A MODIFIED BERNOULLI NUMBER. D. Zagier

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1 A MODIFIED BERNOULLI NUMBER D. Zagier The classical Beroulli umbers B, defied by the geeratig fuctio x e x = = B x!, ( have may famous ad beautiful properties, icludig the followig three: (i B = for odd >. (ii The fractioal part of B is give by B (mod ( >, eve. (p p prime p (iii B is give asymptotically for large eve by the formula B ( /! π (, eve. I this ote we show that the ratioal umbers defied by ( +r B Br = ( > ( r +r r= satisfy the followig amusig variats of the above three properties: (I The value of B for odd is periodic; more precisely, it is give by (mod B 3/4 /4 /4 /4 /4 3/4 (II The fractioal part of the umber B := B B is give by B (mod ( >, eve. p (p+ p prime (III B is asymptotically equal to ( / (! (π for large ad eve, ad is give much more precisely by the approximatio B ( / πy (4π (, eve, where Y (x deotes the th Bessel fuctio of the secod kid.

2 The proofs of (I (III will be give i the ext three sectios, after which we will give the statemet ad proof of a fourth property, a exact formula for B refiig the asymptotic formula (III. The proofs, especially those of (I ad (II, are quite fu ad the reader is ivited to try to fid them him/herself before proceedig. We ed the itroductio with a small table of the umbers B ad B B B Proof of (I. Istead of usig the familiar geeratig fuctio (, we represet the Beroulli umbers by the geeratig fuctio F(x = r= B r r xr Q[[x]]. This formal power series does ot coverge aywhere, but occurs i the asymptotic formula Γ (X logx Γ(X X F( (X X for the logarithmic derivative of the gamma fuctio, ad the fuctioal equatio Γ(X + = XΓ(X of the gamma fuctio implies the fuctioal equatio F ( x = F(x+x+log( x Q[[x]] (3 x of the power series F. A elemetary proof of (3, or of the equivalet but simpler fuctioal equatio G ( x = G(x x for the simpler power series G(x = x r= B rx r+ = x + x F (x, ca be obtaied by otig that either oe of these fuctioal equatios is equivalet to the stadard recursio formula ( B k = ( > (4 k k= for the Beroulli umbers, which is i tur a easy restatemet of the defiitio (. Now itroduce a ew power series F λ (x, depedig o a parameter λ, by F λ (x = F ( x log( λx+x λx+x Q[[x]]. For λ = this specializes to F (x = r= B r r x r ( x r log( x = = B x. (5

3 O the other had, the fuctioal equatio(3 together with the symmetry property F( x = F(x+x, which is a restatemet of (i, give the fuctioal equatios F λ+ (x = F λ (x+ for the power series F λ. We deduce x λx+x = F λ( x F (x F ( x = ( F (x F (x + ( F (x F (x + ( F (x F (x = x x+x + x +x + x +x+x = 3x x3 x 5 +x 7 +x 9 3x x. Statemet (I follows. Proof of (II. Rather surprisigly, this property is a cosequece of the aalogous property (ii for the usual Beroulli umbers, the divisibility by p beig metamorphosed ito the divisibility by p + by the magic of geeratig fuctios. We begi by rewritig the defiitio of B as B = r= (+r+( r +r ( +r r B r = β +β B, (6 where β = ( +r B r (. r r= Fix a prime p. We wat to show that p B is p-itegral for all ad is cogruet to mod p if p + divides ad to otherwise. We suppose that p >. (The case p = is similar but easier. By (ii we kow that pb r is p-itegral ad is cogruet to mod p if p divides r > ad to otherwise. Equatio (6 the immediately gives the p-itegrality of p B ad the cogruece { if (p p B γ γ + otherwise (7 (here ad from ow o all cogrueces are modulo p, where γ = r>, (p r ( +r As usual, we use geeratig fuctios. From the defiitio of γ we have γ x = = r>, (p r x r ( x r+ = 3 r. x p ( x p x p ( x.

