Remarques sur un beau rapport entre les series des puissances tant directes que reciproques

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1 Aug 006 Traslatio with otes of Euler s paper Remarques sur u beau rapport etre les series des puissaces tat directes que reciproques Origially published i Memoires de l'academie des scieces de Berli pp Traslated by Lucas Willis ad Thomas J Osler Mathematics Departmet Rowa Uiversity Glassboro NJ 0808 Osler@rowaedu Itroductio to the traslatio ad otes: This traslatio is the result of a happy collaboratio betwee studet ad professor Lucas Willis is a udergraduate Mathematics Major ad a Frech Mior Tom Osler has bee a mathematics professor for 45 years Together we struggled to uderstad this brilliat work Whe traslatig Euler s words we tried to imagie how he would have writte had he bee fluet i moder Eglish ad familiar with today s mathematical jargo Ofte he used very log seteces ad we frequetly coverted these to several shorter oes However i almost all cases we kept his origial otatio eve though some is very dated We thought this added to the charm of the paper Oe exceptio is Euler s use of lx for our log x the atural logarithm We thought lx was too cofusig Euler was very careful i proof readig his work ad we foud few typos Whe we foud a error we called attetio to it i parethesis ad italics i the body of the traslatio Other errors are probably ours Whe half the traslatio was completed we leared that Professor Robert Stei had made a traslatio of this paper several years earlier He geerously shared his traslatio with us ad we gratefully ackowledge his help The otes that follow this traslatio are a collectio of material that we accumulated while tryig to uderstad ad appreciate Euler s ideas I these otes we completed some steps that Euler omitted added some historical remarks itroduced some moder otatio ad moder mathematical thoughts especially o the use of diverget series Remarks o a beautiful relatio betwee direct as well as reciprocal power series (E 35) By Leohard Euler The relatio which I ited to develop here cocers the sums of these

2 two geeral ifiite series: m m + 3 m 4 m + 5 m 6 m + 7 m 8 m + & c & c The first cotais all the positive powers of the atural umbers of a variable m ad the other egative or reciprocal powers of the same atural umbers of a variable while alteratig the sigs of the terms of both series My pricipal goal is to show that though these two series are etirely differet their sums have a very beautiful relatioship betwee them If we kow the sum of oe of these two series we might deduce the sum of the other series I will show that by kowig the sum of the first series for a variable m we ca almost always determie the sum of the other series for the variable m + It seems importat to remark that while I oly demostrate this relatio for certai special cases my argumet is carried to such a degree of certaity that the reader will coclude it very rigorously show For the series of the first type sice the terms become icreasigly large it is quite true that we could ot create a correct idea of their sum if we uderstad by the sum a value that we all the more approach the more we add terms to the series Thus whe it is said that the sum of this series etc is /4 that must appear paradoxical For by addig 00 terms of this series we get 50 however the sum of 0 terms gives +5 which is quite differet from /4 ad becomes still greater whe oe icreases the umber of terms But I have already oticed at a previous time that it is ecessary to give to the word sum a more exteded meaig We uderstad the sum to be the umerical value or aalytical relatioship which is arrived at accordig to priciples of aalysis that geerate the same series for which we

3 3 seek the sum After havig established this relatioship it is o more doubtful that the sum of this series etc is /4; sice it arises from the expasio of the formula (+ ) whose value is icotestably /4 The idea becomes clearer by cosiderig the geeral series: x+ 3x 4x x + & c that arises while expadig the expressio ( + x) which this series is ideed equal to after we set x 3 It is easy to use the differetial calculus to fid the sums of these series ad we obtai the followig summatios: x+ x x + & c + x 3 3 x 3x 4 x & c x 3 x 4 x & c x 3 x 4 x & c ( + x) x ( + x) 4x+ xx ( + x) 4 x+ xx x x 3 x 4 x & c ( + x) 5 3 6x+ 66xx 6x + x x 3 x 4 x & c ( + x) x+ 30xx 30x + 57x x x+ x x + c & ( + x) 7

4 4 &c We obtai our series of the first type by takig x ad get the followig sums: & c & c & c & c & c & c & c & c & c & c + 04 &c which is 4 As to the series of the other type we previously kew oly the case & c

5 5 whose sum is log First I discovered the sum of the reciprocal series with square powers ad the the sum for all the other eve powers I have show that the sums of all these series deped o π the circumferece of a circle of diameter I foud the followig sums of these series & c Aπ & c Bπ & c Cπ & c Dπ & c Eπ A 6 B 5 A 4 C AB 7 4 D AC B E AD+ BC &c from which I calculated sums of our series of the secod type with alteratig sigs & c Aπ & c Bπ & c Cπ & c Dπ & c Eπ 0

