A History of the 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 Stable URL: http://www.jstor.org/stable/2319162 . Accessed: 09/02/2013 20:07

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This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions A HISTORY OF THE PRIME NUMBER THEOREM

L. J. GOLDSTEIN, Universityof Maryland

The sequenceof primenumbers, which begins 2, 3, 5, 7, 11,13, 17, 19, 23, 29, 31, 37, has held untold fascinationfor mathematicians,both professionalsand amateurs alike. The basic theoremwhich we shall discussin thislecture is knownas theprime numbertheorem and allows one to predict,at leastin grossterms, the way in which theprimes are distributed.Let x be a positivereal number,and let 7r(x)= thenumber of primes

(1) lim (X) = 1, X_0x/logx wherelog x denotesthe naturallog of x. In otherwords, the prime number theorem assertsthat

(2) ( )0log ? (logx) (x- oo), whereo(x/logx) standsfor a functionf(x) withthe property

lim f(x) = 0 l +, x/logx Actually,for reasons whichwill become clear later,it is much betterto replace(1) and (2) by the followingequivalent assertion:

(3) 7t(x)r(x) = Jb doy +?O(lox) log y- log x) To prove that(2) and (3) are equivalent,it sufficesto integrate [x dy

12 log y once by partsto get

(4) f logy logx log2 + log2y

LarryGoldstein received his PrincetonPh.D. underG. Shimura.After a Gibbs instructorship at Yale, he joinedthe Univ. of M arylandas AssociateProfessor and nowis Professor.His research is inAnalytic and AlgebraicNumber Theory and Automorphic Functions. He is theauthor of Analy- tic (Prentice-Hall 1971), and AbstractAlgebra, A First Course (Prentice-Hall, to appear). Editor.

599

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However,for x > 4,

fX dy 'u' dy x dy J2 y l2y J2 + J2_log2x- y

- I 1 1 (5) < 1/x log2 l (1)

- x 0 Vlogx ' wherewe have used the fact that 1 /log2xis monotonedecreasing for x > 1. It is clearthat (4) and (5) show that(2) and (3) are equivalentto one another.The advan- tage of the version(3) is thatthe function

Li(x) = fx 5 ' called the logarithmicintegral, provides a muchcloser numericalapproximation to ir(x)than does x /logx. This is a ratherdeep factand we shall returnto it. In thislecture, I shouldlike to explorethe history of theideas whichled up to the primenumber theorem and to its proof,which was not supplieduntil some 100 years afterthe first conjecture was made. The historyof the prime number theorem provides a beautifulexample of the way in whichgreat ideas developand interrelate,feeding upon one anotherultimately to yield a coherenttheory which rathercompletely explains observedphenomena. The veryconception of a primenumber goes back to antiquity,although it is not possible to say preciselywhen the conceptfirst was clearlyformulated. However, a numberof elementaryfacts concerning the primes were known to the Greeks.Let us cite threeexamples, all of whichappear in 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. (ii) There exist infinitelymany primes. (iii) The primes may be effectivelylisted using the so-called "sieve of Eratosthenes". We willnot commenton (i), (iii) anyfurther, since they are partof thecurriculum of most undergraduatecourses in numbertheory, and hence are probablyfamiliar to mostof you. However,there is a proofof (ii) whichis quitedifferent from Euclid's well-knownproof and whichis verysignificant to the historyof the primenumber theorem.This proofis due to the Swiss mathematicianLeonhard Euler and dates fromthe middleof the 18thcentury. It runsas follows: Assumethat PI, , PN is a completelist of all primes,and considerthe product

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(6) E ( P ) .1(1+H p+T2+.. ' Since everypositive n can be writtenuniquely as a productof primepowers, everyunit fraction1/n appears in the formalexpansion of the product(6). For ... example,if n = p'l p'N then 1/n occurs frommultiplying the terms

1a/ll 1 /pa2, .. lpa,

Therefore,if R is any positiveinteger,

N 1 -1 R (7) I > E,1 In. a-1ru (-' Pi 11=1

However,as R -+ oo, the sum on the righthand side of (7) tendsto infinity,which contradicts(7). Thus, P1, *', PN cannot be a completelist of all primes.We should make two commentsabout Euler's proof: First,it linksthe FundamentalTheorem of Arithmeticwith the infinitudeof primes.Second, it uses an analyticfact, namely the divergenceof the harmonicseries, to conclude an arithmeticresult. It is this latterfeature which became the cornerstoneupon which much of 19th century numbertheory was erected. The firstpublished statement which came close to the primenumber theorem was due to Legendrein 1798 [8]. He assertedthat 7r(x) is of theform x /(Alog x + B) for constantsA and B. On the basis of numericalwork, Legendre refined his con- jecturein 1808 [9] by assertingthat

