NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA. B r = [m = 0] r

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1 NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA MARK WILDON. Beroulli umbers.. Defiiio. We defie he Beroulli umbers B m for m by m ( m + ( B r [m ] r r Beroulli umbers re med fer Joh Beroulli (he mos prolific Beroulli, d he discoverer of he Beroulli effec... Expoeil geerig fucio. If f(z z /! d g(z b z /! he f(zg(z c z /! where he coefficies c re give by c ( r r b r. Thus if we pu β(z B z /! he ( Br [z ]β(z exp z!! r ( r r B r + B! [ ] + B! from which i follows h β(z exp z z + β(z. Therefore ( β(z B z! z exp z. This shows h he rdius of covergece of β(z is π. So lhough i is empig o me subsiuio i log P (e log( e e m m m m e m m m his will o wor, s m is eveully more h π..3. Some vlues. If we dd z/ o boh sides of ( we obi (3 β(z + z z exp z + z/ coh z/, exp z which is eve fucio of z. So B, B / d B if 3 is odd. Some furher vlues re give below B / /6 /3 /4 /3 De: November 6, 4.

2 MARK WILDON Noe h he umerors eed o be ±. For isce B 5/ Coecio wih FLT. Kummer proved h here re o posiive iegrl soluios of x p +y p z p wheever p is odd prime o dividig he umerors of y of he Beroulli umbers B, B 4,..., B p 3. Such primes re sid o be regulr. For isce s B 69/73, he prime 69 is o regulr..5. Esime for B. I [] i is oed h B > 69; ideed he Beroulli umbers re ubouded. The uhors expli h usig Euler s formul for co z (which hs ice elemery proof vi he so-clled Hergloz ric, oe ges z co z z π z β(z + z z coh z z + π + z m ( z for z < π. Hece, comprig coefficies of z, Thus π (! B ( ζ(. (π (π B (! 4 (π. Noe his shows h B /(! is decresig sequece. As hese re coefficies i he coverge geerig fucio β(z, B /(! eds o s eds o..6. Coecio wih uiform disribuio. Le X be disribued uiformly o [, ]. The he mome geerig fucio for X is Ee zx m e z d x (ex β(z. The coefficies of /β(z give us recurrece sisfied by he Beroulli umbers bu s z /β(z ( +! his is jus he defiig recurrece (.

3 NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA 3.7. Sums of powers. We derive from firs priciples specil cse of he Euler summio formul. Le S m ( m + m ( m m. Cosider he followig geerig fucio where m vries d is fixed: Ŝ(z m S m( zm m!. We hve ( Ŝ(z m z m m! (z m exp z m! exp z m m ( ( exp z. B m+ β(z z! z (m +! zm r Comprig coefficies of z m /m! gives m B r (4 S m ( m! r! m r+ m (m r +! r B r m +.8. Beroulli polyomils. Defie B m (z ( m B z m. m ( m + r m+ r. The firs few Beroulli polyomils re B (z, B (z z, B (z z z +, B 6 3(z z 3 3z +, o so o. Noe h B m( B m d h B m+ ( ( m + B m+ (m + S m ( + B m+ so S m ( B m+( B m+ ( m + which is he obvious defiiio for S m (z. Some useful properies: B m ( ( m B m ( m B + B m [m ] + B m so B m ( B m ( B m uless m i which cse B ( B. For m we hve B m (z ( m B (m z m m ( m B z m mb m (z.. The Euler summio formul From ow o le f : R R be fucio wih s my derivives s eeded. Euler s summio formul is:

4 4 MARK WILDON Theorem. (Euler. Le m d le < b be iegers. The b m B (5 f( f(xdx +! f ( (x b + R m <b where he remider R m is give by R m ( m+ b B m ({x} f (m (xdx. m!.. Applicio o sums of powers. Se f(x x r d e m > r. The f (m so he remider erm vishes. Puig d b we obi r r+ r+ r + + which grees wih (4. B r+! r r+ B r + ( r + r+.. Firs proof. As ll he erms elescope icely i is sufficie o prove he formul whe d b. This hs he dvge h we c use B m (x rher h B m ({x}. We proceed by iducio o m. The cse m ses h f( f(xdx (f( f( + (x f (xdx. This follows from simple iegrio by prs: (x f (xdx f( For m > we c wrie he righ-hd-side s m f(xdx + B! (f ( ( f ( ( + B m m! (f (m ( f m ( + ( m+ Applyig he iducive hypohesis gives: f( + ( m+ B m (x (m! f (m (x + B m m! (f (m ( f (m ( + ( m+ f(xdx (f( f(. B m (x f (m (xdx m! B m (x f (m (xdx m! Iegrig he fil erm by prs usig he resuls i.8 gives f( + ( m+ B m (x (m! f (m (x + B m m! (f (m ( f (m ( + ( m+ B m(x f (m (x m! + B m (x ( m (m! f (m (xdx. The ouermos wo erms obviously ccel. Ad so do he ier wo, s if m is odd, he s m >, B m.

