How To Know Ifiitely May Rime Umbers
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1 How Euler Did It Ifiitely may rimes March 2006 Why are there so very may rime umbers? by Ed Sadifer Euclid wodered this more tha 200 years ago, ad his roof that Prime umbers are more tha ay assiged multitude of rime umbers (Elemets IX.20) is ofte cosidered oe of the most beautiful roofs i all of mathematics. Prime umbers, their detectio, their frequecy ad their secial roerties remai at the heart of may of today s most excitig oe questios i umber theory. This moth, we retur to oe of Euler s early aers, Variae observatioes circa series ifiitas, to see what Euler has to say there about rime umbers. This aer has Eeström umber 72, ad we have already see this aer i our colum from February 2005, called Goldbach s series. Euler wrote the aer i 737, ad it was ublished i 744. Here we will fid that Euler gives three aswers to the questio How may rime umbers are there? ad, i a way, hels us uderstad why there are so may of them. Sice we discussed the first art of this aer last February, we will start i the middle of this aer, with Euler s Theorem 7: The roduct cotiued to ifiity of this fractio etc. i which the umerators are rime umbers etc. ad the deomiators are oe less tha the umerators, equals the sum of this ifiite series etc., ad they are both ifiite The factors of the ifiite roduct ca be rewritte as =, so, i moder / otatio, this theorem ca be rewritte as = / the sum-roduct formula for the Riema Zeta fuctio at the value s =. Others may recogize. Some readers will recogize this as It was voted umber 3 i a 988 survey by Mathematical Itelligecer. See [W,. 26]
2 it as art of the cover illustratio of William Duham s woderful boo o Euler ad his wor. [D] Euler offers the followig roof of this theorem. Proof: Let The This leaves x = etc x = etc x = etc Note that there are o eve umbers left i the deomiators o the right had side. Now, to elimiate the deomiators that are divisible by three, we divide both sides by three, to get x = etc Subtractig agai elimiates all remaiig deomiators that are multiles of 3, leavig 2 x = etc This rocess is lie the aciet sieve of Eratosthees because, at each stage, it elimiates a rime deomiator ad all remaiig multiles of that rime deomiator. Evetually, everythig o the right will be elimiated excet the first term,. Euler carries us through oe more iteratio of his rocess. He divides this last equatio by 5, ad does a small rearragemet of the way he writes the roduct of fractios, to get Subtractig leaves 2 x = etc x = etc I the same way, terms with deomiators that are multiles of 7,, ad so forth for all rime umbers, will be elimiated, leavig etc etc. x =. But, sice x is already ow to be the sum of the harmoic series, the desired result is immediate. Q. E. D. From its very first ste, this roof does ot satisfy moder stadards of rigor, but it is t too much wor for a moder mathematicia to recast the theorem to say that the harmoic series 2
3 has a fiite limit if ad oly if the ifiite roduct does, ad the to begi the roof suose that the harmoic series coverges to a value x. Noetheless, if we accet the result, the we have a short roof that there are ifiitely etc. may rimes. For the roduct to diverge it must be a ifiite etc. roduct, hece there must be ifiitely may rime umbers. Though it is t exactly relevat to our toic, the ext theorem, Theorem 8 is extremely imortat: Theorem 8: If we use the series of rime umbers to form the exressio etc ( ) ( ) ( ) ( ) ( ) the its value is equal to the sum of this series etc I moder otatio, this is the familiar sum-roduct formula for the Riema zeta fuctio: ζ ( s) = = s s, / rime ad Euler roof of Theorem 8 is almost exactly lie the roof of Theorem 7, with the exoet icluded. Also, the roof of Theorem 8 is correct by moder stadards because all the series ivolved are absolutely coverget. Occasioally, someoe will isist that, because Euler roved this Theorem 8, the fuctio we call the Riema zeta fuctio ought to be called the Euler zeta fuctio. We disagree. Euler s zeta fuctio is a fuctio of a real variable, ad Euler ever treats its value excet for ositive itegers. Riema exteded the fuctio to comlex values, where its most iterestig ad imortat roerties are foud. If you wat to, you ca try to say the Euler zeta fuctio is a fuctio of a real variable, but the Riema zeta fuctio is comlex. The forces of history robably wo t let you though, ad that is fair. We jum forward to Theorem 9 for Euler s secod result about how may rime umbers there are. His theorem is: 3