= logx + A(x)'

where A(x) is "approximately1.08366 ...". Presumably,by this latterstatement, Legendremeant that lim A(x) = 1.08366. x -I 00

It is preciselyin regardto A(x), whereLegendre was in error,as we shall see below. In his memoir[9] of 1808, Legendreformulated another famous conjecture. Let k and 1 be integerswhich are relativelyprime to one another.Then Legendreasserted thatthere exist infinitely many primes of theform 1 + kn(n = 0, 1,2, 3, ...). In other words,if 7rk,l(x)denotes the numberof primesp of the form1 + kn forwhich p < x, then Legendreconjectured that

(8) 7tk,l(x) -+ oo as x -+ cc.

Actually,the proofof (8) by Dirichletin 1837 [2] providedseveral crucial ideas on how to approachthe primenumber theorem.

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AlthoughLegendre was the firstperson to publish a conjecturalform of the primenumber theorem, Gauss had alreadydone extensivework on the theoryof primesin 1792-3. EvidentlyGauss consideredthe tabulationof primesas some sort of pastimeand amused himselfby compilingextensive tables on how the primes distributethemselves in variousintervals of length1009. We have included some of Gauss' tabulationsas an Appendix.The firsttable, excerpted from [3, p. 436], covers the primesfrom 1 to 50,000.Each entryin the table representsan intervalof length 1000.Thus, for example, there are 168 primesfrom 1 to 1000; 135 from1001 to 2000; 127 from3001 to 4000; and so forth.Gauss suspectedthat the densitywith which primesoccured in theneighborhood of theinteger n was 1 /logn, so thatthe number of primesin the interval[a, b) should be approximatelyequal to Tb dx Jalogx' 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,between 2,600,000 and 2,700,000,Gauss found 6762 primes,whereas 2,700.000 dx -_ = 6761.332. 2,600,000 log x Gauss neverpublished his investigationson the distributionof primes.Never- theless,there is littlereason to doubt Gauss' claim thathe firstundertook his work in 1792-93, well beforethe memoirof Legendrewas written.Indeed, there are severalother known examples of resultsof the firstrank whichGauss proved,but nevercommunicated to anyoneuntil years after the originalwork had been done. This was the case, for example,with the ellipticfunctions, where Gauss preceded Jacobi,and withRiemannian geometry, where Gauss anticipatedRiemann. The only informationbeyond Gauss' tables concerningGauss' work in the distributionof primesis containedin an 1849 letterto the astronomerEncke. We have includeda translationof Gauss' letter. In his letterGauss describeshis numericalexperiments and his conjecturecon- cerningi(x). There are a numberof remarkablefeatures of Gauss' letter.On the second page of the letter,Gauss compareshis approximationto 2r(x),namely Li(x), with Legendre'sformula. The resultsare tabulatedat the top of the second page and Gauss' formulayields a muchlarger numerical error. In a veryprescient statement, Gauss defendshis formulaby notingthat althoughLegendre's formulayields a smallererror, the rateof increaseof Legendre'serror term is muchgreater than his own. We shall see below thatGauss anticipatedwhat is known as the "."Another feature of Gauss' letteris that he casts doubt on Legendre's assertionabout A(x). He assertsthat the numericalevidence does not supportany conjectureabout the limitingvalue of A(x).

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Gauss' calculations are awesome to contemplate,since they were done long before the days of high-speedcomputers. Gauss' persistenceis most impressive. However,Gauss' tables are not error-free.My student,Edward Korn, has checked Gauss' tables using an electroniccomputer and has found a numberof errors.We include the correctedentries in an appendix. In spite of these (remarkablyfew) errors,Gauss' calculationsstill provide overwhelming evidence in favorof theprime numbertheorem. Modern studentsof mathematicsshould take note of the great care with whichdata was compiledby such giants as Gauss. Conjecturesin those days were rarelyidle guesses. They were usuallysupported by piles of laboriously gatheredevidence. The next step toward a proof of the prime numbertheorem was a step in a completelydifferent direction, and was takenby Dirichletin 1837 [2]. In a beautiful memoir,Dirichlet proved Legendre's conjecture(8) concerningthe infinitudeof primesin an arithmeticprogression. Dirichlet's work containedtwo radicallynew ideas, which we should discuss in some detail.

