factand we shall returnto it.
In thislecture,I shouldlike to explorethehistoryof theideas whichled up to the
primenumbertheoremand to itsproof,whichwas notsupplieduntilsome 100 years
afterthefirst
conjecturewas made.The historyoftheprimenumbertheoremprovides
feeding
a beautifulexampleof theway in whichgreatideas developand interrelate,
upon one anotherultimatelyto yield a coherenttheorywhich rathercompletely
explains observedphenomena.
The veryconceptionof a primenumbergoes back to antiquity,althoughit is not
possibleto say preciselywhenthe conceptfirstwas clearlyformulated.However,a
factsconcerningtheprimeswereknownto theGreeks.Let us
numberof elementary
citethreeexamples,all of whichappearin Euclid:
(i) (Fundamental Theorem of Arithmetic):Every positive integern can be
writtenas a productof primes.Moreover,this expressionof n is unique up to a
rearrangementof the factors.
manyprimes.
(ii) There exist infinitely
(iii) The primes may be effectivelylisted using the so-called "sieve of
Eratosthenes".
We willnotcommenton (i), (iii) anyfurther,
sincetheyare partof thecurriculum
of most undergraduate
coursesin numbertheory,and hence are probablyfamiliar
fromEuclid's
to mostof you. However,thereis a proofof (ii) whichis quitedifferent
to the historyof the primenumber
well-knownproofand whichis verysignificant
Leonhard Euler and dates
theorem.This proofis due to the Swiss mathematician
fromthemiddleof the 18thcentury.It runsas follows:
AssumethatPI, , PN is a completelist of all primes,and considerthe product
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A HISTORY OF THE PRIME NUMBER THEOREM
1973]
(6)
E
(
H
.1(1+
P)
601
p+T2+.. '
Since everypositiveintegern can be writtenuniquelyas a productof primepowers,
everyunit fraction1/n appears in the formalexpansionof the product(6). For
the terms
then 1/n occurs frommultiplying
example,if n = p'l ...
p'N
1a/ll
1 /pa2,
..
lpa,
Therefore,if R is any positiveinteger,
I
ru(-'
N
(7)
a-1
-1
1
Pi
R
> 11=1
E,1In.
which
However,as R -+ oo, the sum on the righthand side of (7) tendsto infinity,
contradicts(7). Thus, P1, *', PN cannotbe a completelist of all primes.We should
make two commentsabout Euler's proof: First,it linksthe FundamentalTheorem
of primes.Second,it uses an analyticfact,namely
withtheinfinitude
ofArithmetic
the divergenceof the harmonicseries,to conclude an arithmeticresult.It is this
latterfeaturewhich became the cornerstoneupon which much of 19th century
numbertheorywas erected.
The firstpublishedstatementwhichcame close to the primenumbertheorem
was due to Legendrein 1798[8]. He assertedthat7r(x)is of theformx /(Alog x + B)
forconstantsA and B. On the basis of numericalwork,Legendrerefinedhis conjecturein 1808 [9] by assertingthat
=
logx + A(x)'
where A(x) is "approximately1.08366...".
Legendremeant that
Presumably,by this latterstatement,
lim A(x) = 1.08366.
x -I 00
It is preciselyin regardto A(x), whereLegendrewas in error,as we shall see below.
In his memoir[9] of 1808,Legendreformulatedanotherfamousconjecture.Let k
primeto one another.Then Legendreasserted
and 1be integerswhichare relatively
manyprimesof theform1+ kn(n = 0, 1,2, 3, ...). In other
thatthereexistinfinitely
words,if 7rk,l(x)denotesthenumberof primesp of theform1+ kn forwhichp < x,
thenLegendreconjecturedthat
(8)
7tk,l(x)
-+
oo
as x -+ cc.
Actually,the proofof (8) by Dirichletin 1837 [2] providedseveralcrucialideas on
how to approachtheprimenumbertheorem.
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602
L. J. GOLDSTEIN
[June-July
AlthoughLegendrewas the firstperson to publish a conjecturalformof the
primenumbertheorem,Gauss had alreadydone extensivework on the theoryof
primesin 1792-3. EvidentlyGauss consideredthetabulationof primesas some sort
of pastimeand amused himselfby compilingextensivetables on how the primes
in variousintervalsof length1009. We have included some of
distributethemselves
Gauss' tabulationsas an Appendix.The firsttable,excerptedfrom[3, p. 436], covers
an intervalof length
theprimesfrom1 to 50,000.Each entryin the table represents
168
to
are
from
135
for
there
from1001to 2000;
1
1000.Thus, example,
primes
1000;
127 from3001 to 4000; and so forth.Gauss suspectedthatthe densitywithwhich
primesoccuredin theneighborhoodof theintegern was 1/logn, so thatthenumber
of primesin the interval[a, b) should be approximatelyequal to
Tb dx
x'
Jalog
In the second set of tables, samples from[4, pp. 442-3], Gauss investigatesthe
distributionof primesup to 3,000,000and comparesthe numberof primesfound
withthe above integral.The agreementis striking.For example,between2,600,000
and 2,700,000,Gauss found6762 primes,whereas
2,700.000
2,600,000
dx
-_
log x
= 6761.332.
