WHAT ARE THE BERNOULLI NUMBERS? 1. Introduction
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1 WHAT ARE THE BERNOULLI NUMBERS? C. D. BUENGER Abstract. For the "What is?" semiar today we will be ivestigatig the Beroulli umbers. This surprisig sequece of umbers has may applicatios icludig summig powers of itegers, evaluatig the zeta fuctio, fidig asymptotics of Stirlig s formula, ad estimatig the harmoic series. We ivestigate properties of these umbers ad itroduce Beroulli polyomials, a closely related topic. The, we establish the Euler summatio formula ad use the formula to provide the aforemetioed applicatios. Furthermore, a study of the Fourier coefficiets, yields surprisig results o the zeta fuctio.. Itroductio Beroulli umbers first appeared i a post humorous publicatio of Jaob Beroulli (5-75 i 73, ad were idepedetly discovered by Japaese mathematicia Sei Kōwa i 7. Beroulli observed these umbers i the course of his ivestigatios of sums of powers of itegers. That is sums of squares, cubes, or higher powers. Let S p ( p. We have the followig closed forms of S p ( for small p: S ( S ( S ( S 3 ( 3 + S ( S 5 ( S ( S 7 ( S ( S 9 ( S ( Date: July 3, 3.
2 Through slight maipulatios Beroulli arrived at the followig reformulatio of these sums: S ( ( S ( S ( S 3 ( S ( S 5 ( S ( S 7 ( S ( S 9 ( S ( [ ( [ (3 3 [ ( [ (5 5 [ ( [ (7 7 [ ( [ (9 9 [ ( [ ( ( ( 3 + ( 3 + ( ( 5 + ( 5 + ( 7 + ( ( 9 9 ( ( ( 3 ( 5 ( ( ( + ( ( + ( 3 ( 5 3 ( 5 ( 7 ( 37 ( 9 ( 9 ( ( ( ( ( ( 3 ( 5 ( ( ( Through this reformulatio we otice the repeated occurrece of certai umbers withi the closed form sums. These are the Beroulli umbers. Here are the first few: B, B, B, B 3, B 3, B 5, B, B 7, B 3, B 9, B 5, B. More geerally, via the Euler summatio formula, we will prove that ( m + ( S m ( m+ (Here is the th Beroulli umber. m + We also achieve the results o the value of the zeta fuctio o eve itegers ad the asymptotics of Stirlig s formula ad the partial sums of the harmoic series through ivestigatig the Beroulli umbers ad the Beroulli polyomials.. Defiitio ad elemetary properties Beroulli first discovered through studyig sums of itegers raised to fied powers. This approach hited at above properly defies the Beroulli umbers, but may preset difficulties whe tryig to calculate larger umbers i the sequece sice we would first eed closed forms of S p (. Additioally, to tae this as the defiitio we would eed to prove that the cosistecy of equatio (. The moder approach is to defie the Beroulli umbers through the use of the geeratig fuctio e ad the prove formula (. 5
3 Defiitio. The Beroulli umbers { } are defied as the costats i the power series epasio of the aalytic fuctio e e. Regardig the above equatio as a formal power series, the equatio produces the equatio i+ j From observig the above chart oe sees that B i i!( j + i! Ideed this is true for all odd umbers larger tha ( +!. B 3 B 5 B 7 B 9 B. Lemma.. Let a odd umber larger tha. The B. Proof. e B e + + (e (e (e + (e (e + e (e e e e is odd, e + e is eve, ad is odd. Thus epasio of e B has o otrivial odd terms. e B is a eve fuctio. Thus the power series The Beroulli umbers grow quite quicly. Ideed, we will show i sectio 5 that For ow let us be satisfied with the fact that (πi (as. B I order to achieve the results metioed i the itroductio, we will eed to defie the Beroulli polyomials. 3
4 Defiitio. The Beroulli polyomials are a sequece of polyomials { (y} are defied through the power series epasio of ey e e y e (y. As above we ca derive a closed form for these polyomials by taig the product of the power series for e ad ey. Thus (y Here are the first few Beroulli polyomials: (y B (y e y e i B (y y i+ j ( B i y i i (y B i y j i! j! B (y y y + B 3 (y y 3 3 y + y B (y y y 3 + y 3 Oe may otice that for the listed polyomials B (y (y. This holds i geeral! Lemma.. B (y (y Proof. Let us differetiate the defiig relatio for the Beroulli polyomials with respect to y. Divide through by ad reide e y e e y e B (y. B (y B (y ( +!