Let @,, denotethe ringof residueclasses modulo n, and let En denotethe group

of unitsof 7n. Then En is the so-called "group of reducedresidue classes modulo n" and consistsof thoseresidue classes containingan elementrelatively prime to n. If k is an integer,let us denoteby k its residueclass modulo n. Dirichlet'sfirst brilliant idea was to introducethe characters of the group En; thatis, the homomorphismsof

En into the multiplicativegroup Cx of non-zerocomplex numbers.If x is such a character,then we may associate withx a function(also denotedx) fromthe semi- group Z* of non-zerointegers as follows: Set y(a) = Z(d) if (a, n) = 1 0 otherwise. Then it is clear that x: Z* C' and has the followingproperties:

(i) Z(a + n) = (a), (ii) ,(aa') =&)% (iii) Z(a) = 0 if (a, n) # 1,

(iv) x(i) = 1.

A functionx: E* ?x satisfying(i)-(iv) is called a numericalcharacter modulo n. Dirichlet'smain resultabout such numericalcharacters was the so-calledorthogonal- ity relations,which assertthe following:

(A) E Z(a) = 0(n) if xis identically1, a 0 otherwise, wherea runsover a completesystem of residuesmodulo n;

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(B) X(a) = 4(n) if a -1 (mod n), x 0 otherwise,

whereX runsover all numericalcharacters modulo n. Dirichlet'sideas gave birthto the moderntheory of dualityon locallycompact abelian groups. Dirichiet'ssecond great idea was to associateto each numericalcharacter modulo n and each real numbers > 1, the followinginfinite series x( (9) L(s,X)=- ns n=1

It is clear thatthe seriesconverges absolutely and representsa continuousfunction for s > 1. However,a more delicate analysis shows that the series (9) converges (althoughnot absolutely)for s > 0 and representsa continuousfunction of s in this semi-infiniteinterval provided that X is not identically1. The functionL(s,x) has come to be called a DirichletL-function. Note the followingfacts about L(s, X): FirstL(s, X) has a productformula of the form

(10) L(s, X) = (1- F) ) (s > 1), p p wherethe product is takenover all primesp. The proofof (10) is verysimilar to the argumentgiven above in Euler's proofof the infinityof primenumbers. Therefore, by (10),

logL(s,X) = - E log(1 - ps) (11) ~~~~~~~~p

I Ms p m= 1 mp 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 in- finitudeof primes,by studyingthe function L(s, X)for s near 1. The basic quantityto consideris

XE (a)-'logL(s,X) = - E X(a) X(Pm) (12) X p m=1 X mm

= - Ed Y. ms X(a) X(pm), p m=1 mpjXa 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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X Z(a) 'x(pm)= z X(a*ptm) x x (13) = +(n) if a*pm = 1 (modn) O otherwise.

However,a*pm 1 (mod n) is equivalentto pm_ a (mod n). Therefore,by (12) and (13), we have

(14) Z x(a)llogL(s,Z)-40(n) z m- mPMs x ~~~~~~~~pm=im pnma(mod n) Thus, finally,we have

M - 4>(n)x(n S Z(a)logL(s,Z)- p mm=2 MPmp (15) pm = a(modnt)

- p5 (s >).

p _a(mod n) From (15), we immediatelysee thatin orderto prove thatthere are infinitelymany primesp _ a (mod n), it is enoughto show thatthe function

S1 p_a(mod n) P tendsto + co as s approaches1 fromthe right.But it is fairlyeasy to see that as s-+1+, the sum 1

I Ms p m-2 ItP p m-amod n) remainsbounded. Thus, it sufficesto show that

- 01 Z (a)- log L(s, Z) + oo (s + ) 0(n) , However,if Xodenotes the character which is identically1, thenit is easy to see that

Zn)o(a) 1L(s,Zo)-- + so as s> 1+ .

Therefore,it is enoughto show thatif x # Zo, thenlog L(s, x) remainsbounded as s 1+1. We have alreadymentioned that L(s, X) is continuousfor s > 0 if X# Xo. Therefore,it sufficesto show that L(1, X)# 0. And this is preciselywhat Dirichlet showed.

