A Fourier series approach to calculations related to the evaluation of ζ(2n)

Transcription

1 A Fourier series approach to calculatios related to the evaluatio of ζ Weg Ki Ho Natioal Istitute of Educatio, Nayag Techological Uiversity, Sigapore Tuo Yeog Lee NUS High School of Mathematics & Sciece, Sigapore March 3, 0 Abstract This paper presets a uified approach, based o certai Fourier series, to several iterestig calculatios related to the evaluatio of the Riema ζ-fuctio at eve itegral argumets. Our approach reported herei has two mai advatages: i a lighter mathematical overhead i fact, o more tha freshma calculus, with some very miimal bacgroud owledge i Fourier series, ad ii a emergece of cleaer proofs of ow results. Itroductio I this paper, the Riema Zeta fuctio ζs is defied by { Rs > ζs : s s Rs > 0, s s. The usual aalytic cotiuatio of ζ allows the value of ζ0 to be defied as. Evidet from the title of this article, we oly eed to focus o the Riema Zeta fuctio ζ restricted o the o-egative

2 itegers N. Leohard Euler studied the ζ-fuctio restricted o the positive itegers i 740, ad was the first to aswer the Basel problem, i.e., that of presetig the eact value of ζ i [5]. For a detailed historical accout of Euler s solutio to the Basel problem, we refer to the ecellet wor of L. Debath [4]. The eact values of zeta fuctio at the eve itegral argumets are give by the formula ζ + B π,.! cos 0 where B is the th Beroulli umber. There are several recet wors revisitig the evaluatio of ζ usig elemetary but ot ecessarily easy methods. For eample, [7] employs the Chebyshev s polyomials to obtai the ifiite product represetatio for the sie fuctio which i tur yields a proof of ζ π 6. However, the method reported therei does ot seem to apply to the evaluatio of ζ for higher values of. A recet wor [6] establishes the value of ζ usig the Neville s idetity. A ew proof of Equatio. relyig o the Taylor series epasio for the taget fuctio ca be foud i [3]. No such simple values of the zeta fuctio at odd iteger argumets are ow to date. Most elemetary methods that deal with calculatios related to the ζ-fuctio at oegative itegers argumet are based o special fuctios ad idetities, ad comple aalysis. I this paper, we perform these calculatios ad derive ow results usig a comparatively uiform ad simple approach via Fourier series. Though the idea that the ζ-fuctio is related to Fourier series has bee folloric, this is oly made eplicit very recetly [0]. Our method focuses o a specific class of trigoometric series the associated Clause fuctios ad si 0. Notably, the Fourier cosie series + for m has bee used by N. Robbis i [9] to evaluate ζm. Our preset method differs from [9] i that ot oly are we able to evaluate ζ at its eve itegral argumets but also to produce a eplicit recurrece relatio for ζ see loc. cit. Corollary 3.8. We orgaize our paper as follows. I Sectio, we preset the prelimiary defiitios ad results. The, we employ basic calculus techiques to establish some hady idetities cocerig the associated Clause fuctios. I Sectio 3, a proof of Euler s theorem ζ π à la the ɛ N defiitio is give. I Sectio 4, we establish two ifiite series represetatios cocerig the associated Clause fuctios. The highlight is a forward substitutio method for obtaiig the eact values of ζ, usig a simpler recursively-defied sequece α i place of the Beroulli umbers. I particular, we display the calculatio of the eact values of ζ for,,..., 5. 6

3 Some trigoometric series. Results cocerig sums ad series I this sectio, we highlight some useful results cocerig certai trigoometric series. Propositio.. For ay t R ad Z +, these idetities hold: cos t si + t si t si t. ad cos t cos + t si t. Proof. Applyig the factor formulae ad the method of differece. Corollary.. For ay t 0, π] ad m, Z + where m <, si jt si t. jm+ Proof. By the precedig propositio, we have si jt si +m+ t si m t si t si t. jm+ Theorem.3. If 0, π, the the trigoometric series coverges. Moreover, si π. Proof. See Lemmata 7..5 ad 7..6 i [8]. The followig is a specializatio of Theorem 7.. of [8]: si Lemma.4 Summatio by parts. Let u 0 ad v 0 sequeces of real umbers. The, u v v u j v v + u j j0 be two 3