6 & c &c However i the cases where is a odd umber all my effort to fid their sum is a failure up to ow Nevertheless it is certai that they do ot deped i a similar way o the powers of the umber π Perhaps the followig observatios will spread some light here 5 Sice the umbers A B C D etc are of the highest importace i this subject I will list them here as far as I have calculated them Fπ 0 A 3 B 53 4 C D E F G 5 H I K L 3 3 M N 7 73 O P Q

7 7 R The summatio of the series of the first type i the cases where the variable m is a odd umber also ivolves these same umbers A B C D etc We recall that whe this variable is a eve umber the sum becomes equal to zero A method should be used that reveals this beautiful depedece To achieve this demostratio it is ecessary to use a geeral method that I have previously published to determie the sums of the series of geeral terms Let X be a geeral fuctio of x ad let it be represeted by X f : x Let us cosider the ifiite series f : x+ f :( x+ α) + f :( x+ α) + f :( x+ 3 α) + f :( x+ 4 α) + & c where the followig terms are fuctios of x+ α x+ α x+ 3 α etc Let us call the sum of this series S which is also a fuctio of x If we put x + α i place of x it becomes α d S α dd S α d S α d S S dx dx 3 dx 34 dx & c 3 4 This expressio is the sum of the series f :( x+ α) + f :( x+ α) + f :( x+ 3 α) + f :( x+ 4 α) + & c ad is equal to S - f : x S -X so that α d S α dd S α d S α d S X dx dx 3 dx 34 dx & c 3 4 However from this equatio I previously derived the formula α Ad X α Bd X α Cd X S Xdx X & c α dx dx dx

8 8 where A B C etc are the same umbers which I have just developed By this meas we arrive at the desired sum S usig the itegral Xdx ad the derivatives of every order of the fuctio X 7 Now to obtai the alteratig sigs i place of α let us write α to get the summatio: f : x+ f :( x+ α) + f :( x+ 4 α) + & c Xdx X α αadx α Bd X α Cd X + + & c dx dx dx ad subtractig twice this from the precedig series we get f : x f :( x+ α) + f :( x+ α) f :( x+ 3 α) + f :( x+ 4 α) & c ( ) ( ) ( ) α Ad X 4 α 3 Bd 3 X 6 α 5 Cd 5 X X + + & c dx dx dx where the term which cotais the itegral Xdx has disappeared Let us proceed ow to our goal by lettig f : x X x m ad obtai the followig sum of the series: x m ( x+ α) m + ( x+ α) m ( x+ 3 α) m + ( x+ 4 α) m & c m ( ) α ( ) m Ax m( m )( m ) α Bx m x + 3 ( ) 4 3 m 3 mm ( )( m )( m 3)( m 4) α Cx m 5 ( ) 8 7 m m( m )( m )( m 3)( m 4)( m 5)( m 6) α Dx &c

9 9 which cotais oly a fiite umber of terms whe the variable m is a positive iteger Therefore settig α we will have for our series of the first type m m m m m m x ( x+ ) + ( x+ ) ( x+ 3) + ( x+ 4) ( x+ 5) + & c m m( m )( m ) x ( ) Ax + ( ) Bx m m 4 m 3 mm ( )( m )( m 3)( m 4) ( ) Cx 6 m 5 m( m )( m )( m 3)( m 4)( m 5)( m 6) 8 7 ( m + ) Dx &c 8 Now we have oly to let x to obtai the geeral sum of all our series of the first type However it is easier to fid the sum by lettig x 0 from which we get 0 m m + m 3 m + 4 m 5 m + 6 m 7 m + & c which is the egative of the sum we seek By lettig x 0 all the umbers i the sum disappear except for oe where the power of x is zero This occurs wheever m is a odd umber because whe it is eve all the members disappear ad the sum of the series is reduced to zero Therefore takig the egative of these sums we fid the followig m 0 m m m & c ( ) & c + A & c & c 3 ( 4 ) B 3

10 0 m 4 m 5 m 6 m 7 m 8 m 9 m & c & c 5 ( 6 ) C & c & c 7 ( 8 ) D & c & c 9 ( 0 ) E & c &c Whe these sums are calculated it is foud that they are the same values that I listed above but ow we see their coectio with the values A B C etc 9 We divide each oe of these series of the first type by that series of the secod type which cotais the same umber A B C D etc to obtai the followig equatios ( ) ( ) π & c & c & c & c ( ) 3 4 ( ) π & c & c & c & c

