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 Stable URL: http://www.jstor.org/stable/2319162 . Accessed: 09/02/2013 20:07 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org 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 <x. Then the primenumber theorem asserts that (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 Number Theory(Prentice-Hall 1971), and AbstractAlgebra, A First Course (Prentice-Hall, to appear). Editor. 599 This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions 600 L. J. GOLDSTEIN [June-July 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) thandoes 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 This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions 1973] A HISTORY OF THE PRIME NUMBER THEOREM 601 (6) E ( P ) .1(1+H p+T2+.. ' Since everypositive integer 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. This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions 602 L. J. GOLDSTEIN [June-July 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 "Riemann hypothesis."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). This content downloaded on Sat, 9 Feb 2013 20:07:45 PM All use subject to JSTOR Terms and Conditions 1973] A HISTORY OF THE PRIME NUMBER THEOREM 603 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.