4 Theorem 9: The sum of the series of recirocals of rime umbers etc is ifiitely large, ad is ifiitely less tha the harmoic series, etc Moreover, the first sum is almost the logarithm of the secod sum. 2 Euler s roof of this agai o addig ad subtractig series that do ot coverge, ad so does ot meet moder stadards of rigor. Ufortuately, Euler s roof ca t really be reaired. This time, we omit the roof. Note that the first statemet i the roof, that the series of recirocals of rimes diverges, ot oly tells us that there are ifiitely may rimes (sice a fiite series caot diverge), but tells us that the rimes are dese eough that the sum of their recirocals diverges. The sum of 2 π the recirocals of the square umbers, o the other had, coverges to, as Euler showed a 6 few years earlier whe he solved the Basel roblem. I this sese, there are more rime umbers tha there are square umbers. Fially, we come to Euler s third measure of how may rimes there are. This comes from the secod art of Theorem 9. This will require some iterretatio ad some aalysis, but we hoe to show that this remar is closely related to the so-called Prime Number Theorem. Earlier i this aer, Euler made a crytic remar about the value of the harmoic series, if the absolute ifiity is tae to be = the this exressio will have the a value =l, which is the smallest of all the ifiite owers. The art about =l is fairly easy to iterret. I moder terms, it robably meas that, for large values of, l. The art about smallest of all the ifiite owers is t as clear, but I thi it is about what we ow call -series, that the smallest value of for which the -series diverges is = -, the case of the harmoic series. That is, the harmoic series is the smallest ower that gives a ifiity. 2 I Euler s Lati: Atque illius summa est huius summae quasi logarithmus. 4
5 Iterretig the rest of Euler s claim i the same way, it would seem to traslate ito the aroximatio l( l ). rime < We ca fid this as a theorem i moder umber theory boos, for examle theorem 427 i [H+W]. We ca differetiate this to fid out how much the sum is exected to icrease if we roceed from to +. We fid that the derivative is. We ca aly some ideas from l robability. If is rime, the the sum will icrease by, ad if is ot rime, the the sum will ot icrease. Hece, the robability that is rime is about l. Accordig to MacTutor [McT] The statemet that the desity of rimes is /log is ow as the Prime Number Theorem. Moreover, Legedre observed this fact about the desity of rimes i 798, ad Gauss claimed to have observed it i 793, but it was ot roved util 896 whe Hadamard ad de la Vallée Poussi ideedetly discovered roofs. Precedece is usually give to Gauss s observatio. However, as we have just see, the Prime Number Theorem is a easy cosequece of Euler s Theorem 9. Refereces: Euler scooed Gauss by more tha fifty years. [E] Euclid, Elemets, Sir Thomas Heath, tr., 3 vol., Dover, 956. [E72] Euler, Leohard, Variae observatios circa series ifiitas, Commetarii academiae scietiarum Petroolitaae 9 (737), 744, Rerited i Oera Omia Series I volume 4, Also available o lie at [H+W] Hardy, G. H, ad E. M. Wright, A Itroductio to the Theory of Numbers, 5ed, Oxford Uiv. Press, 979. [S] Sadifer, Edward, Goldbach s series, How Euler Did It, MAA OLie, htt:// February [D] Duham, William, Euler The Master of Us All, Dolciai Mathematical Exositios vol. 22, Mathematical Associatio of America, Washigto, DC, 999. [McT] Prime Numbers, MacTutor History of Mathematics Archive, htt://www-grous.dcs.stad.ac.u/~history/histtoics/prime_umbers.html. [W] Wells, David, The Pegui Boo of Curious ad Iterestig Mathematics, Pegui, Lodo, 997. ( ) Ed Sadifer (SadiferE@wcsu.edu) is Professor of Mathematics at Wester Coecticut State Uiversity i Dabury, CT. He is a avid maratho ruer, with 33 Bosto Marathos o his shoes, ad he is Secretary of The Euler Society ( How Euler Did It is udated each moth. Coyright 2006 Ed Sadifer 5
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