on the distributionof primes.NeverGauss neverpublishedhis investigations
theless,thereis littlereasonto doubt Gauss' claim thathe firstundertookhis work
in 1792-93, well beforethe memoirof Legendrewas written.Indeed, thereare
severalotherknownexamplesof resultsof the firstrank whichGauss proved,but
nevercommunicatedto anyoneuntilyearsafterthe originalworkhad been done.
This was the case, for example,withthe ellipticfunctions,whereGauss preceded
whereGauss anticipatedRiemann.The only
Jacobi,and withRiemanniangeometry,
informationbeyond Gauss' tables concerningGauss' work in the distributionof
primesis containedin an 1849 letterto theastronomerEncke. We have includeda
translationof Gauss' letter.
In his letterGauss describeshis numericalexperiments
and his conjectureconcerningi(x). There are a numberof remarkablefeaturesof Gauss' letter.On the
secondpage of theletter,Gauss compareshis approximationto 2r(x),namelyLi(x),
with Legendre'sformula.The resultsare tabulatedat the top of the second page
and Gauss' formulayieldsa muchlargernumericalerror.In a veryprescient
statement,
Gauss defendshis formulaby notingthat althoughLegendre'sformulayields a
smallererror,therateof increaseof Legendre'serrortermis muchgreaterthanhis
own. We shall see below thatGauss anticipatedwhat is knownas the "Riemann
hypothesis."Anotherfeatureof Gauss' letteris thathe casts doubt on Legendre's
assertionabout A(x). He assertsthatthe numericalevidencedoes not supportany
conjectureabout the limitingvalue of A(x).
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1973]
A HISTORY OF THE PRIME NUMBER THEOREM
603
Gauss' calculations are awesome to contemplate,since they were done long
beforethe days of high-speedcomputers.Gauss' persistenceis most impressive.
My student,Edward Korn, has checked
However,Gauss' tablesare not error-free.
Gauss' tables usingan electroniccomputerand has founda numberof errors.We
include the correctedentriesin an appendix. In spite of these (remarkablyfew)
evidencein favorof theprime
errors,Gauss' calculationsstillprovideoverwhelming
numbertheorem.Modern studentsof mathematicsshould take note of the great
care withwhichdata was compiledby such giantsas Gauss. Conjecturesin those
days were rarelyidle guesses.They were usuallysupportedby piles of laboriously
gatheredevidence.
The next step toward a proof of the prime numbertheoremwas a step in a
different
completely
direction,and was takenby Dirichletin 1837 [2]. In a beautiful
memoir,Dirichletproved Legendre's conjecture(8) concerningthe infinitudeof
primesin an arithmeticprogression.Dirichlet'sworkcontainedtwo radicallynew
ideas, whichwe should discuss in some detail.
denotethe ringof residueclasses modulo n, and let En denotethe group
Let
of unitsof 7n. Then En is theso-called"group of reducedresidueclassesmodulo n"
and consistsof thoseresidueclassescontainingan elementrelatively
primeto n. If k
is an integer,let us denoteby k its residueclass modulo n. Dirichlet'sfirstbrilliant
of the groupEn; thatis, thehomomorphisms
idea was to introducethecharacters
of
En into the multiplicative
group Cx of non-zerocomplexnumbers.If x is such a
character,thenwe may associate withx a function(also denotedx) fromthe semigroup Z* of non-zerointegersas follows: Set
@,,
y(a) = Z(d) if (a, n) = 1
0 otherwise.
Then it is clear thatx: Z*
C' and has the followingproperties:
(i)
Z(a + n)
=
(ii)
,(aa')
=&)%
(a),
(iii) Z(a)
=
0 if (a, n) # 1,
(iv) x(i)
=
1.
charactermodulo n.
A functionx: E* ?x satisfying
(i)-(iv) is called a numerical
about
such
numerical
characters
was
the
so-calledorthogonalDirichlet'smain result
ityrelations,which assertthe following:
(A)
E Z(a) = 0(n) if xis identically1,
a
0
otherwise,
wherea runsover a completesystemof residuesmodulo n;
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604
[June-July
L. J. GOLDSTEIN
(B)
x
X(a)
= 4(n) if a -1 (mod n),
0
otherwise,
whereX runsoverall numericalcharactersmodulo n. Dirichlet'sideas gave birthto
the moderntheoryof dualityon locallycompactabelian groups.
