How Euler Did It. sharpens these relations by using the fact that the function 1/x naturally expresses the terms of the harmonic series and that

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1 How Euler Did It A flse logrithm series December 007 by Ed Sdifer Solvig good reserch questio should ope more doors th it closes. Oe of Euler s lesser ppers, Methodus geerlis summdi progressioes ( Geerl methods of summig progressios ) [E5] is more oteworthy for the thigs it strted th the thigs it fiished. The pricipl role of the pper is s oe of sequece of ppers tht led to Euler s developmet of the Euler-Mcluri summtio formul. Tht sequece beg i 79 with letter to Goldbch cotiig results tht Euler lter published i 738 i [E0], d cotiued through [E5], [E43], [E46], [E47] up to [E55], Methodus uiverslis series summdi ulterius promot, writte i 736 d published i 74. This sequece of ppers hs woderful plot. First Euler emies the reltios betwee the prtil sums of the hrmoic series, d the logrithm fuctio, l. The he 3 4 shrpes these reltios by usig the fct tht the fuctio / turlly epresses the terms of the hrmoic series d tht d= l. He eteds his results first to other more geerl series of reciprocls like , the betwee prtil sums of other series of reciprocls d k k k k 3 4 their correspodig itegrls d, d the to fuctios i geerl, k f ( i) d f ( ) d. It is i= delightful to wtch the youg Euler shrpe his tools d his isights from oe yer to the et betwee the yers 79 d 736. At the ed of E5, though, Euler csts glce i other directio, d poses two series for which the methods he hd used to evlute so my other series do ot seem to work: is α d The ptter for the secod of these series is ot obvious, but Euler eplis tht the geerl term where d α deote itegers greter th oe. Goldbch hd show tht this series sums to

2 , d Euler epded o Goldbch s techique i [E7], oe of Euler s gretest ppers, to discover the Euler product formul. See [S, BPV, D]. This moth s topic, though is the first of these series. It comes up gi lmost 5 yers lter i [E90], Cosidertio qurumdm serierum que sigulribus propriettibus sut predite ( Cosidertio of some series which re distiguished by specil properties ). I E90, Euler studies the series ( )( ) ( )( )( ) ( )( )( )( 3 ) s = Here, the umertors dd fctor of t ech term, d the deomitors ivolve epoets tht re trigulr umbers. I moder ottio, oe might write s = = 0 k ( ) k= 0 T( ) T( + ) where T() is the th trigulr umber, give by T() = ( + )/. It tkes some thought to recogize wht specil properties might distiguish this series, eve if we follow 8 th cetury style d igore questios of covergece. Adveturous reders my wt to ply with the series for few miutes before redig o. Note tht if =, the the first term equls, d ll the rest of the terms re zero, sice they hve fctor of i their umertors. Hece s( ) =. Further, if = the the first term reduces to ll the other terms vish, so s( ) =. Similrly, Euler observes tht ( ) + d the secod term reduces to, while s = for positive whole umbers, though Euler oly shows us the clcultios up to = 3 d clims to hve doe them s = whe is positive iteger, himself up to = 5. Lter i the rticle, Euler gives proof tht ( ) but t this poit he gives oly evidece. This evidece, though, turlly leds to the cojecture tht s =. ( ) log But it is t. Euler demostrtes this by showig tht, for = 0, s(9) = but log 9 = (though the editors of the Oper omi ote tht Euler mde error i his clcultio of s(9). It should be It still is t log 9.) It is etertiig to check this usig your fvorite computer lgebr system. Euler ppretly picked = 0 d = 9 rther chritbly. I fct your computer lgebr system c show tht most of the time, s() d log re ot very close together t ll. Euler did ot hve computer lgebr system. He does hve esier wy, ot ivolvig pproimtios, to show tht s() is ot logrithm fuctio. He tkes = 0, d he kows tht log 0 =. O the other hd, for = 0, his series gives

3 This series hs fiite sum, so s() cot be the logrithm fuctio. Note, though, tht this series is the egtive of oe of the series tht Euler proposed t the ed of E5, d here gi he tells us, this series cot be summed. It must hve both disppoited d ecited Euler tht s() ws ot the logrithm fuctio. O the oe hd, if the series were the logrithm fuctio, the it would hve provided uusully fstcovergig mes of clcultig logrithms. O the other hd, sice s() is ot the logrithm fuctio, it chlleged oe of his bsic ssumptios. He hd two fuctios, s() d log tht turlly epressed the sme sequece of umbers; tht is, they greed t ifiitely my vlues of, yet they were ot the sme fuctio. This is i sectio 4 of this 3-sectio rticle. Euler speds most of the rest of this rticle studyig properties of his series, showig how much it relly does differ from the logrithm fuctios, d lso showig rigorously tht it does gree with the logrithm fuctio t iteger powers of. We ll omit tht, d refer the iterested reder to the Mttmueller trsltio vilble o The Euler Archive. [E90] Isted, we will lep forwrd to sectio 8, where Euler returs to the series from E5, Of this, Euler writes (i the Mttmueller trsltio), for >, eve though it is fiite d c esily be determied by pproimtios, [it] cot be epressed either i rtiol or i irrtiol umbers. It ppers therefore especilly worth the effort tht mthemticis ivestigte the ture of tht trscedetl qutity by which its sum is epressed. Here, Euler clls irrtiol wht we would cll lgebric, though he uses the word trscedetl i its moder sese. Uble to epress the series ectly, he sets out to give good pproimtios. He defies ew series s, ot the sme oe he lso deoted by s erlier i this rticle, s s = z z z z z The the series Euler wts to pproimte is the vlue of this series s whe z =. He sets to work o this ew series. Euler skips few steps here tht we will put i. The first term of this series c be epded ito geometric series s 3 z z z = = z z Likewise, the secod term epds s 3