11 6 ( ) 5 6 ( ) π & c & c & c & c ( ) 7 8 ( ) π & c & c & c & c ( ) 9 0 ( ) π & c & c &c (Misprit i the 3 rd equatio above: the term 4 6 is mistakely prited as 5 6 ) But the equatio which precedes these is & c & c log whose coectio with the followig oes is etirely hidde 0 Cosideratio of these equatios leads me to this geeral formula: & c ( ) ( ) π 3 ( ) + N & c where we must ow proceed to precisely determie the coefficiet N of the

12 variable To achieve this evaluatio let us cosider the values of this coefficiet N which correspod to each variable that I have just examied: N etc etc Sice wheever is a odd umber the letter N must disappear ad for the case 4i+ it is ecessary that it is N ; but for the case 4i it becomes N it is obvious that we ca satisfy these coditios by takig N cos( π / ) : For this reaso I veture to propose the followig cojecture that for ay variable the followig equatio is always valid: & c ( ) 3 ( ) π cos & c ( ) π This cojecture will udoubtedly stad out as bold but sice it has bee show to be true for the case where is a positive iteger larger tha oe I shall ext prove this relatio for the case ad the for 0 After that I will show that if this cojecture is proved for the cases where is a positive iteger it will also be true whe is a egative iteger Fially I will demostrate some cases where we give to a fractioal value First let ad get the expressio & c & c whose value is /(log ) However for this case our cojectured relatio cotais the expressios 3 ( ) ad π π while the two other expressios cos( π / )

13 3 ad are both zero with oe dividig the other This is why I write the expressio that our cojecture gives i this case as: π cos π π cos where it is a questio of determiig the value of the fractio where the umerator ad the deomiator are both zero Now let us treat the letter as a variable ad sice the differetial of the umerator is πd π si ad that of the deomiator is log d our fractio for this case will be the same as π π si Lettig this is reduced to log π so that the value that we log seek will be cos( π / ) π log Thus our cojecture also holds for the case which iitially appeared to etirely deviate from the rule of the prior cases This is already a type of proof for the truth of this cojecture for it seems impossible that a false suppositio could support this test We may already look at our cojecture as very firmly established However I am goig to brig eve more covicig evidece Let 0 ad ow we must cosider the expressio & c & c

14 4 whose value is obviously equal to log However from our cojecture we have the expressios cos( π / ) ad πⁿ as well as 3 (-)(ⁿ-) The factor 3 (-) is ifiite ad the other expressio ⁿ- is zero from which we see that our cojecture is ot yet cotradicted i this case But to proceed to a proof I otice that 3 ( ) 3 ad i the case 0 we have 3 Therefore i this same case 3 ( ) ad the value from our cojecture ( ) where sice the umerator ad deomiator disappear whe puttig 0 we have oly to substitute for them their differetials Thus we have aother fractio d log log d equivalet to the oe for the case 0 Now this oe gives us the same value log that the ature of the series demads Here is therefore a ew verificatio which beig joied to the precedig oe will be able to give us of a more complete demostratio of our cojecture Nevertheless we have ot give a direct demostratio that cotais at oce all the possible cases 3 Our cojecture beig verified oly for all the case where is a positive iteger I am goig to prove ow that it is equally true whe we take for ay egative iteger I these cases the value of the expressio 3 (-) is ifiity ad this seems to ivalidate the cojecture that I have i mid; however a evaluatio that I made previously will overcome this obstacle Take this otatio [ λ ] to represet the

15 5 product: 3 λ I have show previously that it is always true that λπ [ λ] [ λ] Thus lettig m or m+ we get the expressio si λπ m m m m m & c m m m m m & c m+ m ( ) m m+ ( ) π 3 ( ) ( m) π cos mπ where sice 3 ( m) [ m] ad [ m] [ m] we will have si mπ mπ π 3 ( m) 3 msimπ 3 ( m )simπ The sice cos(( m) π / ) si( mπ / ) the expressio from our cojecture takes the followig form upo makig these substitutios: m ( ) m ( ) m π mπ si 3 ( m )si mπ m ( m ) π m m ( ) mπ 3 ( ) cos where we have used si mπ si( mπ / )cos( mπ / ) Now we have oly to ivert the equatio foud by puttig o top the deomiator ad o bottom the umerator ad we will obtai the equatio: m m m m m etc m m m m m etc m m ( ) ( m m ) π 3 ( ) mπ cos which is the same oe that was cojectured It is see clearly that if the