Dirichiet'ssecondgreatidea wasto associateto each numericalcharactermodulo n
and each real numbers > 1, the followinginfiniteseries
L(s,X)=-
(9)
x(
ns
n=1
a continuousfunction
It is clear thatthe seriesconvergesabsolutelyand represents
for s > 1. However,a more delicate analysisshows that the series(9) converges
a continuousfunctionof s in this
(althoughnot absolutely)fors > 0 and represents
that
is
not
intervalprovided
semi-infinite
identically1. The functionL(s,x) has
X
come to be called a DirichletL-function.
Note thefollowingfactsabout L(s, X): FirstL(s, X) has a productformulaof the
form
L(s, X) =
(10)
(1-
p
p
(s > 1),
F) )
wheretheproductis takenoverall primesp. The proofof (10) is verysimilarto the
of primenumbers.Therefore,
argumentgivenabove in Euler's proofof theinfinity
by (10),
logL(s,X)
(11)
=
-
E
log(1 -
p
m= 1 mp
~~~~~~~~p
I
ps)
Ms
Dirichlet'sidea in provingthe infinitudeof primesin the arithmeticprogression
a, a + n, a + 2n,*, (a, n) = 1, was to imitate,somehow,Euler's proof of the infinitude
of primes,by studyingthefunctionL(s, X)fors near 1. The basic quantityto
consideris
(12)
X
E (a)-'logL(s,X)
X
=
-
E
p
=
-
m=1
Ed Y.
p
m=1
X
X(a) X(Pm)
mm
ms
X(a)
mpjXa
X(pm),
wherewe have used (11). Let a* be an integersuch that aa* =_1 (mod n). Then
X(a*) = X(a)' l by (i)-(iv). Moreover,
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1973]
605
A HISTORY OF THE PRIME NUMBER THEOREM
X Z(a) 'x(pm)=
x
(13)
=
z X(a*ptm)
x
+(n) if a*pm
= 1 (modn)
O otherwise.
However,a*pm
(13), we have
(14)
by (12) and
1 (mod n) is equivalentto pm_ a (mod n). Therefore,
Z x(a)llogL(s,Z)-40(n)
x
z
m-
mPMs
m=im
~~~~~~~~p
n)
pnma(mod
Thus, finally,we have
-
(15)
4>(n)
x(n S Z(a)logL(s,Z)-
p
M
mp
MP
m
m=2
pm = a(modnt)
(s >).
p5
-
p _a(mod n)
many
see thatin orderto prove thatthereare infinitely
From (15), we immediately
primesp _ a (mod n), it is enoughto showthatthefunction
p_a(mod n)
P
S1
tendsto + co as s approaches1 fromtheright.But it is fairlyeasy to see thatas
s-+1+, the sum
1
p
p m-amod
I
n)
m-2
Ms
ItP
to show that
remainsbounded.Thus, it suffices
01
0(n) ,
Z (a)-
log L(s, Z) - + oo
(s
+ )
However,if Xodenotesthecharacterwhichis identically1, thenit is easy to see that
Zn)o(a) 1L(s,Zo)-- + so as
s>
1+ .
it is enoughto show thatif x#Zo, thenlog L(s, x) remainsbounded as
Therefore,
s 1+1. We have alreadymentionedthatL(s, X) is continuousfors > 0 if X# Xo.
Therefore,it sufficesto show that L(1, X)# 0. And this is preciselywhat Dirichlet
showed.
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606
L. J. GOLDSTEIN
[June-July
Dirichlet'stheoremon primesin arithmeticprogressionswas one of the major
achievementsof 19thcenturymathematics,
because it introduceda fertilenew idea
into numbertheory-that analyticmethods(in thiscase the studyof the Dirichlet
L-series)could be fruitfully
applied to arithmetic
problems(in thiscase theproblem
of primesin arithmetic
progressions).To thenovice,such an applicationof analysis
to numbertheorywould seemto be a wasteof time.Afterall, numbertheoryis the
studyof the discrete,whereasanalysisis the studyof the continuous;and what
should one have to do with the other! However,Dirichlet's 1837 paper was the
beginningof a revolutionin number-theoretic
thought,thesubstanceof whichwas to
apply analysisto numbertheory.At first,undoubtedly,mathematicianswere very
uncomfortable
with Dirichlet'sideas. They regardedthemas verycleverdevices,
whichwould eventuallybe supplantedby completelyarithmetic
ideas. For although
analysismightbe usefulin provingresultsabout the integers,surelythe analytic
toolswerenotintrinsic.
Rather,theyenteredthetheoryoftheintegers
in an inessential
wayand could be eliminatedbytheuse of suitablysophisticated
arithmetic.
However,
thehistoryof numbertheoryin the 19thcenturyshowsthatthisidea was eventually
connectionbetweenanalysisand numbertheorycame to
repudiatedand therightful
be recognized.