5 NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA 5.3. A lerive proof. There used o be secio here climig lerig proof by ig sequece of polyomils covergig uiformly o f o he iervl [, b] d usig uiform covergece o ierchge he iegrl wih he limi of he polyomils: his reduces o he cse where f(x x r. However i is o loger possible o reduce o he cse where d b, d he mi iducive sep i he proof is o subsilly simplified from he previous secio. I is however worh oig h if m > r d we e d b N he he resul comes ou very esily. I his cse Euler s formul sys h <N Usig he ideiy s ( r r N r+ r + + N r+ r + + r+ r r B! r N r ( ( r+, he righ-hd side c be rewrie B r + which is equl o <N r by (4. ( r + N r+.4. Esimes for error erm. We c se Euler s summio formul i he followig form (6 <b f( b f(xdx (f(b f(+ m B (! f ( (x b +R m. I c be show h B m ({x} B m so rough esime for he error erm is give by.5: R m B m (m! b f (m (x dx. If f (m is posiive he his shows h he mgiude of he error erm is mos he mgiude of he fil erm i he sum. Noe h very ofe he remider erm R m will o ed o s he upper limi i he summio, b eds o. However i will ofe hve some o-zero limi. I his cse we hve R m R m ( b B m ({x} f (m (xdx (m! where he righ-hd-side eds o R m ( s b. I is proved i [] p475 h if f (m+ d f (m+4 re posiive for x [, b] he B m+ R m θ m (m +! f m+ (x b where θ m. So he remider erm lies somewhere bewee d wh would hve bee he ex erm i he sum.

6 6 MARK WILDON 3. Exmples of Euler summio 3.. Firs exmple. We shll emp o use Euler summio o fid S where <. Of course we lredy ow he swer, S ( /(, d i fc Euler summio urs ou o be very ieffecive wy of fidig i! Sill here re some pois of ieres. If f(x x he f (r (x (log r x so by equio (6 we hve ( m x B dx ( + (! (log x + R m ( m log B (log + R m (! for d m. Now somehig lile suprisig hppes: if > e π he log < π d so we c use he expoeil geerig fucio for he Beroulli umbers o obi B (! (log log + d so + ( m+ B (! (log + R m As we ow he swer, he ls equio implies h R m s m. Less rificilly, we c emp o prove his direcly. We fid R m (log m B m ({x} x dx (m! (log m B m (m! ( (log m 4 (π m ( So he error erm R m s m, provided e π < < e π, which holds s we hve lredy ssumed h e π < <. The limiig behviour wih respec o is lso of ieres. Kowig he limi of S llows us o see h lim R m m+ B (log (! This is exmple of he poi mde erlier, h while he remider erm will o usully ed o s eds o, i my well hve some limiig vlue.

7 NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA Summig squre roos. Le f(x x. Noe h f(xdx ( 3 d h 3 ( f (r (x r! x r. r Euler s summio formul gives ( m ( B ( 3 + R m. Seig m we ge ( ( + R. We c esime R by usig he boud B ({x} < B 6 : So we hve R 6.8 x 3 4 ( 3 ( C + O( for some cos C. I geerl he remr o he previous pge shows h he error is give by ( B m+ R m θ m ( m C(m + O( m m + m where θ m [, ] d C(m does o deped o, This gives ( 3 m ( B C(m O( m. There is smll simplificio: C(m c be deermied by ig limi wih respec o, so ( C lim ( 3 3 d so does o deped o m. (Exercise 9.7 i [] revels h C ζ( ; i fc he defiiio of ζ(α for α > A esime for P (x. log P (e log( e m m e m m e m. This suggess wys we migh pply Euler summio. (Bu hey ur ou o be equivle.

8 8 MARK WILDON Te from Kuh fscicle Exercise 5. Le f(x log( e x e mx m. m The we hve log P (e f(. The iegrl of f is give by he Li fucio: x f(udu m e mu m x m e m m m e mx m Li (e Li (e x. The derivives of f re coeced wih he Euleri umbers: le m deoe he umber of permuios i S wih excly m sces (or desces. The we clim Lemm 3.. For we hve f ( (x e x ( ( e x e x. Proof. By iducio o m. If m he he formul is redily verified. Suppose rue for m. The f (m+ (x me x ( m ( e x m+ e x + e x ( ( ( e x e x e x ( + ( e x + e x ( + ( e x + ( ( ( To fiish we pply he ideiy ( m m + ( e x + + (m + m e x e x m. We re ow redy o pply Euler summio. Tig m i (6 we ge f( Li (e Li (e (log( e + log( e + e x ( e x + R. This ideiy c be proved s follows. Le g S hve excly m sces. If ppers i posiio... b... or... he removig gives permuio i S wih m sces. Coversely, give such permuio, here re m desces, d puig bewee y umbers ivolved i desce, or he ed, gives permuio i S wih m sces. Oherwise we hve... b... or... i which cse removig does o chge he umber of sces. Coversely give permuio i S wih m sces, puig bewee he wo umbers ivolved i sce, or he sr, does o chge he umber of sces.

9 NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA 9 where s he eve derivives re posiive e x ( R e x So leig we ge e e. e log P (e Li (e log( e + O( where O( sds for quiy lwys lyig i [, ]. If we use he ideiy suggesed by Kuh, mely or, equivlely, we ge Li (x + Li ( x ζ( log x log( x Li (e log P (e ζ( ζ( ζ( Li ( e Li ( e + ζ( + log( e 3 log( e + O( log( e + O( ; where O( [, ] log + O( ; where O( [ log, log ]. As Li (y log( y gives Li ( e d log( e log + log( / +... log + O(. I priculr by beig lile more creful wih he O( errors we c ge lower boud for log P (e : π 6 + log log log P (e π 6 + log + log. I fc he righ cos is log( π.989; his follows from log P (e π 6 + log log π + O( where he O( erm is (by he fuciol equio, 4 + log P (e 4π /. Noe h.693 <.989 < Alerive. We migh lso ry o use Euler summio o sum log( e. Perhps surprisigly his urs ou o be equivle o he previous pproch, sice d dx log( e x e x e x e x which is esseilly he fucio we summed erlier. Refereces [] Rold L. Grhm, Dold E. Kuh, d Ore Pshi. Cocree Mhemics. Addiso Wesley, 994.

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