16 6 cojectured expressio is correct for the case where is a positive umber it will be true also whe is a egative umber because of m + fractio 4 A remarkable case is foud by settig / which leads to this & c & c whose umerator ad deomiator beig equal gives the value : We must fid the value of the expressio which the cojecture suggests it is equal to: ( ) 3 π cos + 4 π π π However I have show before by examiig the factorial progressio ; ; 3 34 ; whose geeral term is 3 [ ] that settig / we have [/ ] π / Because [/ ] (/ )[ / ] it is evidet that [ /] π / (this should be π ) which ideed leaves our expressio equal to There should ot be ay further doubt about our cojecture havig verified it ot oly for all the cases where the variable is a iteger be it is positive or is egative but also for the case / For the other cases ivolvig fractioal umbers that we would like to use istead of we caot claim a particular proof cosiderig that o oe has yet discovered a correct method to determie the sum of a series whe the variable is a fractio I these cases it is ecessary to be satisfied with umerical approximatios: however we will see that our cojecture remais true

17 7 5 To carry out a test let 3/ Because 3 ( ) this fractio π ad 3π cos 4 must be equal to this quatity & c & c π π ( ) But o calculatig the first 9 terms of the series i the umerator we get from which it was ecessary ad ifiitum to cut off the sum of the followig terms & c From sectio 7 this is 4 ( ) A ( ) 3 B ( ) C 3579 ( ) D + & c A 3 5 B C D & c ad after substitutig the values

18 8 A 6 B 90 C 945 D 9450 E &c we calculate which is about 5460 (should be ) ad makig the umerator series : (should be ) Now for the lower series the first 9 terms gives (should be ) from which it is ecessary to cut off the sum from all of the followig which is 3 3 A B C + & c ad is about ad thus the sum of this ifiite series will be (should be ) Now let us see if the first series divided by this oe which is the fractio (should be / ) is equal to the value The differece is so small beig oly two hudred-thousadths of the uit (should be te millioths) that oe could ot doubt i the least that this matter is true 6 Sice our cojecture has bee demostrated to the highest degree of certaity so that there remais o more doubt of its validity whe is a fractio we list results for the case where is a fractio of the form (i + ) / : ( ) ( ) π & c & c ( ) ( ) π & c & c 4

19 ( ) ( ) π & c & c ( ) ( ) π & c & c ( ) ( ) π & c & c ( ) ( ) π & c & c ( ) ( ) π & c & c 8 It should be oted that λ λ + ( λ ) ca be reduced to λ λ Therefore with each pair of these series as soo as we have foud the sum of oe we will fid from it the sum of the other i a relatio ivolvig the umber π 7 Regardig the reciprocal series of the powers & c I have already observed that their sums have bee foud oly whe is a eve iteger ad that for the case where is a odd iteger all of my efforts have bee completely useless Now havig related the sum of these reciprocal series to that of the direct series ad sice we i geeral kow the sum of

20 & c + + we could expect to fid some way to achieve our goal but it is ufortuate that wheever is a odd umber the sum of this direct series is zero Thus we could ot coclude aythig because lettig λ + by our cojecture we have: + + & c λ+ λ+ λ+ λ ( ) π λ+ λ ( ) λ λ λ λ λ λ & c 3 λ + cos π However i this last expressio the value of both the umerator λ λ λ λ & c λ + ad the deomiator cos π si λπ are zero whe λ is a iteger It is true that we ca easily discover the value of such a fractio by substitutig istead of the umerator ad deomiator their differetials however this techique is ot successful as I am goig to demostrate 8 To show this we fid that the differetial of the umerator is λ λ λ λ ( c) dλ log log + 3 log 3 4 log 4 + & ad that of the deomiator is πdλcosλπ We get for our case the sum expressed i the form + + & c λ+ λ+ λ+ λ λ λ ( ) π λ+ λ( ) 3 cosλπ λ λ λ λ ( log log 3 log 3 4 log 4 & c) + + Upo substitutig for λ the umbers 3 we obtai the followig summatios:

21 3 π ( log log + 3 log3 4 log 4 + & c) c & π ( log log + 3 log3 4 log 4 + & c) c & ( log log 3 log3 4 log 4 & ) c & π + + c π ( log log + 3 log3 4 log 4 + & c) c & π ( log log + 3 log 3 4 log 4 + & c) c & Thus it is ecessary that we fid the sums of the series of the form λ λ λ λ log log + 3 log 3 4 log 4 + & c But this evaluatio is perhaps more difficult tha that which we have i mid ad I do ot foresee ay method that ca lead us to the desired goal 9 These equatios become a bit simpler upo cosiderig that the series & c is equal to this oe m m m m & m m ( ) + + m m m m c Usig the previous methods we fid the geeral sum & c λ+ λ+ λ+ λ λ π λcosλπ ad the list the particular cases: λ λ λ λ ( log 3 log3 4 log 4 5 log5 & ) c π ( log 3 log 3+ 4 log 4 & c) c & +