The firstmajor progresstoward a proof of the prime numbertheoremafter
Dirichletwas due to theRussianmathematician
in two memoirs[12,13]
Tchebycheff
writtenin 1851 and 1852. Tchebycheff
introducedthe followingtwo functionsof a
real variable x:
0(x) =
V(x) =
X
logp,
?
logp,
p<x
pen < X
wherep runsoverprimesand m overpositiveintegers.Tchebycheff
provedthatthe
primenumbertheorem(1) is equivalentto eitherof the two statements
lim 0(x) = l
(16)
x-00
(17)
lim (
xo+
x
)
1.
Moreover,Tchebycheff
provedthatiflimx 0.(0(x) /x)exists,thenitsvalue mustbe 1.
Furthermore,Tchebycheff
proved that
(18)
.92129
= 1 _ lim sup
? lim inf x/log
-() x <_
(x)x < 1.10555.
x /log
methodswere of an elementary,
Tchebycheff's
combinatorialnature,and as such
werenot powerfulenoughto provetheprimenumbertheorem.
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1973]
A HISTORY OF THE PRIME NUMBER THEOREM
607
The firstgiantstridestowarda proofof theprimenumbertheoryweretakenby
B. Riemannin a memoir[10] writtenin 1860. RiemannfollowedDirichletin connectingproblemsof an arithmeticnaturewith the propertiesof a functionof a
continuousvariable.However,whereDirichlet consideredthe functionsL(s, X) as
functionsofa realvariables, Riemanntook thedecisivestepin connecting
arithmetic
of a complexvariable.Riemannintroducedthefollowing
withthetheoryoffunctions
function:
1
(19)
c(n)
n
=1
It is reasonablyeasy to
whichhas come to be known as the Riemannzeta function.
fors in a compactsubset
see thatthe series(19) convergesabsolutelyand uniformly
of thehalf-planeRe (s) > 1. Thus, C(s) is analyticforRe (s) > 1. Moreover,by using
of primes,it is easy
thesame sortof argumentused in Euler's proofof theinfinitude
to provethat
(20)
C(s) =
171(1p
I
p )
(Re (s) > 1),
of
wherethe productis extendedover all primesp. Euler's proofof the infinitude
primessuggeststhatthe behaviorof C(s) for s = 1 is somehowconnectedwiththe
of primes.And, indeed,thisis the case.
distribution
Riemannprovedthat C(s) can be analyticallycontinuedto a functionwhichis
in the whole s-plane.The onlysingularity
of C(s) occursat s = 1 and
meromorphic
the Laurentseriesabout s = 1 looks like
(21)
sa + a1(s-1)
C(s) =
+
Moreover,if we set
(22)
R(s)
=
s(s- 1)7sI2r(S /2)4(s),
thenR(s) is an entirefunctionof s and satisfiesthefunctionalequation
(23)
R(s) = R(1 - s).
To see the immediateconnectionbetweenthe Riemannzeta functionand the
distributionof primes,let us returnto Euler's proof of the infinitudeof primes.
A variationon theidea of Euler's proofis as follows: Suppose thattherewereonly
finitely
manyprimesPl, ', PN. Then by (20), 4(s) would be boundedas s tendsto 1,
whichcontradictsequation (21). Thus, the presenceof a pole of C(s) at s = 1 immediatelyimpliesthatthereare infinitely
manyprimes.But the connectionbetween
the zeta functionand the distributionof primesrunseven deeper.
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608
[June-July
L. J. GOLDSTEIN
Let us considerthe followingheuristicargument:From equation(20), it is easy
to deduce that
(24)
C()
=
E
p
m=1
(Re (s) > 1).
(logp)pmS
Moreover,by residuecalculus,it is easy to verifythat
1-
lrn
(25)
T-o2
2+Tas
J2-iT
-ds-=
_
s
1, x< 1
0,x>1.
we see that
of limitand summationis justified,
assumingthatinterchange
Therefore,
have
forx not equal to an integer,we
lim _
T o2m
+
h-iT
X C'()
S
C(O)
ds
=
=
(26)
E
p
m 1
pm;iX
(logp) lim
T
1 J2
o27ri
~
pm
J-iT
-ds
logp (by equation (25))
= x).
Thus, we see thatthereis an intimateconnectionbetweenthefunctionf(x) and C(s).
This connectionwas firstexploitedby Riemann,in his 1860 paper.
