# THE ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM: AN HISTORICAL PERSPECTIVE (by D. Goldfeld)

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1 THE ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM: AN HISTORICAL PERSPECTIVE (by D. Goldfeld) The study of the distribution of prime numbers has fascinated mathematicians since antiquity. It is only in modern times, however, that a precise asymptotic law for the number of primes in arbitrarily long intervals has been obtained. For a real number >1, let π() denote the number of primes less than. The prime number theorem is the assertion that / lim π() log() = 1. This theorem was conjectured independently by Legendre and Gauss. The approimation π() = A log()+b was formulated by Legendre in 1798 [Le1] and made more precise in [Le2] where he provided the values A =1,B = On August 4, 1823 (see [La1], page 6) Abel, in a letter to Holmboe, characterizes the prime number theorem (referring to Legendre) as perhaps the most remarkable theorem in all mathematics. Gauss, in his well known letter to the astronomer Encke, (see [La1], page 37) written on Christmas eve 1849 remarks that his attention to the problem of finding an asymptotic formula for π() dates back to 1792 or 1793 (when he was fifteen or siteen), and at that time noticed that the density of primes in a chiliad (i.e. [, ]) decreased approimately as 1/ log() leading to the approimation π() Li() = 2 dt log(t). The remarkable part is the continuation of this letter, in which he said (referring to Legendre s log() A() approimation and Legendre s value A() = ) that whether the quantity A() tends to 1 or to a limit close to 1, he does not dare conjecture. The first paper in which something was proved at all regarding the asymptotic distribution of primes was Tchebychef s first memoir ([Tch1]) which was read before the Imperial Academy of St. Petersburg in In that paper Tchebychef proved that if any approimation to π() held to order / log() N (with some fied large positive integer N) then that approimation had to be Li(). It followed from this that Legendre s conjecture that lim A() = was false, and that if the limit eisted it had to be 1. The first person to show that π() has the order of magnitude log() was Tchebychef in 1852 [Tch2]. His argument was entirely elementary and made use of properties of factorials. It is easy to see that the highest power of a prime p which divides! (we assume is an integer) is simply [ ] [ ] [ ] + p p 2 + p 3 + 1

2 where [t] denotes the greatest integer less than or equal to t. It immediately follows that! = p [/p]+[/p2 ]+ and log(!) = ([ ] [ ] [ ] ) + p p 2 + p 3 + log(p). Now log(!) is asymptotic to log() by Stirling s asymptotic formula, and, since squares, cubes,... of primes are comparatively rare, and [/p] is almost the same as /p, one may easily infer that log(p) = log() +O() p from which one can deduce that π() is of order Tchebychef, who actually proved that [Tch2] for all sufficiently large numbers, where / B < π() log() < 6B 5 log(). This was essentially the method of B = log log log 5 5 log and 6B Unfortunately, however, he was unable to prove the prime number theorem itself this way, and the question remained as to whether an elementary proof of the prime number theorem could be found. Over the years there were various improvements on Tchebychef s bound, and in 1892 Sylvester [Syl1], [Syl2] was able to show that / < π() log() < for all sufficiently large. We quote from Harold Diamond s ecellent survey article [D]: The approach of Sylvester was ad hoc and computationally comple; it offered no hope of leading to a proof of the P.N.T. Indeed, Sylvester concluded in his article with the lament that...we shall probably have to wait [for a proof of the P.N.T. ] until someone is born into the world so far surpassing Tchebychef in insight and penetration as Tchebychef has proved himself superior in these qualities to the ordinary run of mankind. 2

5 Quote from Selberg: Letter to H. Weyl Sept. 16, 1948 I found the fundamental formula... in March this year... I had found a more complicated formula with similar properties still earlier. April 1948: Recall that ϑ() = log(p). Define a = lim inf ϑ() ϑ(), A = lim sup. Sylvester s estimates guarantee that a A In his letter to H. Weyl, Sept. 16, 1948, Selberg writes: I got rather early the result that a + A = 2, The proof that a + A = 2 is given as follows. Choose large so that ϑ() =a + o(). Then since ϑ() A + o() it follows from Selberg s fundamental formula that a log()+ A log(p) 2 log()+o( log()). p Using Tchebychef s result that log(p) p log() it is immediate that a + A 2. On the other hand, we can choose large so that ϑ() =A + o(). Then since ϑ() a + o() it immediately follows as before that A log()+ a log(p) 2 log()+o( log()), p from which we get a + A 2. Thus a + A =2. Remark: Selberg was aware of the fact (already in April 1948) that a + A = 2, and that the prime number theorem would immediately follow if one could prove either a =1or A =1. 5