22 π ( log 3 log 3+ 4 log 4 & c) c & π ( log 3 log3+ 4 log 4 & c) c & + π & &c ( log 3 log3 4 log 4 & ) c (Misprit i the d equatio above: the term 5 3 appears as 4 3 ) However here it should be oticed that the geeral sum i these two previous paragraphs is true oly whe the variable λ is a positive iteger sice it is based o the coditio that the sum of the series c λ λ λ & c is zero This sum is ot zero aymore i the case λ 0 therefore we ca use for λ oly the umbers 3 4 I ote that the series log log 3 + log 4 log 5 + etc has the sum π log which gives us reaso to hope for success i fidig the sum of the series that lead us here 0 I the same way we ca compare the sums of these two ifiite series + + ad + + & c & c ad obtai the similar cojecture & 3 ( ) si c π & c π Wheever is a positive eve iteger the umerator sum of the series disappears ad i these cases also the sie of the agle becomes zero Therefore lettig λ we have:

23 3 λ λ λ λ π (3 log 3 5 log log 7 & c) c λ λ λ λ (λ ) cosλπ + + & Lettig be a positive iteger we calculate the followig summatios: π (3 log3 5 log5 + 7 log 7 & c) + + & c π (3 log3 5 log5 + 7 log 7 & c) + + & c π & c (3 log3 5 log5 7 log 7 & ) c π (3 log 3 5 log log 7 & c) + + & c π (3 log3 5 log5 + 7 log 7 & c) + + & c This last cojecture cotais a expressio simpler tha the precedig oe therefore there is hope that further work will brig success Fidig a demostratio of it will ot fail to spread much light o a umber of other problems of this ature Traslator s Notes These otes are keyed to the 0 sectios of Euler s paper Sectio The mai iterest i this paper for moder readers is that Euler fids a relatio equivalet to the fuctioal equatio for the zeta fuctio The zeta fuctio defied by the series ζ () s Re( s ) > s is oe of the most importat special fuctios i mathematics The fuctioal equatio for the zeta fuctio is

24 4 s π s () ζ( s) ( π) cos Γ( s) ζ( s) Istead of usig ζ ( s) Euler uses the alteratig series [ p 807] + ( ) η() s s which is defied for the wider regio Re( s ) > 0 The fuctio η( s) is oe simple step removed from ζ ( s) as show by the relatio s ( ) η() s ζ() s The fuctioal equatio ow becomes s s s π s η( s) π cos Γ( s) η( s) () ( ) ( ) This is easily maipulated ito Euler s relatio whe s writes as (3) a atural umber which he ( ) ( ) π ( )! π cos Whe is a atural umber the series i the umerator diverges ad the moder reader is severely troubled Euler however the grad master of series maipulatio is udauted Oe of the most iterestig features of this paper is Euler s excitig evaluatio of diverget series Throughout his paper Euler refers to the direct series m m m m m m m m (4) ad the reciprocal series (5) Euler had received much acclaim for fidig exact closed form values for the series ζ () s for s a positive eve iteger (see sectio 4 aad 5) So why did he choose to write about the alteratig series η ( s) rather tha ζ () s? Because the alteratig series ca be summed by his methods (see sectio 5) eve for values of s where the series diverges This is ot true of ζ () s Why was he iterested i a fuctioal equatio?

25 5 Probably because he was tryig to sum ζ () s for s a odd positive iteger a attractive problem that has ever bee solved At the ed of his paper Euler cosiders aother fuctio of iterest ζ( s) + η( s) Ls () + s s s 3 5 valid for Re( s ) > 0 Euler fids the fuctioal equatio s s s (5) L( s) π si π Γ( s) L( s) Hardy [4 p 3] writes These results have usually bee attributed to Riema Malmste ad Schlomilch It was comparatively recetly that it was observed first by Cahe ad the by Ladau that both () which is equivalet to () ad (6) stad i a paper of Euler writte i 749 over 00 years before Riema Helpful summaries of Euler s ideas from this paper are i Ayoub s article [] ad Hardy s classic book [4 pp 3-6] Sectio Euler explais his idea of the sum of a diverget series He uses the example of the series etc which arises by first expadig ( + x) x+ x x + x + the settig x We coclude that etc / 4 I moder terms this is the Abel summatio of the series which Hardy [4 p 7] describes essetially as follows: Defiitio of Abel sum If ax f( x) is coverget for x < ad 0 lim f( x) s x the we call s the Abel sum of a 0 Few moder mathematicias would try usig diverget series as a tool to discover mathematical truth As early as 86 Niels Heirik Abel wrote: The diverget series are the ivetio of the devil ad it is a shame to base o them ay demostratio whatsoever By usig them oe may draw ay coclusio he pleases ad that is why these series have produced so may fallacies ad so may paradoxes