4 Hece (7 gives p B x (+x > (+x = The expressio o the right simplifies to γ x + xp x p x p ( x ( x p x p ( x + xp x p. x p+ xp+, completig the proof. Proof of (III. As i the proofs of (I ad (II, we deduce property (III from its classical Beroulli aalogue. Suppose is eve ad large. The for r = k with k fixed (ad eve we have ( +r r Br +r = ( / (! (π ( ( k/ (4π k k! +O (, ad sice ( k/ (4π k = cos(4π = this gives the asymptotic formula i (III. k! The same argumet i cojuctio with the biomial coefficiet idetity ( +r r +r = ( h k h ( h! (r = k r! h!(k h! h k/ (which holds because both sides express the coefficiet of x k i ( x+x r /r lets us replace the asymptotic formula for B by the full asymptotic expasio ( / B h ( h! h! (π +h. Cosultig stadard referece works, we discover that the expressio o the right is also the asymptotic developmet of πy (4π, sice Y is defied by ( h! πy (x = x +h + (c h +c +h logx ( h x h+ h! h!(+h! h= with c h = h γ = O(logh. (Cf. [], 7.(3. This proves the secod assertio i (III, though without ay estimate of the error. Here are a few umerical values to illustrate the accuracy of the two approximatios for B i (III : 3 5 (!(π πy (4π ( / B The pooress of the approximatios i the first row is explaied by the above asymptotic expasio, which shows that the ratio of B to (!(π is +C +O( with C = 4π 4. 4 h=

5 A exact formula. I the case of the usual Beroulli umbers, the rough asymptotic formula (iii ca be replaced by the exact formula ( B = ( /! π ( >, eve (8 due to Euler. It is reasoable to look for a correspodig exact formula for B. We start with umerical data. From the table above we have B5 +πy 5 (4π,884 9, the exact value beig Guessig that this differece might be related to the value πy 5 (8π, we compute this latter umber, which turs out to be , suggestig that we are o the right track. Goig oe step further, we fid that the differece B5 + πy 5 (4π + πy 5 (8π equals , rather close to the value πy 5 (π = This suggests that B might be very well approximated by, or eve equal to, the umber U = l= ( / πy (4πl. However, this sum diverges, sice ( / πy (4πl behaves like l / for l large, so we must reormalize it, settig U := ( l= ( / πy (4πl + l ζ(. (9 The series coverges oly like l 3/, but ca be replaced by the expressio ( K U = ( / c,k K πy (4πl + c l k+/,k ζ ( k +, l= k= k= c,k = ( [k+ ] (8π k k! (+k (+k 3 ( k + for ay K, where the lth term is O(l K 3/, so the umerical value of U ca be computed easily. Comparig it with B, we fid the followig table: B U suggestig a formula of the form B + = U +ε ( >, eve, ( where ε is positive ad goes to rapidly as teds to ifiity. (Notice that the quatity B +/ occurrig o the left is give by the same expressio as i (, but with B r replaced by ( r B r. We ow prove this ad give a exact formula for the error term ε. (IV Defie B by ( ad U by (9. The equatio ( holds with ( k + k +4 ε = ( ( k(k

6 Proof of (IV. Usig the relatio (8 betwee Beroulli umbers ad zeta values, we fid B = 4 ( +r (r! r (πi r ζ(r ad hece where r r eve B U = 4 + ζ( + b,l, ( b,l := ( / πy (4πl r r eve l= ( +r (r! r (πil r l. O the other had, stadard formulas for Bessel fuctios (cf. [], 7.(5 ad 7.3(6 give ( / πy (x = R ( K (ix π/ e ix (t/+ix +e ix (t/ ix = Γ(+ (ix e t t dt for eve ad positive, ad hece, after some simple maipulatios, the formula b,l = Γ( Γ(+ where f (x is defied for x > by ( t e t (8πl t f dt (3 t 4 πl f (x = (+ix +( ix (ix r r eve ( (ix r. r Note that f(x = (x / + O(x 3/ as x, so that b,l = O(l 3/, as we already kow. Clearly f (x exteds to C := C ( i, i] [i,i as a eve holomorphic fuctio, ad the biomial theorem gives the Taylor expasio f (x = ( ( r x r ( x <. +r r The beta itegral idetity ( = +r π Γ(+ Γ(r+ Γ(+r+ 6 = π u r ( u du

7 ow gives the itegral represetatio f (x = π u ( u +x u du for x <, ad by aalytic cotiuatio this holds for all x C. Cosider the more geeral itegral f (x,s = π u s ( u +x u du (x C, s C, < R(s <, so that f (x = f (x,. It ca be estimated for large x by f (x,s = π u s du +x u + O(x s = x s si(πs/ + O(x s. I particular, l f (lx,s coverges for R(s > ad all x >, ad usig the Poisso summatio idetity +l t = π/t tahπ/t = π ( + e πk/t (t > t l= we fid f (lx,s = x l= ( Γ(s Γ(+ Γ(s+ + l= u s ( u e πk/xu du for R(s >. Writig the left-had side of this idetity as f (,s+ (f (lx,s + (lx s ζ(sx s si(πs/ si(πs/ gives its aalytic cotiuatio to R(s >, ad settig s =, x = 8π we obtai t ( (8πl t f = Γ(+ t 4 πl 8Γ( l= Γ( t ζ( 4 Γ(+ π t Γ( Γ(+ + t u 3 ( u e kt/4u du. 8π Combiig this with (3 ad performig the itegratios over t we get b,l = 4 ζ( + l= Γ(+ 4Γ( Γ(+ u 3 ( u du. (+k/4u + The desired result ow follows by isertig this ito ( ad usig the idetity Γ(+ ( u 3 ( u 4Γ( Γ(+ (+k/4u + du = k + k +4, k(k +4 which ca be proved either from stadard hypergeometric formulas or by expadig both sides as power series i /k for k > 4 ad the usig aalytic cotiuatio. 7