7 You must have made a mistake because with this result I can get an elementary proof of the prime number theorem, and I have convinced myself that my inequality is not powerful enough for that. Quote from Weyl s letter to Selberg August 31, 1948 Is it not true that you were in possession of what Erdős calls the fundamental inequality and of the equation a + A = 2 for several months but could not prove a = A = 1 until Erdős deduced +1 1 from your inequality? Here is Selberg s response in his letter to Weyl, Sept. 16, Turán had mentioned to Erdős after my return from Montreal he told me he was trying to prove +1 1 from my formula. Actually, I didn t like that somebody else started working on my unpublished results before I considered myself through with them. But though I felt rather unhappy about the situation, I didn t say anything since after all Erdős was trying to do something different from what I was interested in. In spite of this, I became... rather concerned that Erdős was working on these things... I, therefore, started very feverishly to work on my own ideas. On Friday evening Erdős had his proof ready (that +1 1 ) and he told it to me. On Sunday afternoon I got my first proof of the prime number theorem. I was rather unsatisfied with the first proof because it was long and indirect. After a few days (my wife says two) I succeeded in giving a different proof. Quote from Erdős s paper, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat. Acad. Scis. 1949: Using (1) (fundamental formula) I proved that +1 1asn.In fact, I proved the following slightly stronger result: To every ɛ there eists a positive δ(ɛ) so that for sufficiently large we have π ( (1 + ɛ) ) π() >δ(ɛ)/ log() where π() is the number of primes not eceeding. I communicated this proof to Selberg, who, two days later... deduced the prime number theorem. Recently, Selberg sent me a letter which more precisely specifies the actual dates of events. Quote from Selberg s letter to D. Goldfeld, January 6, 1998: July 14, 1948 was a Wednesday, and on Thursday, July 15 I met Erdős and heard that he was trying to prove I believe Turán left the 7

9 Selberg s first proof that the prime number theorem followed from the fundamental formula is given both in [E] and [S]. The cru of the matter goes something like this. We may write the fundamental formula in the form ϑ() + ϑ(/p) /p log(p) p log() = 2+O ( ) 1. log() Recall that a and A are the limit inferior and limit superior, respectively, of ϑ(). Now, choose large so that ϑ() is near A. Since a + A = 2, it follows from the fundamental formula and log(p) p log() 1, that ϑ(/p) /p p S must be near a for most primes p. If S denotes the set of eceptional primes, then we have / log(p) log(p) 0. p p Now, choose a small prime q S such that ϑ(/q) /q is near a. Rewriting the fundamental formula with replaced by /q, the same argument as above leads one to conclude that ϑ(/pq) /pq is near A for most primes p /q. It follows that ϑ(/p) a/p for most primes p and that ϑ(/pq) A/pq for most p /q. A contradiction is obtained (using Erdős s idea of nonoverlapping intervals) unless a = A = 1. The Erdős-Selberg dispute arose over the question of whether a joint paper (on the entire proof) or seperate papers (on each individual contribution) should appear on the elementary proof of the PNT. August 20, 1948: Quote from a letter of Selberg to Erdős. What I propose is the only fair thing: each of us can publish what he has actually done and get the credit for that, and not for what the other has done. You proved that +1 lim =1. n I would never have dreamed of forcing you to write a joint paper on this in spite of the fact that the essential thing in the proof of the result was mine. Since there can be no reason for a joint paper, I am going to publish my proof as it now is. I have the opinion,... that I do you full justice by telling in the paper that my original proof depended on your result. 9