Note that the function
xs C'()
s cs
(27)
has poles at s = 0 and at all zeroesp of C(s). Moreover,note thatby equation (20),
we see that C(s) : 0 forRe (s) > 1. Therefore,all zeroes of C(s) lie in thehalf-plane
Re (s) < 1. Further,since R(s) is entireand C(s) : 0 for Re(s) > 1, the functional
equation (23) implies that the only zeroes of C(s) for which Re (s) < 0 are at
s = -2, -4, -6, -8, ..., and theseare all simplezeroes and are called the trivial
zeroes of C(s) lie in the strip
zeroesof C(s).Thus, we have shownthatall non-trivial
0 ? Re (s) < 1. This stripis called the criticalstrip.The residue of (27) at a nontrivialzero p is
xP
p
Thug,if ai is a largenegativenumber,and if CG,T denotesthe rectanglewithvertices
a + iT, 2 + iT, thenCauchy's theoremimpliesthat
(28)
1
I2+iT
-s
-c
2 2ri
J2iTS
xs
1
(s)ds =--I
(
X)
2
Fic+IT
iT
-T
+
2+iT
.iT
2
+
ff~
_iTj Xs
Is
-iT
C'(s)
-ds
C(S)
+ R(u, T),
whereR(u, T) denotesthesumof theresiduesof thefunction(27) at thepoles inside
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1973]
A HISTORY OF THE PRIME NUMBER THEOREM
609
CG,T. By lettinga and T tend to infinity
and by applyingequations (26) and (28),
Riemannarrivedat thefollowingremarkableformula,known as Riemann'sexplicit
formula
(29)
/X)= x -
X
,,
p -?4'i22)- 1og(1 p
CM0 2
x-2),
wherep runs over all non-trivialzeroes of the Riemannzeta function.Riemann's
formulais surprisingfor at least two reasons.First,it connectsthe function*(x),
which is connectedwith the distributionof primes,with the distributionof the
zeroes of the Riemannzeta function.That thereshould be any connectionat all is
amazing. But, secondly,the formula(29) explicitlyputs in evidencea formof the
primenumbertheoremby equating*(x) withx plus an errortermwhichdepends
on thezeroesof thezeta function.If we denotethiserrortermby E(x), thenwe see
thatthe primenumbertheoremis equivalentto theassertion
(30)
lim
x+ 00
x
0,
=
which,in turn,is equivalentto the assertion
(31)
lim
X-c
x
E
p
p
= 0.
Riemannwas unableto prove(31), but he made a numberof conjecturesconcerning
of thezeroesp fromwhichthestatement
thedistributions
(31) followsimmediately.
The most famous of Riemann's conjecturesis the so-called Riemannhypothesis,
whichassertsthatall non-trivial
zeroesof C(s) lie on theline Re (s) = i, whichis the
line of symmetry
of the functionalequation (23). This conjecturehas resistedall
attemptsto proveit formorethana centuryand is one of themostcelebratedopen
problemsin all of mathematics.However,if the Riemannhypothesisis true,then
'XPI
1- =
I
I I
and fromthisfactand equation (29), it is possibleto provethat
(32)
t(x) = x + O(xJ+)
foreverye > 0, whereO(xi + ?) denotesa functionf(x)suchthatf(x)/xi? is bounded
forall largex. Thus,theRiemannhypothesis
implies(31) in a trivialway,and hence
the primenumbertheoremfollowsfromthe Riemannhypothesis.What is perhaps
is thefactthatif(32) holdsthentheRiemannhypothesis
morestriking
is true.Thus,
the primenumbertheoremin the sharp form(32) is equivalentto the Riemann
thattheconnectionbetweenthezeta functionand the
hypothesis.We see, therefore,
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610
L. J. GOLDSTEIN
[June-July
distribution
ofprimesis no accidentalaffair,
butsomehowis wovenintothefabricof
nature.
In his memoir,Riemannmade many otherconjectures.For example,if N(T)
denotesthe numberof non-trivial
zeroesp of C(s) such that - T ? Im(p) _ T, then
Riemannconjecturedthat
(33)
N(T)
=
1
-T
27t
logT-
1+log (27t)
+ TO(log T).
27t
The formula(33) was firstprovenby von-Mangoldtin 1895 [14]. An interesting
line
of researchhas been involvedin obtainingestimatesforthe numberof non-trivial
zeroes p on the line Re(s) = -. Let M(T) denotethe numberof p such thatRe(s)
j, - T< Im(s) ? T. ThenHardy[6] in 1912,provedthatM(T) tendsto infinity
as
T tendsto infinity.