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Dirichlet'stheorem on primesin arithmeticprogressions was one of the major achievementsof 19thcentury mathematics, because it introduceda fertilenew idea into numbertheory -that analyticmethods (in thiscase the studyof the Dirichlet L-series)could be fruitfullyapplied to arithmeticproblems (in thiscase the problem of primesin arithmeticprogressions). To the novice,such an applicationof analysis to numbertheory would seem to be a waste of time.After all, numbertheory is the studyof the discrete,whereas analysis is 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-theoreticthought, the substance of whichwas to apply analysisto numbertheory. At first,undoubtedly, mathematicians were very uncomfortablewith Dirichlet's ideas. They regardedthem as veryclever devices, whichwould eventuallybe supplantedby completelyarithmetic ideas. For although analysis mightbe usefulin provingresults about the ,surely the analytic toolswere not intrinsic. Rather, they entered the theory of the integers in an inessential wayand could be eliminatedby theuse of suitablysophisticated arithmetic. However, the historyof numbertheory in the 19thcentury shows that this idea was eventually repudiatedand therightful connection between analysis and numbertheory came to be recognized. The firstmajor progresstoward a proof of the prime numbertheorem after Dirichletwas due to theRussian mathematician Tchebycheff in two memoirs[12,13] writtenin 1851 and 1852. Tchebycheffintroduced the followingtwo functionsof a real variable x: 0(x) = X logp, p

V(x) = ? logp, pen < X wherep runsover primesand m over positiveintegers. Tchebycheff proved that the primenumber theorem (1) is equivalentto eitherof the two statements

(16) lim 0(x) = l x-00

(17) lim ( ) 1. xo+ x Moreover,Tchebycheff proved that if limx 0. (0(x) /x)exists, then its value mustbe 1. Furthermore,Tchebycheff proved that

(18) .92129 lim inf = 1 lim sup < 1.10555. ? x/log-() x <_ _ x /log(x) x Tchebycheff'smethods were of an elementary,combinatorial nature, and as such werenot powerfulenough to provethe prime number theorem.

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The firstgiant strides toward a proofof the primenumber theory were taken by B. Riemannin a memoir[10] writtenin 1860. Riemannfollowed Dirichlet in con- nectingproblems of an arithmeticnature with the propertiesof a functionof a continuousvariable. However, where Dirichlet consideredthe functionsL(s, X) as functionsof a real variables, Riemanntook thedecisive step in connectingarithmetic withthe theory of functionsof a complexvariable. Riemann introduced the following function: 1

(19) c(n) =1 n

whichhas come to be known as the Riemannzeta function.It is reasonablyeasy to see thatthe series(19) convergesabsolutely and uniformlyfor s in a compactsubset of the half-planeRe (s) > 1. Thus, C(s) is analyticfor Re (s) > 1. Moreover,by using the same sortof argumentused in Euler's proofof theinfinitude of primes,it is easy to provethat

I (20) C(s) = 171(1- p ) (Re (s) > 1), p wherethe productis extendedover all primesp. Euler's proofof the infinitudeof primessuggests that the behaviorof C(s) for s = 1 is somehowconnected with the distributionof primes.And, indeed,this is the case. Riemannproved that C(s) can be analyticallycontinued to a functionwhich is meromorphicin the whole s-plane.The only singularityof C(s) occursat s = 1 and the Laurentseries about s = 1 looks like

(21) C(s) = sa + a1(s-1) +

Moreover,if we set

(22) R(s) = s(s- 1)7sI2r(S /2)4(s),

thenR(s) is an entirefunction of s and satisfiesthe functional equation

(23) R(s) = R(1 - s).

To see the immediateconnection between the Riemannzeta functionand the distributionof primes,let us returnto Euler's proof of the infinitudeof primes. A variationon the idea of Euler's proofis as follows: Suppose thatthere were only finitelymany primes Pl, ', PN. Then by (20), 4(s) would be boundedas s tendsto 1, whichcontradicts equation (21). Thus, the presenceof a pole of C(s) at s = 1 im- mediatelyimplies that there are infinitelymany primes. But the connectionbetween the zeta functionand the distributionof primesruns even deeper.