5 The equatig the powers of gives the desired relatio. Additioally there are may other iterestig facts cocerig these polyomials. I ecourage you to try to prove the followig relatios: ( y ( (y ( B ( B ( ( B B (for eve (ydy (if (y + (y + y (yb l (ydy ( l +l ( +l (if, l Fially let us mae the Beroulli polyomials restricted to [, ito periodic fuctio defied o R with period. That is Defiitio 3. For N, let B (y : B(y y. 3. Euler-Maclauri summatio formula Here we aim to prove the followig summatio formula which will be critical i the course of our aalysis. Theorem 3.. Let a < b Z ad let f be a smooth fuctio o [a, b. The for all m b b b f (i f (d + f ( ( + R m Where ia a R m ( m+ b Proof. By the fudametal theorem of calculus, a f ( f ( + Itegratig with respect to gives us the followig: f ( f ( + f ( + f ( + f ( + B m( a f (m (d. f (tdt f (t f (tdtd t ddt f (t( tdt f (t(tdt 5
6 By addig the last two equatios we fid that f (d f ( + f ( + ( t f (tdt. After dividig by f (d Through a quic maipulatio we fid that f ( + f ( + ( t f (tdt. f ( f (d + f ( f ( + ( f (d. I other words, ( f ( f (d + B f ( + B ( f (d. Now we prove (3 f ( f ( + f ( ( + ( m+ B m ( f (m (d. Observe that ( is the base case of (3. Assume that we have proved equatio (3 for all m. By the fact that B ( (, ( f ( (d +( + f ( ( + ( f (+ (d + By the fact that (3 holds for m ad the prior calculatio, f ( f ( + f ( ( + (m+ ( Bm+ ( m + f (m ( If m is odd, ( m+. If m is eve B m+ ( B m+ (. Thus ( m+ B m+ ( m + f (m ( B m+ (m +! f (m (, ad B m+ ( f (m+ (d. m + f ( m+ f ( + f ( ( + (m+ B m+ ( f (m+ (d. (m +! Applyig this result to f ( j + for all a j < b ad summig gives the desired formula.
7 . Applicatios of the summatio formula Now we retur to our origial eample, the sum of itegers raised to a power. Corollary.. S m ( m + ( m + m+. Proof. Fi p. We will apply the Euler summatio formula to the fuctio f ( p, with a, b ad m p. For l p Thus f (p p! ad Thus R m ( m+ b m i f (l ( m(m... (m l + ml. a m d + m+ m + + m + B m(d ( m+ (b a B m (d m(m... (m + ml+ ( m + ml+ m + ( m + ml+ + R m Net we apply the summatio formula i the case of the harmoic series. Let f (, a, b, ad m. The Thus i i l( + l( + B ( f (l ( ( l l! l+. ( (! + ( m ( B m( B m( d m+ ( m d m+ Let Thus i i γ B + l( + γ + B B m( m+. 7 B B m( d. m+
8 Fially a similar calculatio ca be made i the case f ( l( to gai results o Stirlig s formula. Let a ad b ad let m be ay iteger larger tha. Followig the same logic as above, we arrive at the formula: l(! ( + l( + C + B ( B m( m m d 5. Fourier aalysis of Beroulli polyomials Let us ow calculate the Fourier coefficiets of the periodic fuctios B (y. We will deote the lth Fourier coefficiet of f ( by f ˆ(l. Bˆ ( B (d. For l, Bˆ (l ( eπil d πil The usig the fact that for all smooth fuctios f we have that ˆf (l πil ˆ f (l ad the fact that B ( (. We get that π i l ˆ B (l ˆ B (l We will tae for grated that the Fourier series acts icely for (for our fuctios are C. So for I particular for ad For eve B ( l l (π i l eπ i l (π i l (π i l (π i ζ( l Thus we ca calculate the values of ζ( for eve values of very quicly. For eample ζ( π ζ( π ζ( π ζ( π Furthermore sice lim ζ(, we fid the asymptotic relatio metioed i sectio : (πi (as.
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