Later,Hardy[7] improvedhisargument
to provethatM(T) > AT,
whereA is a positiveconstant,not dependingon T. The ultimateresultof thissort
was obtainedby AtleSelbergin 1943[11]. He provedthatM(T) > ATlog T forsome
positiveconstantA. In viewofequation(33), Selberg'sresultshowsthata positivepercentage of thezeroesof C(s) actuallylie on theline Re(s) = i. This resultrepresents
thebestprogressmade to date in attempting
to prove the Riemannhypothesis.
it
Fortunately, is not necessaryto prove the Riemann hypothesisin order to
provetheprimenumbertheoremin theform(17). However,it is necessaryto obtain
about thedistribution
someinformation
of thezeroesof C(s). Such information
was
obtained independently
by Hadamard [5] and de la Vallee Poussin [1] in 1896,
therebyprovidingthefirstcompleteproofsof theprimenumbertheorem.Although
in detail,theybothestablishtheexistenceof a zero-free
theirproofsdiffer
regionfor
C(s),the existenceof whichservesas a substituteforthe Riemannhypothesesin the
reasoningpresentedabove. More specifically,
theyprovedthatthereexist constants
a, to such that C(o + it) : 0 if ? 1 - 1/a log It|, I t| to. This zero-freeregion
allows one to prove the primenumbertheoremin the form
-
(34)
= x + O(xe- C(log X) 1/14)
tfr(x)
Please note,however,thattheerrortermin (34) is muchlargerthantheerrorterm
predictedby the Riemannhypothesis.
Thus,theprimenumbertheoremwas finallyproved aftera centuryof hardwork
by manyof theworld'sbestmathematicians.
It is grosslyunfairto attribute
proofof
sucha theoremto thegeniusof a singleindividual.For, as we have seen,each stepin
the directionof a proof was conditionedhistoricallyby the work of preceding
generations.On the otherhand, to denythatthereis geniusin th&workwhichled
up to the ultimateproofwould be equally unfair.For at each step in the chain of
discovery,brilliantand fertileideas werediscovered,and providedthe materialout
of whichto fashionthe nextlink.
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1973]
611
A HISTORY OF THE PRIME NUMBER THEOREM
APPENDIX A: Samples from Gauss' Tables. TABLE 1
1
168
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
(Werke,II, p. 436)
26
135
127
120
119
114
117
107
110
112
106
103
109
105
102
108
98
104
94
102
98
104
100
1C4
94
98
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
101
94
98
92
95
92
106
100
94
92
99
94
90
96
88
101
102
85
96
86
90
95
89
98
The frequencyof primes.TABLE 2 (Werke,II, p. 443) 2000000..3000000
210
0
1
2
3
4
5
6
7
8
9
10
11
12
13
3
10
32
69
119
197
204
157
115
63
21
8
2
14
15
16
220
2
9
27
69
146
183
201
168
109
52
18
9
4
230
2
9
29
73
138
179
205
168
113
44
30
10
3
240
4
11
32
86
136
176
194
158
112
55
28
7
1
250
1
9
37
78
147
193
189
151
102
58
23
7
5
260
270
3
5
1
4
10
35
88
136
194
180
170
88
58
24
13
6
28
71
158
195
201
142
96
53
22
17
1
280
2
7
43
95
135
195
188
145
87
67
24
9
2
1
17
290
300
2
15
2
13
30
85
140
179
222
132
109
53
18
8
5
44
64
153
187
214
134
103
58
15
11
1
2
6857
6849
6787
6766
6804
6762
6714
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337
778
1408
1878
1998
1525
1034
561
223
99
27
1
6
1
6705
6862
1
6874
1
25
98
1
6744
612
[June-July
L. J. GOLDSTEIN
APPENDIX
B: Gauss' Letterto Enke.
My distinguished
friend:
Your remarks
concerning
thefrequency
ofprimeswereofinterest
to mein morewaysthanone.
me of myownendeavorsin thisfieldwhichbeganin theverydistantpast,in
You havereminded
1792or 1793,afterI had acquiredtheLambertsupplements
to thelogarithmic
tables.Evenbefore
intohigherarithmetic,
one of myfirstprojectswas
I had begunmymoredetailedinvestigations
to turnmyattention
to thedecreasing
of primes,to whichend I countedtheprimesin
frequency
severalchiliads(intervals
of length1000; Trans.)and recordedthe resultson theattachedwhite
thatbehindall ofitsfluctuations,
thisfrequency
pages.I soonrecognized
is on theaverageinversely
to thelogarithm,
so thatthenumberofprimesbelowa givenboundn is approximately
proportional
equal to
I
dn
n
Jlog
is understood
wherethelogarithm
to be hyperbolic.
Lateron, whenI becameacquaintedwiththe
listin Vega,'stables(1796)goingup to 400031,I extended
that
mycomputation
further,
confirming
In 1811,theappearanceofChernau'scribrum
estimate.
gavememuchpleasureandI havefrequently
(sinceI lack thepatiencefora continuous
count)spentan idlequartr ofan hourto countanother
chiliadhereand there;althoughI eventually
gaveit up withoutquite gettingthrougha million.