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Let us considerthe followingheuristic argument: From equation (20), it is easy to deduce that

(24) = E (logp)pmS (Re (s) > 1). C() p m=1 Moreover,by residuecalculus, it is easy to verifythat

1 2+Tas _ 1, x< 1 (25) lrn - -ds-= T-o2 J2-iT s 0,x>1. Therefore,assuming that interchange of limitand summationis justified,we see that forx not equal to an integer,we have

lim _ + X C'() ds = E (logp) lim 1 J2 -ds T o2m h-iT S C(O) p m 1 T o27ri J-iT pm~ (26) = logp (by equation (25)) pm;iX

= x). Thus, we see thatthere is an intimateconnection between the functionf(x) and C(s). This connectionwas firstexploited by Riemann,in his 1860 paper. Note that the function

(27) xs C'() s cs has poles at s = 0 and at all zeroes p of C(s). Moreover,note thatby equation (20), we see that C(s) : 0 for Re (s) > 1. Therefore,all zeroes of C(s) lie in the half-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 zeroesof C(s).Thus, we have shownthat all non-trivialzeroes of C(s) lie in the strip 0 ? Re (s) < 1. This stripis called the criticalstrip. The residue of (27) at a non- trivialzero p is xP p

Thug,if ai is a largenegative number, and if CG,T denotesthe rectanglewith vertices a + iT, 2 + iT, then Cauchy's theoremimplies that

1 I2+iT xs 1 Fic+IT 2+iT + ff~_iTj Xs C'(s) (28) -s-c (s) ds =--I + I- -ds + R(u, T), 2 2riJ2iTS ( X) 2 -Ti T .iT 2 -iT s C(S) whereR(u, T) denotesthe sum of theresidues of thefunction (27) at thepoles inside

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CG,T. By lettinga and T tend to infinityand by applyingequations (26) and (28), Riemannarrived at thefollowing remarkable formula, known as Riemann'sexplicit formula

(29) /X)= x - X p -?4'i22)- 1og(1 - x-2), ,, p CM0 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 primenumber theorem by equating*(x) withx plus an errorterm which depends on the zeroes of the zeta function.If we denotethis error term by E(x), thenwe see thatthe primenumber theorem is equivalentto the assertion

(30) lim = 0, x+ 00 x which,in turn,is equivalentto the assertion

(31) lim - E = 0. X-c x p p

Riemannwas unable to prove(31), but he made a numberof conjecturesconcerning the distributionsof the zeroesp fromwhich the statement(31) followsimmediately. The most famous of Riemann's conjecturesis the so-called Riemannhypothesis, whichasserts that all non-trivialzeroes of C(s) lie on theline Re (s) = i, whichis the line of symmetryof the functionalequation (23). This conjecturehas resistedall attemptsto proveit formore than a centuryand is one of the mostcelebrated open problemsin all of mathematics.However, if the Riemannhypothesis is true,then

'XPI I 1- = I I

and fromthis factand equation (29), it is possibleto prove that

(32) t(x) = x + O(xJ+)

forevery e > 0, whereO(xi + ?) denotesa functionf(x)such thatf(x) /xi ? is bounded forall largex. Thus, the Riemannhypothesis implies (31) in a trivialway, and hence the primenumber theorem follows from the Riemannhypothesis. What is perhaps morestriking is thefact that if (32) holds thenthe Riemann hypothesis is true.Thus, the prime numbertheorem in the sharp form(32) is equivalentto the Riemann hypothesis.We see, therefore,that the connectionbetween the zeta functionand the

This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions 610 L. J. GOLDSTEIN [June-July distributionof primesis no accidentalaffair, but somehowis woveninto the fabric of nature. In his memoir,Riemann made many otherconjectures. For example,if N(T) denotesthe numberof non-trivialzeroes p of C(s) such that - T ? Im(p) _ T, then Riemannconjectured that

1 1+log (27t) (33) N(T) = -T + 27t logT- 27t TO(log T).