11 that the FUND. LEMMA does not imply the PNT (prime number theorem). If you would have told me about what you know about a and A, I would have finished the proof of PNT on the spot. Does it occur to you that if I would have kept the proof of +1 1to myself (as you did with a + A = 2) and continued to work on PNT... I would soon have succeeded and then your share of PNT would have only been the beautiful FUNDAMENTAL LEMMA. Sept. 27, 1948: Quote from Erdős s letter to Selberg. I completely reject the idea of publishing only +1 lim =1. n and feel just as strongly as before that I am fully entitled to a joint paper. So if you insist on publishing your new proof all I can do is to publish our simplified proof, giving you of course full credit for your share (stating that you first obtained the PNT, using some of my ideas and my theorem). Also, I will of course gladly submit the paper to Weyl first, if he is willing to take the trouble of seeing that I am scrupulously fair to you. Quote from E.G. Straus: The elementary proof of the prime number theorem. It was Weyl who caused the Annals to reject Erdős s paper and published only a version by Selberg which circumvented Erdős s contribution, without mentioning the vital part played by Erdős in the first elementary proof, or even the discovery of the fact that such a proof was possible. Quote from Selberg s letter to D. Goldfeld, January 6, 1998: This is wrong on several points, my paper mentioned and sketched in some detail how Erdős s result played a part and was used in the first elementary proof of PNT, but that first proof was mine as surely as Erdo s result was his. Also the discovery that such a proof was possible was surely mine. After all, you don t know that it is possible to prove something until you have done so! Ecerpt: Handwritten Note by Erdős: It was agreed that Selberg s proof should be in the Annals of Math., mine in the Bulletin. Weyl was supposed to be the referee. To my great surprise Jacobson the referee... The Bulletin wrote that the referee does not recommend my paper for publication.... I immediately withdrew the paper and planned to publish it in the JLMS but... had it published in the Proc. Nat. Acad. 11

12 Feb. 15, 1949: Quote from H. Weyl s letter to Jacobson I had questioned whether Erdős has the right to publish things which are admittedly Selberg s...ireally think that Erdős s behavior is quite unreasonable, and if I were the responsible editor I think I would not be afraid of rejecting his paper in this form. But there is another aspect of the matter. It is probably not as easy as Erdős imagines to have his paper published in time in this country if the Bulletin rejects it... Soitmaybebetter to let Erdős have his way. No great harm can be done by that. Selberg may feel offended and protest (and that would be his right), but I am quite sure that the two papers Selberg s and Erdős s together will speak in unmistakable language, and that the one who has really done harm to himself will be Erdős. Quote from E.G. Straus: The elementary proof of the prime number theorem. The elementary proof has so far not produced the eciting innovations in number theory that many of us epected to follow. So, what we witnessed in 1948, may in the course of time prove to have been a brilliant but somewhat incidental achievement without the historic significance it then appeared to have. Quote from Selberg s letter to D. Goldfeld, January 6, 1998: With this last quote from Straus, I am in agreement (actually I did not myself epect any revolution from this). The idea of the local sieve, however, has produced many things that have not been done by other methods. Remark: To this date, there have been no results obtained from the elementary proof of the PNT that cannot be obtained in stronger form by other methods. Other elementary methods introduced by both Selberg and Erdős have, however, led to many important results in number theory not attainable by any other technique. Dec. 4, 1997: Letter from Selberg to D. Goldfeld. The material I have is nearly all from Herman Weyl s files, and was given to me probably in 1952 or 1953 as he was cleaning out much of his stuff in Princeton, taking some to Zurich and probably discarding some. The letters from Weyl to myself was all that I kept when I left Syracuse in 1949, all the rest I discarded. Thus there are gaps. Missing is my first letter to Erdős as well as his reply to it... I did not save anything ecept letters from Weyl because I was rather disgusted with the whole thing. I never lectured on the elementary proof of the PNT after the lecture in Syracuse, mentioned in the first letter to Herman Weyl. However, I did at Cornell U. early in 1949 and later at 12