26 6 O the other had Hardy writes: The defiitios of covergece ad divergece are ow commoplaces of elemetary aalysis The ideas were familiar to mathematicias before Newto ad Leibiz (ideed to Archimedes); ad all the great mathematicias of the seveteeth ad eighteeth ceturies however recklessly they may seem to have maipulated series kew well eough whether the series which they used coverged The great electrical egieer Oliver Heaviside (850 95) wrote: The series is diverget; therefore we may be able to do somethig with it It is the surprisig success of Euler s masterful maipulatios with diverget series that gives this paper its distictive flavor ad itese iterest Sectio 3 Euler lists seve special cases ( 0 to 6) of the followig closed form summatio of power series: cx+ cx cx + + ( ) cx x x x ( + x ) + Euler skips the derivatio so we outlie ours here We start with the case 0 which is the geometric series x+ x x + etc + x 3 To derive the remaiig relatios simply multiply the last oe by x ad differetiate For example if we have foud (3) cx+ cx cx + + ( ) cx x x x ( + x ) + the multiplyig by x we get x cx + cx c3x + + ( ) cx x x + 3 x 4 x + + ( + x) ad differetiatig we have the ext relatio cx + + cx + cx ( ) + c+ x x+ 3 x 4 x + + ( + x) A simple examiatio of the coefficiets c r i Euler s list reveals the recurrece relatio c ( r+ ) c + ( r+ ) c + r r r i actio We also ote that c r r c Now let x i (3) ad obtai the Abel summatio of the diverget series

27 7 c+ c c3+ + ( ) c (3) Sectios 4 ad 5 I previous work Euler foud closed form expressios for ζ ( z) whe z is a eve atural umber He showed that π ζ () 6 π ζ (4) π ζ (6) ad today we write i geeral (4) ( ) ζ( p) p+ p p p π p ( p)! s (See Kopp [5 page 37]) Usig η s ( ) (4) Here the umbers B () ζ() s we have p ( ) + p+ ( ) ( ) B p p π p ( p)! η( p) B are called Beroulli s umbers ad they are all ratioal The first few are B0 B B B4 B6 ad B3 B5 B ad their geeratig fuctio is (43) x B x x e! 0 These ca all be calculated recursively by startig with B 0 B B B B 0 for ad usig

28 8 These umbers first appeared i 73 i Jakob Beroulli s posthumous book The Art of Cojecturig [3 vol 3 pp 64 67] They arise i a formula which Beroulli cojectures for the sum p s( p ) k where p is a atural umber k The Beroulli umbers were used by Euler i some of his publicatios but he does ot use them explicitly here Rather Euler uses cosecutive letters of the alphabet i 4 6 the form ζ() Aπ ζ(4) Bπ ζ(6) Cπ I these otes we will use the otatio A ( ) whe we refer to Euler s otatio so that A() A A() Thus from (4) we have Euler s otatio i terms of Beroulli umbers B A(3) C (44) + ( ) B A ( ) ( )! Euler also summed other alteratig series related to ζ ( z) These iclude ( ) π L() 0 ( + ) 4 3 ( ) π L(3) 3 0 ( + ) 3 ( ) L(5) π 5 0 ( + ) ad i geeral ( ) p ( ) p p p p 0 ( ) ( p)! π L(p+ ) (See Kopp [5 page 40]) Here the E E are called Euler s umbers They are all itegers ad the first few are E0 E E4 5 E6 6 E8 385 ad E E3 E5 0 The E ca all be calculated recursively by startig with E 0 ad the usig E + E + E E0 0 for 3 4 Sectio 6 Euler uses a versio of the Euler Maclauri summatio formula I moder times we frequetly write this formula as