Onlysometimelaterdid I makeuse of thediligenceof Goldschmidt
to fillsomeof theremaining
thecomputation
gaps in thefirstmillionand tocontinue
to Burkhardt's
tables.Thus(for
according
manyyearsnow) thefirstthreemillionhave beencountedand checkedagainstthe integral.A
smallexcerptfollows:
TABLE
Below
500000
1000000
1500000
2000000
2500000
3000000
Here are
Prime
41556
78501
114112
148883
183016
216745
A
Integral
ln
Error
log n
41606.4+
79627.5+
114263.1+
149054.8+
183245.0+
216970.6+
50.4
126.5
151.1
171.8
229.0
225.6
Your
Formula
Error
41596.9+ 40.9
78672.7-+ 171.7
114374.0+ 264.0
149233.0+ 350.0
183495.1+ 479.1
217308.5+ 563.5
I was notawarethatLegendrehad also workedon thissubject;yourlettercausedmeto look
in his Thioriedes Nombres,
and in thesecondeditionI founda fewpageson thesubjEctwhichI
musthavepreviously
overlooked(or,bynow,forgotten).
Legendreusedtheformula
n
log n -A
whereA is a constant
whichhe setsequal to 1.08366.After
a hastycomputation,
I findin theabove
cases the deviations
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19731-
A HISTORYOF THE PRIME NUMBERTHEOREM
613
TABLE B
23,3
+ 42,2
? 68,1
-
1- 92,8
+159,1
+167,6
buttheyseemto growfasterwithi
are evensmallerthanthosefromtheintegral,
Thesedifferences
so thatit is quitepossibletheymaysurpassthem.To makethecountand theformulaagree,one
numbers:
insteadof A = 1.08366,thefollowing
wouldhaveto use,respectively,
TABLE C
1,09040
1,07682
1,07582
1,07529
1,07179
1,07297
I darenotconjecn,the(average)valueofA decreases;however,
It appearsthat,withincreasing
from1. I cannotsay that
is I or a numberdifferent
thelimitas n approachesinfinity
turewhether
forexpecting
a verysimplelimiting
value;on theotherhand,theexcessof
thereis anyjustification
oftheorderof1/logn.I wouldbe inclinedto believethatthe differwellbe a quantity
A over1 might
thanthefuLnction
mustbe simpler
itself.
entialofthefunction
would suggestthatthedifferential
forhe function,
Legendre'sformula
n is postulated
If dn/log
of theformdnl(log n- (A -1)). By theway,forlargen,yourformightbe something
function
mula could be consideredto coincidewith
n
log n-(i
/2k)
if we put A = 1/2k
formula,
thatis,withLegendre's
wherek is themodulusofBrigg'slogarithms;
-
1.1513.
between
yourcountsandmine.
Finally,I wanttoremarkthatI noticeda coupleofdisagreements
Between 59000 and 60000,you have 95, whileI have 94
94
93.
102000
101000
theprime59023
possiblyresultsfromthefactthat,in Lambert'sSupplement,
The firstdifference
with
is virtually
crawling
occurstwice.The chiliadfrom101000- 102000in Lambert'sSupplement
errors;in mycopy,I have indicatedsevennumberswhichare notprimesat all, and suppliedtwo
missingones.Woulditnotbe possibleto induceyoungMr.Dase tocounttheprimesinthefollowing
forpublicdisusingthetablesat theAcademywhich,I am afraid,are notintended
(few)millions,
to my
In thiscase,letme remarkthatin the2ndand3rdmillion,thecountis,according
tribution?.
million.
thefirst
haveemployedin counting
based on a specialschemewhichI myself
instructions,
on a singlepage in 10 columns,each columnbelonging
The countsforeach 100000are irndicated
10000;Trans.);an additionalcolumnin front(left)and another
to one myriad(an interval
oflength
columns
columnandthetwoadditional
iton theright;forexamplehereisa vertical
columnfollowing
forthe interval10000000to 11000000 --
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614
[June-July
L. J. GOLDSTEIN
takethefirst
verticalcolumn.In themyriad1000000to 1010000thereare 100
As an illustration,
a singleprime,noneconHecatontades;(intervals
oflength
100; Trans.)amongthemone containing
foureach; elevencontaining
5 each,etc.,yielding
altotainingtwoor threeprimes;twocontaining
gether752= 1.1 + 4.2 + 5.11 + 6.14 + * primes.The last columncontainsthetotalsfromthe
verticalcolumnare superfluous
otherten.The numbers14,15, 16 in thefirst
sinceno hecatontades
thatmanyprimes;but on the following
occurcontaining
pages theyare needed.Finallythe 10
pages are againcombinedintoone and thuscomprisetheentiresecondmillion.
Withmostcordialwishesforyourgood health
It is hightimeto quit---.
