How Euler Did It. 1- n p. 1 k pprime. For the readers unfamiliar with the zeta function, we ll give a brief introduction.

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1 Multi-eta fuctios Jauary 008 How Euler Did It by Ed Sadifer Two of Euler s best kow ad most ifluetial discoveries ivolve what we ow call the Riema eta fuctio. The first of these discoveries made him famous whe he solved the Basel problem. He showed [E4] that the sum of the reciprocals of the square umbers was π etc. = Euler s secod great result [E7] o this topic was what we ow call the Euler product formula, ad we write it as =. k pprime - p For the readers ufamiliar with the eta fuctio, we ll give a brief itroductio. It has log bee kow that the harmoic series, these terms to some power, as , diverges, but that if we take , the the series will coverge wheever >. The 4 value to which it coverges depeds o, ad is ow deoted ζ ( ). π 6 writte either as a ifiite sum or as a ifiite product. Euler s first result showed that ζ ( ) =, ad his secod result showed that ( ) ζ ca be Sice Euler s time, the eta fuctio has captured the imagiatios of may great mathematicias. I particular, i 859 Berhard Riema, showed that eed ot be a real umber, ad that the eta fuctio has a atural aalytic cotiuatio as a fuctio of a complex variable. Hece the fuctio is traditioally called the Riema eta fuctio ad defied i terms of a complex variable s as

2 ζ ( s) = s k. Riema surmised that his fuctio is ero for ifiitely may values of s, ad that all its complex roots share the property that their real part is ½. For some reaso that is still uclear, Riema s cojecture is kow as the Riema hypothesis istead of the Riema cojecture. Though it is badly amed, it is oe of the most importat usolved problems i mathematics today. Over the years, eta fuctios have evolved a umber of variatios. For example, istead of takig the sum over the ordiary itegers, oe could take the sum over the itegers i some umber field. This leads to a topic kow as L-series. We could also chage the umerators i the sum, ad look at sums like χ ( k), k where χ ( k) is some fuctio of k. We saw Euler himself do somethig like this i last moth s colum, where we describe a series Euler ivestigated the ed of [E90], 4 4 s = etc a a a a a a a a a This is ot exactly a L-series, because the deomiators form a geometric series, ot a arithmetic series, but the th umerator is give by the umber-theoretic fuctio This is very much i the spirit of a moder L-series. χ ( ) = the umber of divisors of. At a recet sectio meetig of the MAA, Michael Hoffma of the US Naval Academy i Aapolis brought to my attetio aother moder variatio of the eta fuctio, ad showed how that variatio derived from Euler s work. Most of the remaider of this colum is based o what he showed me. [H99, H007] I the moder way, a multiple eta value is defied as ζ ( i, i, ik ) = i i. ik > k k Both the motivatio ad the otatio are obscure here. Let s try to utagle both of them at the same time. Let s ask, what would happe if we multiplied together two ordiary eta fuctios, say m ζ? As series, we would get ζ ( ) ad ( ) ζ = m k k ( m) ζ ( )

3 Euler would ot have used the Sigma otatio, so he might have writte this as etc. etc. m m m The he probably would have expaded this to get somethig like () m m m m m m m m m m m m m m m Now we ca take this apart ad put it back together a differet way. First, ote the terms o the diagoal of this sum, the first term i the first row, the secod i the secod row, etc. They sum to form a ordiary eta fuctio as follows: = ζ ( m+ m m m m ) O the other had, the terms below the diagoal sum as etc. m m m m m m m This might be a little clearer (or maybe ot) if we explicitly iclude the factors of i the products. This gives etc. m m m m m m m Now we ca see clearly that i each deomiator, moder Sigma otatio as m a b we have a > b >, so we ca rewrite the sum i a> b Glacig back up the page, we see that is exactly what Hoffma defies as the multi-eta value ζ ( m, ) m ab Likewise, the terms above the diagoal i our product sum to ζ ( m, ). This gives oe of the motivatios for multi-eta values. They arise i multiplyig values of the eta fuctio, ad lead to the formula

4 () ζ ( m) ζ ( ) = ζ ( m+ ) + ζ ( m, ) + ζ ( m, ) as A slightly differet approach ivolves defiig a differet multi-eta value, usig > istead of >, ζ *( m, ) =. m ab a b This icludes the diagoal terms i the big summatio, so ζ *( m, ) ζ ( m ) ζ ( m, ) leads to a similar formula about products of eta values: () ζ ( m) ζ ( ) ζ *( m, ) ζ *( m, ) ζ ( m ) = + +. = + + ad it All these are moder ideas ad moder otatios, ad they are well documeted i the fie bibliography maitaied by Michael Hoffma. I was surprised to lear that these ideas are ot of moder origi, but first came from Christia Goldbach i a letter to Euler dated December [J+W] There, Goldbach uses 8th cetury otatio to fid that π 7 6 9π + =, 5670 ζ *, ( ) = ad ζ *5, ( ) ζ *4, ( ) though he does ot claim to kow either ζ *5, ( ) or *4, ( ) ζ. Euler respoded quickly, though at this time Euler was i Berli ad Goldbach was i Moscow, ad the mail was perhaps slower i the middle of the witer. Noetheless, Euler s letter dated Jauary 9, 74 cotaied some additios to Goldbach s results, providig equatios for ζ *, ( ), ζ *5, ( ), ζ *7, ( ) ad ζ *9, ( ) i terms of products of the ordiary eta fuctio. Not all of Euler s claims are correct, though. Hoffma poits out that his claim that ζ * 6, ζ ζ 5 ζ 4 ζ 8 4 ( ) = ( ) ( ) ( ) + ( ) is false, ad so were a few others. I fact, this oe is t eve very close. Accordig to Maple, the left had side is about.6557 ad the right had side is about , ad Michael Hoffma tells me that obody has yet foud a way to write ζ *6, ( ) as a polyomial fuctio of ordiary eta values with ratioal coefficiets, ad whether or ot oe exists is a ope ad active research questio. Euler ad Goldbach exchaged a total of five letters o this subject. I the last oe, dated February 6, Euler used properties of these multi-eta fuctios to give eightee-place decimal ζ through = 6. approximatios to ( ) 4