29 9 (6) k 0 x0 + α 0 α 0 0 α x0 k k [ α ] f( x + k) f( x) dx+ f( x ) + f( x + ) + B kα ( k)! k k f ( x0 + α) f ( x0) Usually it appears with α We ca thik of this as a geeralizatio of the trapezoidal rule for umerical itegratio of the fuctio f( x ) over the iterval from x x0 to x x + α where α is the legth of the icremet step 0 Euler uses a variatio of (6) which we ow describe Rather tha itegratig from left to right we ca itegrate from right to left We achieve this modificatio by replacig the icremet α by α ad ow thik of x 0 is the right most poit ad x x α is the left most poit We get 0 (6) k 0 x0 α α x0 k k [ ] f( x α+ αk) f( x) dx+ f( x ) + f( x α) B kα ( k)! k k f ( x0 α) f ( x0) This form of the summatio formula ca be easily modified if we wat to sum over the ifiite rage of umbers x x+ α x+ α For this purpose write x x0 α ad let x0 ad i such a way that x remais fixed to get f( x+ αk) f( x) dx+ [ f( x) + f( ) ] α k B kα k k f ( x) f ( ) ( k)! k 0 x k If we call becomes f ( ) B α ( k)! k k k ( ) + f ( ) βα k the the above summatio formula (63) x k B kα k ( + α ) ( ) + ( ) ( ) + β( α) f x k f x dx f x f x α ( k)! k 0 k Sectio 7 We must sum alteratig series so he modifies (63) We replace the icremet α by α to get

30 30 (7) x k k Bk α k f( x+ αk) f( x) dx+ f( x) f ( x) + β( α) k 0 α k ( k)! Next double (7) ad subtract the previous summatio (63) from it to get (7) ( k ) k B k α k k ( ) f( x+ αk) f( x) f ( x) β( α) + β( α) k 0 k ( k)! Fially let α ad get the geeral alteratig ifiite series ( k Bk k ) k ( ) f( x+ k) f( x) f ( x) β() + β() ( k)! k 0 k If the series o the left coverges the we see that the costat term β() + β() 0 by imagiig x We get (73) ( k Bk k ) k ( ) f( x+ k) f( x) f ( x) ( k)! k 0 k Sice Euler does ot use the Beroulli umbers rather he uses cosecutive letters of the alphabet which we describe as (74) + ( ) B A ( ) he writes (66) i the form ( )! k k ( ) Ak ( )( k ) k ( ) f( x+ k) f( x) + f ( x) k k 0 k This is the form of the summatio formula used by Euler throughout the remaider of the paper m To obtai series of the first type Euler sets f( x) x ad α i (74) to get m m m m m m x ( x+ ) + ( x+ ) ( x+ 3) + ( x+ 4) ( x+ 5) + m m( m )( m ) x ( ) A() x + ( ) A() x m m 4 m 3 mm m m m + ( ) ( )( )( 3)( 4) 6 (3) m 5 A x where the sum o the right is fiite if m is a o-egative iteger Sectio 8 Euler ow lets x 0 i this last result to obtai series of the first type () O the right all the terms disappear if m is eve ad all but oe term disappears if m is odd

31 3 (8) m m m m m m I terms of Beroulli umbers this is (8) 0 if m is eve ( m )/ ( )! ( m+ m ) A (( m + )/) if m is odd m m+ ( ) B m m m m m m m m + These sums agree with the results of sectio but oly ow is the depedece o A() revealed Comparig (8) with (3) we have (83) m+ ( ) m mc+ mc mc3 + + ( ) B mcm m+ m + Hardy [4 p 4] gives a derivatio of (8) that does ot use the Euler Maclauri summatio formula ad proves that the result is the Abel summatio of the diverget series Hardy begis with the geometric series Now let x x + 3 x x + x x e y with y > 0 to get e e e y e + y y 3y + Differetiatig m times we get (84) for x < m d m y m y m 3y e e + 3 e y dy e for m 0 + m+ Cosider the followig elemetary maipulatios to write terms of Beroulli umbers: y e + as a power series i y i y e y e + y e y e + + y y y e e e + ( )( )

32 3 + y y e e y y + + y e y e y y y y y y y y y e y e Usig the geeratig fuctio for the Beroulli umbers (43) we ow have Bm m Bm ( y) + y y e y m! y m! + Thus we have show that e m m m+ m y Bm+ y + m ( m+ )! Differetiatig m times we get m m k+ d d k B y k + y dy e + dy k ( k + )! Comparig this with (84) we get m k ( ) k + m+! k m Bk + y k m( k+ )! ( k m)! k+ m y m y m 3y m+ k! k m e e + e Bk + y k m( k+ )! ( k m)! 3 ( ) Write x e y to get (85) 3 ( ) ( log ) k+ m m m 3 m+ k! k m x x + x Bk + x k m( k+ )! ( k m)! This last result allows us to fid the eeded Abel summatio of the series Let x ad oly the first term i the series o the right remais We get m+ m m m m+ + B m + 3 ( ) m + Now B / ad all the remaiig Beroulli umbers with odd subscripts are zero Therefore for m 0 we get