8 Remarks. We ed this paper with a umber of remarks.. The formula give i (i, which ca be rewritte i the form B = ( ( 4 ( odd, occurred origially i [4] i the cotext of the Eichler-Selberg trace formula for the traces of the Hecke operator T l actig o modular forms o SL (Z. The method of prooftheregaveaformulaforthesetraceswhichhadasomewhatdifferetformfrom the classical oe, ad i particular ivolved Beroulli umbers. The specializatio oftheformulatothecasel = gavethedimesioofm k (SL (ZitermsofB k, ad the equality of this expressio with the stadard dimesio formula required the periodicity property (I.. The same idea as was used to prove (I leads to aother simple expressio for the modified Beroulli umbers B. Specifically, from F (x = F (x+ ad the biomial formula we get the idetity B = ( 3 + x x+x + x +x ( +r ( +r r r= Br +r, whose secod term has a pleasig similarity to the origial formula ( defiig B. 3. Next, we metio that the defiitio ( ca also be iverted to express the ordiary Beroulli umbers i terms of the modified oes, should we for ay reaso wish to do so. Ideed, from (5 we have B x = log ( +4x + x = ad comparig coefficiets of B we get ( B = ( + r= B r ( r +x +4x, x ( ( r r Br. r r= 4. Fially, we metio that there are (at least two ways of calculatig Beroulli umbers which are faster tha the stadard recursio (4. The first is due to M. Kaeko [3]. The secod I oticed myself, but Kaeko has iformed me that it is i fact a classical idetity goig back to Kroecker. (See [] for a historical survey. Nevertheless, these formulas are both pretty ad useful, so for the sake of popularizatio we reproduce them here. 8

9 a. Oe ca replace (4 by a recursio of the same type, but with oly half as may terms, amely, settig b = (+B, b = ( + b +i (4 + i i= together with the coditios b = ad b + = for >. Equatio (4 ca be see as a special case of the followig fact: Defie a ivolutio o the set of sequeces {a, a, a,...} by A (x = e x A( x, where A(x := = a x /( +!, or more explicitly by a = ( ( + i= i+ ai. The the expressio i=( i a+i ( is ati-ivariat uder ad hece vaishes if A = A. (Note that A(x = x/(e x = A (x for a =b. b. The Beroulli umbers ca be calculated directly, rather tha recursively, by the closed formula k ( k + B k = ( k + + k. (5 + + = To prove this, we apply Beroulli s famous formula for k + + k to get k + + k + = B k+(+ B k+ (k +(+ = (polyomial of degree k i + B k +, where B r (x deotes the rth Beroulli polyomial; takig the (k +st differece of both sides kills the polyomial o the right, leavig oly a easily computed multiple of B k. Formula (5 is much more coveiet for umerical computatios tha the recursio formula, at least if oe wats to compute idividual Beroulli umbers rather tha a table up to some limit, sice the umber of steps eeded to compute B k is O(k rather tha O(k (each term i the sum ca be computed from its predecessor i O( steps. Ideed, eve for computig a table, (5 is sometimes more useful tha the recursio (4, sice the time required is about the same but the storage requiremets are reduced from O(k to O(. Here is a oe-lie PARI program implemetig the formula (5 (for k > : B(k=h=;s=;c=k+;for(=,k+,c=c*(-k-/;h=h+c*s/;s=s+^k;h Refereces. [] A. Erdélyi, W. Magus, F. Oberhettiger ad F. Tricomi, Higher Trascedetal Fuctios (Batema Mauscript Project, Vol. II, McGraw-Hill, New York (953 [] H.W. Gould, Explicit formulas for Beroulli umbers, Amer. Math. Mothly 79 ( [3] M. Kaeko, A recurrece formula for the Beroulli umbers, Proc. J. Acad. 7 ( [4] D. Zagier, Hecke operators ad periods of modular forms, Israel Math. Cof. Proc. 3 (