5 As usual, Euler could ot leave to a letter what he could expad ito a paper, but Euler apparetly let early 0 years pass before he retured to multiple eta values. His work became [E477], writte i 77 ad published i 776. There he begis by citig his letters with Goldbach, ad ζ * m,. He wrote describig the series that is ow deoted by ( ) I commercio litterario, quod olim com Illustrissimo Goldbachio coluerum, iter alias varii argumeti speculatioes circa series i hac forma geeralis + etc. m + + m m Euler used to deote what we ow write as ζ m ( m). He writes P for ζ *( m, ) ad Q for ζ *( m, ). I Euler s otatio, ad i the origial 776 publicatio, he wrote his versio of formula like this: + etc. m + + m m = P etc. Q + m m m m m m + = 4 4 ex pricipio supra stabilito habebimus P+ Q= + + m m A few paragraphs later, Euler improves his otatio, ad deotes *( m, ) istead of by P. The formula becomes + = + y y + ζ by m m m m y Later i the paper, Euler develops some geeral results about multi-eta values, especially those that he wrote as ad we would write as ζ * (, ). He shows easily that =, that is ζ *, ( ) ζ ( ) =. 5

6 It takes a bit more work for him to show that ad the still more work to fid that 5 = 5 =,, that is ζ * (, ) ζ ( ) ζ ( 4 ) 4 = 4, that is ζ *, ( ) ζ ( 5 ) ζ ( ) ζ ( ) 5 Beig Euler, he cotiues for more tha pages, stoppig with =. = We wo t traslate this ito moder otatio. I this work, the patters are ot evidet, but he applies formula several times ad trasforms the results ito ad fially 4 = y 5 = 4 y 6 4 = 6 y = From this, the patter is evidet. I moder otatio, it reads ( ) = ( + ) ( + ) ( i) ( i+ ) ζ *, ζ ζ ζ This is a form of what is ow kow as Euler's decompositio formula for the double eta fuctio, ad more tha two ceturies later it is still a iterestig result, Refereces: Special thaks to Michael E. Hoffma for ispirig ad helpig with this colum. i= [E4] [E7] Euler, Leohard, De summis serierum reciprocarum, Commetarii academiae scietiarum imperialis Petropolitaae 7 (74/5) 740, pp. -4. Reprited i Opera omia I.4 pp Available olie at EulerArchive.org. Euler, Leohard, Variae observatioes circa series ifiitas, Commetarii academiae scietiarum imperialis Petropolitaae 9 (77) 744, pp Reprited i Opera omia I.4 pp Available olie at EulerArchive.org. 6

7 [E90] Euler, Leohard, Cosideratio quarudam serierum quae sigularibus proprietatibus sut praeditae, Novi commetarii academiae scietiarum imperialis Petropolitaae (750/5) 75, pp. 0-, Reprited i Opera omia I.4 pp Available olie at EulerArchive.org, where oe also fids a Eglish traslatio by Marti Mattmueller. [E477] Euler, Leohard, Meditatioes circa sigulare serierum geus, Novi commetarii academiae scietiarum imperialis Petropolitaae, 0 (775) 776, pp. 4-5, Reprited i Opera omia I.5 pp Available olie at EulerArchive.org [H99] Hoffma, Michael E., Multiple Harmoic Series, Pacific Joural of Mathematics, 5. (99) pp [H007] Hoffma, Michael E., Multiple Zeta Values: From Euler to the Preset, preseted at the MD-DC-VA Sectio meetig of the MAA, Aapolis, MD, November 0, 007. Notes available olie at [J+W] [S] Juskevic, A. P., ad E. Witer, eds., Leohard Euler ud Christia Goldbach: Briefwechsel , Akademie- Verlag, Berli, 965. Sadifer, C. Edward, The Early Mathematics of Leohard Euler, Mathematical Associatio of America, Washigto, DC, 007. Ed Sadifer (SadiferE@wcsu.edu) is Professor of Mathematics at Wester Coecticut State Uiversity i Dabury, CT. He is a avid maratho ruer, with 5 Bosto Marathos o his shoes, ad he is Secretary of The Euler Society ( His first book, The Early Mathematics of Leohard Euler, was published by the MAA i December 006, as part of the celebratios of Euler s terceteial i 007. The MAA published a collectio of forty How Euler Did It colums i Jue 007. How Euler Did It is updated each moth. Copyright 008 Ed Sadifer 7