33 / ad for m 3 we ca igore get (86) m+ B m + m m m + 3 m + ( ) m+ sice it is egative oly whe B + 0 We This last relatio is the same as (83) ad we have rigorously demostrated the Abel m m m summatio of the diverget series + 3 whe m is a positive iteger Sectios 9 I this sectio Euler prepares for his mai cojecture Whe p is a positive iteger he has obtaied the followig two results: ( p ) p p p p p p ( ) (p )! A( p) p ad m p ( ) p Apπ p p p p p p p Dividig the two he elimiates Ap ( ) ad gets (9) He also has + p p ( ) p p ( ) π p p p p p p ( ) ( )! p p p p p p p p p p p (9) p+ p+ p+ p+ p+ p Euler lists this pair (9) ad (9) for p to 5 Sectio 0 Reflectig o the above results Euler cojectures the geeral formula (0) ( ) ( ) π ( )! π cos which he has show to be true for 3 4 This is the mai result of the paper Today we would write this as (0) ( ) ( ) ( ) π η( ) Γ π cos η( ) Sectio I this sectio Euler proves that his cojecture (0) is valid for

34 34 Sectio Now Euler verifies his cojecture (0) for 0 Lookig at the cojecture i the form (0) we see at oce a difficulty sice Γ( x) has a pole at x 0 This he overcomes by multiplyig ad dividig by x to get Γ( xx ) Γ ( x+ ) This last x x expressio approaches x as x 0 which simplifies the ivestigatio of the limits that Euler ecouters Sectio 3 The cojecture (0) has bee demostrated for 0 3 He ow proves the cojecture for a egative iteger Euler itroduces his otatio for the geeralized factorial He writes [x] for the moder x! ad so for arbitrary x we have i moder terms [ x] Γ ( x+ ) Lookig at the cojecture i the form (0) ad recallig that Γ ( x) has poles at x 0 we see a problem This is overcome by usig the π x idetity Γ( x) Γ ( + x) which he has foud i a previous publicatio siπ x Sectio 4 Euler verifies his cojecture (0) for / which is / / / / / / Γ(/ ) ( ) π cos / / ( ) π / / / / / This is the oly case cosidered by Euler i which both the series i the umerator ad the deomiator coverge He uses Γ (/ ) π ad remarks that his success i this case makes his cojecture very covicig Sectio 5 Next Euler tests his cojecture for 3/ which is π He makes the verificatio umerically This meas Euler must fid a umerical value for the diverget series i the umerator! A task that would frighte the best aalysis He iitially calculates the sum of the first ie terms ad gets From this he must subtract the remaider of the series which is For this secod calculatio he calls o the alteratig

35 35 series versio of the Euler-Maclauri sum formula (73) from which he uses the first 5 terms This is about 5460 Thus the umerator series equals He calculates the sum of the coverget series i the deomiator i the same way (9 terms of the series followed by 5 terms of the Euler Maclauri formula) ad gets Dividig these two values he fids agreemet with the cojectured value to 4 decimal places (Euler actually has 6 decimal place accuracy but he made a few arithmetical errors) Euler eds by sayig oe could ot doubt i the least that this matter is true Usig Mathematica we tried the same calculatios with the first 6000 terms of the origial series followed by 350 terms of the Euler Maclauri formula for the remaider of the series Whe we compared the umerical results with the cojectured value we had over 000 accurate decimal places i less tha two miutes Sectios 7 to 9 Euler cotiues his search to fid the sum of the series whe is a odd iteger Agai he fails He shows that for λ a positive iteger + + λ+ 3 λ+ 4 λ+ 5 λ+ λ λ ( ) π λ+ λ( ) 3 cosλπ ad observes that summig the series λ λ λ λ log log 3 log 3 4 log etc is probably more difficult tha his origial problem Sectio λ λ λ λ ( log log 3 log 3 4 log 4 ) Euler states that he has foud the similar cojecture ( ) si π π usig the same methods I moder otio we write ζ( s) + η( s) Ls () + s s s 3 5 valid for Re( s ) > 0 We have the moder fuctioal equatio

36 36 s Γ() s π s L( s) si L( s) s π Refereces [] Abramowitz M ad Stegu I A Hadbook of Mathematical Fuctios With Formulas Graphs ad Mathematical Tables Courier Dover Publicatios 965 [] Ayoub Raymod Euler ad the Zeta fuctio America Mathematical Mothly 8(974) pp [3] Beroulli J DieWerke vo Jakob Beroulli Basel: Naturforschede Gesellschaft i Basel Birkhäuser Verlag 975 [4] Hardy G H Diverget Series Oxford Uiversity Press 949 [5] Kopp Korad Theory ad Applicatio of Ifiite Series Dover Publicatios New York 990 (A traslatio by R C H Youg of the 4 th Germa additio of 947)