4 3 z z z z z = = The other terms epd similrly. Euler reverses the order of summtio to gther together like powers of z, d gives us s i differet form s s = z z This doe, Euler uses trick tht he hd lso used i E5. He kows tht most of the error i the pproimtio of the sum of series occurs i the first few terms of the series. To reduce this effect, he tkes A to be the sum of the first terms of s; tht is This leves. A = z z z z z = s A z + z + z + z Euler gi epds this ito geometric series d collects like terms to get s= A z z Ech lie i this epressio is geometric series, so he c sum those to get 3 z z z s= A ( ) ( ) ( ) ( ) For the series Euler proposed i E5, the cse = d z =, this gives s= A He tkes = 4 d so sums the first four terms of the series to get 4

5 A = = This is icorrect i the lst deciml plce, which should be 3. Tht mkes Euler sums the first 5 terms of this series to get s= A s= A = This grees ectly with my computer lgebr system for the ifiite series. This is the best Euler c do with tht series from E5. He hs oe lst remrk, though. If we go bck to the series from the begiig of sectio 8, d if we epd ech of the terms s geometric series, the collect like powers of, we get the form s = , where the th umertor couts the umber of divisors of. Euler does ot try to epli why this is 4 true (though it is), but he does tell us tht the umertor i the term is 4 becuse the epoet 6 6 hs four divisors, mely,, 3 d 6. For prime epoets, the umertor will lwys be, d for composite epoets it will lwys be greter th. These umertors re esy to clculte, d for the specil cse = 0, it is esy for us to sum the series. Euler gives the sum to 30 deciml plces: s = s = It is cler tht the series hs umber theoretic properties, but Euler did ot pursue them y frther. Those properties re relted to wht we ow cll q-series. They were etesively studied by such gret mthemticis s Guss, Cuchy, Jcobi, Sylvester d Rmuj d re still of gret iterest tody. 5

6 I d like to thk Wrre Johso, ow t Coecticut College, for brigig this rticle to my ttetio, for helpig me uderstd its coectios with q-series, d for helpful commets o the tet itself. Refereces: [BPV] Bibiloi, Lluís, Jume Prdis d Pelegri Vider, O Series of Goldbch d Euler, The Americ Mthemticl Mothly, v. 3 o. 3, Mrch 006, pp This rticle wo the Lester R. Ford Awrd i 006. [D] Duhm, Willim, Euler: The Mster of Us All, MAA, Wshigto, DC, 999. [E0] Euler, Leohrd, De summtioe iumerbilium progressioum, Commetrii cdemie scietirum imperilis Petropolite 5 (730/3) 738, pp Reprited i Oper omi I.4, pp Avilble olie t EulerArchive.org. [E5] Euler, Leohrd, Methodus geerlis summdi progressioes, Commetrii cdemie scietirum imperilis Petropolite 6 (73/33) 738, pp Reprited i Oper omi I.4 pp Avilble olie t EulerArchive.org. [E90] Euler, Leohrd, Cosidertio qurudm serierum que sigulribus propriettibus sut predite, Novi commetrii cdemie scietirum imperilis Petropolite 3 (750/5) 753, pp. 0-, Reprited i Oper omi I.4 pp Avilble olie t EulerArchive.org, where oe lso fids Eglish trsltio by Mrti Mttmueller. [S] Sdifer, C. Edwrd, The Erly Mthemtics of Leohrd Euler, MAA, Wshigto, DC, 007. Ed Sdifer (SdiferE@wcsu.edu) is Professor of Mthemtics t Wester Coecticut Stte Uiversity i Dbury, CT. He is vid mrtho ruer, with 35 Bosto Mrthos o his shoes, d he is Secretry of The Euler Society ( His ew book, The Erly Mthemtics of Leohrd Euler, ws published by the MAA i December 006, s prt of the celebrtios of Euler s terceteil i 007. The MAA published collectio of the How Euler Did It colums durig the summer of 007. How Euler Did It is updted ech moth. Copyright 007 Ed Sdifer 6