The formula(33) was firstproven by von-Mangoldtin 1895 [14]. An interestingline of researchhas been involvedin obtainingestimates for the numberof non-trivial zeroes p on the line Re(s) = -. Let M(T) denote the numberof p such that Re(s) - j, - T< Im(s) ? T. ThenHardy [6] in 1912,proved that M(T) tendsto infinityas T tendsto infinity.Later, Hardy [7] improvedhis argumentto provethat M(T) > AT, whereA is a positiveconstant, not dependingon T. The ultimateresult of thissort was obtainedby AtleSelberg in 1943[11]. He provedthat M(T) > ATlog T forsome positiveconstant A. In viewof equation (33), Selberg'sresult shows that a positiveper- centage of the zeroesof C(s) actuallylie on the line Re(s) = i. This resultrepresents the bestprogress made to date in attemptingto prove the Riemannhypothesis. Fortunately,it is not necessaryto prove the Riemann hypothesisin order to provethe prime number theorem in theform (17). However,it is necessaryto obtain some informationabout thedistribution of the zeroes of C(s). Such informationwas obtained independentlyby Hadamard [5] and de la Vallee Poussin [1] in 1896, therebyproviding the firstcomplete proofs of the primenumber theorem. Although theirproofs differ in detail,they both establish the existence of a zero-freeregion for C(s), the existenceof whichserves as a substitutefor the Riemannhypotheses in the reasoningpresented above. More specifically,they proved that there exist 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 primenumber theorem in the form

(34) tfr(x)= x + O(xe- C(log X) 1/14)

Please note,however, that the errorterm in (34) is muchlarger than the errorterm predictedby the Riemannhypothesis. Thus, theprime number theorem was finallyproved aftera centuryof hard work by manyof theworld's best mathematicians. It is grosslyunfair to attributeproof of such a theoremto thegenius of a singleindividual. For, as we have seen,each stepin the directionof a proof was conditionedhistorically by the work of preceding generations.On the otherhand, to denythat there is geniusin th&work whichled up to the ultimateproof would be equally unfair.For at each step in the chain of discovery,brilliant and fertileideas werediscovered, and providedthe materialout of whichto fashionthe nextlink.

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APPENDIX A: Samples from Gauss' Tables. TABLE 1 (Werke,II, p. 436)

1 168 26 98 2 135 27 101 3 127 28 94 4 120 29 98 5 119 30 92 6 114 31 95 7 117 32 92 8 107 33 106 9 110 34 100 10 112 35 94 11 106 36 92 12 103 37 99 13 109 38 94 14 105 39 90 15 102 40 96 16 108 41 88 17 98 42 101 18 104 43 102 19 94 44 85 20 102 45 96 21 98 46 86 22 104 47 90 23 100 48 95 24 1C4 49 89 25 94 50 98

The frequencyof primes.TABLE 2 (Werke,II, p. 443) 2000000..3000000

210 220 230 240 250 260 270 280 290 300 0 1 1 1 3 2 2 4 1 3 4 2 2 2 25 2 10 9 9 11 9 5 10 7 15 13 98 3 32 27 29 32 37 35 28 43 30 44 337 4 69 69 73 86 78 88 71 95 85 64 778 5 119 146 138 136 147 136 158 135 140 153 1408 6 197 183 179 176 193 194 195 195 179 187 1878 7 204 201 205 194 189 180 201 188 222 214 1998 8 157 168 168 158 151 170 142 145 132 134 1525 9 115 109 113 112 102 88 96 87 109 103 1034 10 63 52 44 55 58 58 53 67 53 58 561 11 21 18 30 28 23 24 22 24 18 15 223 12 8 9 10 7 7 13 17 9 8 11 99 13 2 4 1 5 6 1 2 5 1 27 14 3 1 2 6 15 1 1 16 17 1 1 6874 6857 6849 6787 6766 6804 6762 6714 6744 6705 6862

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APPENDIX B: Gauss' Letterto Enke.

My distinguishedfriend: Your remarksconcerning the frequency of primes were of interest to me in moreways than one. You have remindedme of myown endeavorsin thisfield which began in thevery distant past, in 1792or 1793,after I had acquiredthe Lambert supplements to thelogarithmic tables. Even before I had begunmy more detailed investigations into higher arithmetic, one of myfirst projects was to turnmy attention to thedecreasing frequency of primes,to whichend I countedthe primes in severalchiliads (intervals of length1000; Trans.)and recordedthe resultson theattached white pages.I soon recognizedthat behind all ofits fluctuations, this frequency is on theaverage inversely proportionalto thelogarithm, so thatthe number of primes below a givenbound n is approximately equal to I dn Jlogn

wherethe logarithm is understoodto be hyperbolic.Later on, whenI becameacquainted with the listin Vega,'stables (1796) going up to 400031,I extendedmy computation further, confirming that estimate.In 1811, theappearance of Chernau's cribrum gave me much pleasure and I havefrequently (sinceI lack thepatience for a continuouscount) spent an idlequart r of an hourto countanother chiliadhere and there;although I eventuallygave it up withoutquite gettingthrough a million. Onlysome timelater did I makeuse of thediligence of Goldschmidtto fillsome of theremaining gaps in thefirst million and to continuethe computation according to Burkhardt'stables. Thus (for manyyears now) thefirst three million have been countedand checkedagainst the integral.A smallexcerpt follows:

TABLE A

Below Here are Integral Your Error Prime ln Error Formula log n 500000 41556 41606.4+ 50.4 41596.9+ 40.9 1000000 78501 79627.5+ 126.5 78672.7-+ 171.7 1500000 114112 114263.1+ 151.1 114374.0+ 264.0 2000000 148883 149054.8+ 171.8 149233.0+ 350.0 2500000 183016 183245.0+ 229.0 183495.1+ 479.1 3000000 216745 216970.6+ 225.6 217308.5+ 563.5

I was not awarethat Legendre had also workedon thissubject; your letter caused me to look in his Thioriedes Nombres,and in thesecond edition I founda fewpages on the subjEctwhich I musthave previouslyoverlooked (or, by now,forgotten). Legendre used theformula

n log n -A whereA is a constantwhich he setsequal to 1.08366.After a hastycomputation, I find in theabove cases the deviations

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TABLE B

- 23,3 + 42,2 ? 68,1 1- 92,8 +159,1 +167,6

Thesedifferences are evensmaller than those from the integral, but they seem to growfaster with i so thatit is quitepossible they may surpass them. To makethe count and the formulaagree, one wouldhave to use,respectively, instead of A = 1.08366,the following numbers:

TABLE C 1,09040 1,07682 1,07582 1,07529 1,07179 1,07297

It appearsthat, with increasing n, the (average) value of A decreases;however, I dare not conjec- turewhether the limit as n approachesinfinity is I or a numberdifferent from 1. I cannotsay that thereis anyjustification for expecting a verysimple limiting value; on theother hand, the excess of A over1 mightwell be a quantityof the order of 1/logn. I wouldbe inclinedto believethat the differ- entialof the function must be simplerthan the fuLnction itself. If dn/logn is postulatedfor he function,Legendre's formula would suggestthat the differential functionmight be somethingof theform dnl (log n- (A -1)). By theway, for large n, yourfor- mula could be consideredto coincidewith n log n-(i /2k)

wherek is themodulus of Brigg's logarithms; that is, with Legendre's formula, if we put A = 1/2k - 1.1513. Finally,I wantto remarkthat I noticeda coupleof disagreements between your counts and mine. Between 59000 and 60000, you have 95, whileI have 94 101000 102000 94 93. The firstdifference possibly results from the fact that, in Lambert'sSupplement, the prime 59023 occurstwice. The chiliadfrom 101000- 102000in Lambert'sSupplement is virtuallycrawling with errors;in mycopy, I have indicatedseven numbers which are not primesat all, and suppliedtwo missingones. Would it not be possibleto induceyoung Mr. Dase to countthe primes in thefollowing (few)millions, using the tables at theAcademy which, I am afraid,are notintended for public dis- tribution?.In thiscase, let me remarkthat in the2nd and 3rdmillion, the count is, accordingto my instructions,based on a specialscheme which I myselfhave employed in countingthe first million. The countsfor each 100000are irndicatedon a singlepage in 10 columns,each columnbelonging to one myriad(an intervalof length 10000; Trans.); an additionalcolumn in front(left) and another columnfollowing it on theright; for example here is a verticalcolumn and the two additional columns for the interval10000000 to 11000000 --

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As an illustration,take the first vertical column. In themyriad 1000000 to 1010000there are 100 Hecatontades;(intervals of length 100; Trans.)among them one containinga singleprime, none con- tainingtwo or threeprimes; two containing four each; elevencontaining 5 each,etc., yielding alto- gether752 = 1.1 + 4.2 + 5.11 + 6.14 + * primes.The last columncontains the totals from the otherten. The numbers14, 15, 16 in thefirst vertical column are superfluoussince no hecatontades occurcontaining that many primes; but on the followingpages theyare needed.Finally the 10 pages are again combinedinto one and thuscomprise the entire second million. It is hightime to quit ---. Withmost cordial wishes for your good health Yours, as ever, C. F. Gauss Gbttingen,24 December1849.