13 an AMS meeting in Baltimore gave a lecture with an elementary proof of (using the notation of Beurling generalized primes & integers) the fact that if ( ) N() =A + o log 2 () then References Π() = Beurling has the same conclusion if ( N() =A + O ( ) log() + o. log() (log()) α with α> 3 2. I never published this. Erdős of course lectured etensively in Amsterdam, Paris, and other places in Europe. After his lecture in Amsterdam, Oct. 30, 1948, v.d. Corput wrote up a paper, Scriptum 1, Mathematisch Centrum, which was the first published version! [A B G] K.E. Aubert, E. Bombieri, D. Goldfeld, Number theory, trace formulas and discrete groups, symposium in honor of Atle Selberg, Academic Press Inc. Boston (1989). [A S] N. Alon, J. Spencer, The probabilistic method, John Wiley & Sons Inc., New York (1992). [B] H. Bohr, Address of Professor Harold Bohr, Proc. Internat. Congr. Math. (Cambridge, 1950) vol 1, Amer. Math. Soc., Providence, R.I., 1952, [Ch] J. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), [C G] F. Chung, R. Graham, Erdős on graphs: his legacy of unsolved problems, A.K. Peters, Ltd., Wellesley, Massachusetts (1998). [C] L.W. Cohen, The annual meeting of the society, Bull. Amer. Math. Soc 58 (1952), [D] H.G. Diamond, Elementary methods in the study of the distribution of prime numbers, Bull. Amer. Math. Soc. vol. 7 number 3 (1982), [E] P. Erdős, On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Nat. Acad. Scis. U.S.A. 35 (1949), [G] D. Goldfeld, The Erdős Selberg dispute: file of letters and documents, to appear. [H1] J. Hadamard, Étude sur les proprietés des fonctions entiéres et en particulier d une fonction considérée par Riemann, J. de Math. Pures Appl. (4) 9 (1893), ; reprinted in Oeuvres de Jacques Hadamard, C.N.R.S., Paris, 1968, vol 1, ),

14 [H2] J. Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bull. Soc. Math. France 24 (1896), ; reprinted in Oeuvres de Jacques Hadamard, C.N.R.S., Paris, 1968, vol 1, [Ho] P. Hoffman, The man who loves only numbers, The Atlantic, November (1987). [I] A.E. Ingham, Review of the two papers: An elementary proof of the prime number theorem, by A. Selberg and On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, by P. Erdős. Reviews in Number Theory as printed in Mathematical Reviews , Amer. Math. Soc. Providence, RI (1974). See N20-3, Vol. 4, [La1] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig (1909), 2 volumes, reprinted by Chelsea Publishing Company, New York (1953). [La2] E. Landau, Über den Wienerschen neuen Weg zum Primzahlsatz, Sitzber. Preuss. Akad. Wiss., 1932, [Le1] A.M. Legendre, Essai sur la théorie des nombres, 1. Aufl. Paris (Duprat) (1798). [Le2] A.M. Legendre, Essai sur la théorie des nombres, 2. Aufl. Paris (Courcier) (1808). [S] A. Selberg, An elementary proof of the prime number theorem, Ann. of Math. (2) 50 (1949), ; reprinted in Atle Selberg Collected Papers, Springer Verlag, Berlin Heidelberg New York, 1989, vol 1, [Syl1] J.J. Sylvester, On Tchebycheff s theorem of the totality of prime numbers comprised within given limits, Amer. J. Math. 4 (1881), [Syl2] J.J. Sylvester, On arithmetical series, Messenger of Math. (2) 21 (1892), 1 19 and [Tch1] P.L. Tchebychef, Sur la fonction qui détermine la totalité des nombres premiers inférieurs à une limite donnée, Mémoires présentés à l Académie Impériale des Sciences de St.-Pétersbourg par divers Savants et lus dans ses Assemblées, Bd. 6, S. (1851), [Tch2] P.L. Tchebychef, Mémoire sur les nombres premiers, J. de Math. Pures Appl. (1) 17 (1852), ; reprinted in Oeuvres 1 (1899), [VP] C.J. de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers, Ann. Soc. Sci. Bruelles 20 (1896), [W1] N. Wiener, A new method in Tauberian theorems, J. Math. Physics M.I.T. 7 ( ), [W2] N. Wiener, Tauberian theorems, Ann. of Math. (2) 33 (1932),

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