Yours, as ever,
C. F. Gauss
24 December1849.
Gbttingen,
APPENDIX C:
THOUSANDS
20
159
199
206
245
289
290
334
352
354
500
UP
Corrections
to Gauss' Tables
GAUSS
ACTUAL
102
87
96
85
78
85
84
80
80
79
104
77
86
83
88
77
85
81
81
76
TO
41,556
78,501
114,112
148,883
183,016
216,745
-2
d-10
+10
+2
-10
+8
-1
-1
-1
+3
+18
HERE
A
TOTALS
500,000
1,000,000
1,500,000
2,000,000
2,500,000
3,000,000
A
41,538
78,498*
114,156*
148,934*
*
183,073
216,817*
+18
+3
-44
-51
-57
-72
* from
ListofPrimeNumbers
from1 to 10,006,771,
by D. N. Lehmer,(adjusted:He counts1 as
a prime).
totheHistory
Researchsupported
ofMathebyNSF GrantGP 31280X.Thisarticlewaspresented
of Marylandon March20,1972.Theauthorwishesto thankPromaticsSeminarat theUniversity
Ehrlichforpreparing
thetranslation
ofGauss' letterwhichappearsin AppendixB.
fessorGertrude
thecalculations
He also wishesto thankMr.EdwardKornforproviding
ofAppendixC.
References
1. Ch.de la ValleePoussin,Recherches
surla theorie
desnombres
analytiques
premiers.
Premiere
en general.Deuxiemepartie:Les foncpartie.La fonction
C(s)de Riemannet les nombres
premiers
This content downloaded on Sat, 9 Feb 2013 20:07:45 PM
All use subject to JSTOR Terms and Conditions
1973]
615
DIFFERENTIATION UNDER THE INTEGRAL SIGN
tionsde Dirichlet
et lesnombres
de la formelineaireMx+N. Troisieme
premiers
partie:Les formes
quadratiques
de determinant
negatif,
Ann.Soc. Sci. Bruxelles,
20 (1896)183-256,281-397.
2. L. Dirichlet,Uber den Satz: das jede unbegrenzte
arithmetische
Progression,
derenerstes
Glied und Differenz
keinengemeinschaftlichen
Factorsind,unendlichen
vielePrimzahlen
enthalt,
1837; Mathematische
Abhandlungen,
Bd. 1, (1889) 313-342.
3. C. F. Gauss,TafelderFrequenzderPrimzahlen,
Werke,11(1872)436-442.
4. , Gauss an Enke, Werke, 11(1872) 444-447.
5. J.Hadamard,Surla distribution
des zerosde la fonction
arithmetiC(s) et ses consequences
ques,Bull. Soc. Math.de France,24 (1896) 199-220.
6. G. H. Hardy,Sur les zerosde la fonctionC(s) de Riemann,ComptesRendus,158 (1914)
1012-14.
7.
The zerosofRiemann'szetafunction
, andJ.E. Littlewood,
on thecriticalline,Math.
Zeit.,10 (1921)283-317.
8. A. M. Legendre,Essai surla theoriede Nombres,1sted., 1798,Paris,p. 19.
9. , Essai surla Theoriede Nombres,2nd ed. 1808,Paris,p. 394.
10. B. Riemann,fiberdie AnzahlderPrimzahlen
untereinergegebenen
Grosse,Gesammelte
Mathematische
Werke,2ndAufl.,(1892) 145-155.
11. A. Selberg,On thezerosof Riemann'szeta function,
Skr.NorskeVid. Akad.,Oslo (1942)
no. 10.
12. P. Tchebycheff,
Sur la fonction
la totalitede nombrespremiersinferieurs
qui determine
A une limitedonnee,Oeuvres,1 (1899)27-48.
13.
-, Memoiresurles nombrespremiers,
Oeuvres,1 (1899)49-70.
14. H. von Mangoldt,Auszugaus einerArbeitunterdem Titel: Zu Riemann'sAbhandlung
untereinergegebenenGrbsse,Sitz. Konig. Preus.Akad. Wiss.
uberdie Anzahlder Primzahlen
zu Berlin,(1894) 337-350,883-895.
DIFFERENTIATION
UNDER THE INTEGRAL SIGN*
HARLEY FLANDERS, Tel-AvivUniversity
+(fh(t)
1. Introduction.
Everyoneknowsthe Leibniz rulefordifferentiating
an integral:
(1.1)
gt
=
(t)
)
F(x, t) dx
-F[h(t),t(t)
-F[g(t), t]g(t) +
()
at
dx.
We are all fondof thisformula,althoughit is seldomifeverused in suchgenerality.
Usually,eitherthelimitsare constants,or the integrandis independentof thetime
t. Frequent cases are
dt
F(x) dx = F(t), dt fF(x,
t)dx=
f
(t),
dx.
PresentedMay 5, 1972to theRockyMountainSectionmeeting,SouthernColorado State
College,Pueblo,CO.
*
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