APPENDIX C: Correctionsto Gauss' Tables

THOUSANDS GAUSS ACTUAL A

20 102 104 -2 159 87 77 d-10 199 96 86 +10 206 85 83 +2 245 78 88 -10 289 85 77 +8 290 84 85 -1 334 80 81 -1 352 80 81 -1 354 79 76 +3

500 UP TO HERE +18

TOTALS A

500,000 41,556 41,538 +18 1,000,000 78,501 78,498* +3 1,500,000 114,112 114,156* -44 2,000,000 148,883 148,934* -51 2,500,000 183,016 183,073* -57 3,000,000 216,745 216,817* -72

* fromList of Prime Numbers from 1 to 10,006,771,by D. N. Lehmer,(adjusted: He counts1 as a prime).

Researchsupported by NSF GrantGP 31280X.This article was presented to the History of Mathe- maticsSeminar at theUniversity of Marylandon March20, 1972.The authorwishes to thankPro- fessorGertrude Ehrlich for preparing the translation of Gauss' letterwhich appears in AppendixB. He also wishesto thankMr. Edward Korn for providing the calculations of Appendix C.

References 1. Ch. de la ValleePoussin, Recherches analytiques sur la theoriedes nombres premiers. Premiere partie.La fonctionC(s) de Riemannet les nombrespremiers en general.Deuxieme partie: Les fonc-

This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions 1973] DIFFERENTIATION UNDER THE INTEGRAL SIGN 615 tionsde Dirichletet les nombrespremiers de la formelineaire Mx+N. Troisiemepartie: Les formes quadratiquesde determinantnegatif, Ann. Soc. Sci. Bruxelles,20 (1896) 183-256,281-397. 2. L. Dirichlet,Uber den Satz: das jede unbegrenztearithmetische Progression, deren erstes Glied und Differenzkeinen gemeinschaftlichen Factor sind, unendlichen viele Primzahlen enthalt, 1837; MathematischeAbhandlungen, Bd. 1, (1889) 313-342. 3. C. F. Gauss,Tafel der Frequenz der Primzahlen, Werke, 11 (1872) 436-442. 4. - , Gauss an Enke, Werke, 11(1872) 444-447. 5. J. Hadamard,Sur la distributiondes zerosde la fonctionC(s) et ses consequencesarithmeti- ques, Bull. Soc. Math. de France,24 (1896) 199-220. 6. G. H. Hardy,Sur les zeros de la fonctionC(s) de Riemann,Comptes Rendus, 158 (1914) 1012-14. 7. , and J.E. Littlewood,The zerosof Riemann's zeta function on thecritical line, Math. Zeit.,10 (1921)283-317. 8. A. M. Legendre,Essai sur la theoriede Nombres,1st ed., 1798,Paris, p. 19. 9. - , Essai surla Theoriede Nombres,2nd ed. 1808,Paris, p. 394. 10. B. Riemann,fiber die Anzahlder Primzahlenunter einer gegebenen Grosse, Gesammelte MathematischeWerke, 2nd Aufl., (1892) 145-155. 11. A. Selberg,On thezeros of Riemann'szeta function,Skr. Norske Vid. Akad.,Oslo (1942) no. 10. 12. P. Tchebycheff,Sur la fonctionqui determinela totalitede nombrespremiers inferieurs A une limitedonnee, Oeuvres, 1 (1899) 27-48. 13. -, Memoiresur les nombrespremiers, Oeuvres, 1 (1899)49-70. 14. H. von Mangoldt,Auszug aus einerArbeit unter dem Titel: Zu Riemann'sAbhandlung uberdie Anzahlder Primzahlenunter einer gegebenen Grbsse, Sitz. Konig. Preus.Akad. Wiss. zu Berlin,(1894) 337-350,883-895.

DIFFERENTIATION UNDER THE INTEGRAL SIGN*

HARLEY FLANDERS, Tel-AvivUniversity

1. Introduction.Everyone knows the Leibniz rule fordifferentiating an integral: +(fh(t) F(x, t) dx (1.1) gt (t) )

= -F[h(t),t(t) -F[g(t), t]g(t) + () at dx.

We are all fondof thisformula, although it is seldomif ever used in such generality. Usually,either the limitsare 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, 1972 to the Rocky MountainSection meeting, Southern Colorado State College,Pueblo, CO.

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