4 Lemma.5. Let t 0, π] ad Z +. The, the followig holds: j+ si jt..3 Proof. Let N Z + be arbitrary. N + si jt + + j+ N N + N + si jt N + + si t + N si t, + j+ + si jt by Lemma.4 j+ N + N + si jt j+ N + + by Corollary. N 3 Evaluatio of ζ The tas of obtaiig the eact values of ζ begis with the simplest case of ζ, which is also ow as the Basel problem. Our approach based o trigoometric series higes crucially upo the followig observatio: Lemma 3.. Let Z +. The, the th partial sum of ζ ca be writte as π t. 3. π 0 Proof. Follows directly from itegratio by parts. Recallig from Theorem.3 that π t, oe may be tempted to reaso aively that by limitig, π π t ζ π π t π

5 si Alas! This reasoig is flawed as is ot uiformly coverget to π o [0, π]. A more subtle argumet i the style of a ɛ N proof, as we shall ow demostrate, is required to secure the desired result: Theorem 3. Euler, 740. π Proof. By virtue of Lemma 3., for every positive iteger, we have π 6 t π π t π 0 0. This i tur, by the triagle iequality, is bouded above by π t π t 0 0 } {{ } U To boud U, cosider the iequalities: π t U + 0 π + π 4. [ π t π t π t + π t π t }{{} V 0 π t To boud V, first eploit Theorem.3 ad Lemma.5 to rewrite the differece of the itegrals as ] π + π t + + j+ ad the use Corollary. to boud this above by π π si t + si jt, + π l π + l. +. 5

6 So, all i all, we have π 6 π + π 4 + π l π + l l Havig gaied success o the eact evaluatio of ζ usig trigoometric series such as si, it is atural to as if similar series ca be cosidered i the eact evaluatio of ζ for positive itegers. I the esuig developmet, we carry out this pla by cosiderig the followig two trigoometric series: cos si C : ad S :, [0, π], Z+ + They are sometimes referred to as the associated Clause fuctios i the eistig literature see, for eample, p.0 of []. We record here a elemetary property cocerig these series: Propositio 3.3. Let Z +. The, the series cos si ad, [0, π] + coverge absolutely ad hece uiformly o [0, π]. Our iductive approach bears upo us to begi with. Theorem 3.4. If [0, π], the cos C cos ζ coverges ad π Proof. I view of Theorem 3., we may assume that 0, π. Sice for ay positive iteger, it holds that π π t + si jt + + j+ + si t, 6

7 it follows that cos π Fially, because 0 π t ζ, we obtai the desired result. π. 4 Remar 3.5. Notice that the proof techique used i the precedig theorem caot be cheaply eteded to accout for the covergece at 0 sice the boudig techique ivolves the fuctio si t which has a sigularity at t 0. Theorem 3.6. If [0, π], the si 3 S : si 3 ζ coverges ad π 3 π. 3.4 Proof. The result follows immediately from cos t cos t 0 as. ad Theorem 3.4. To fid the eact value of ζ4, oe could have proceeded by usig the result that S ζ π 3 π ad the followig fact aalogous to Lemma 3. that 4 π t π 3 0 ad arguig alog the lie of reasoig used i Theorem 3.. However, we choose ot to do so. Istead, we opt for a more elegat method that would derive the evaluatio of ζ at all positive eve iteger argumets. This ecessarily calls for geeralizatios of Theorems 3.4 ad 3.6, which appear as follows. Theorem 3.7. For each positive iteger ad [0, π], the followig hold: cos r+ π r r ζ r r! si + r0 r r0 3.5 π r+ r ζ r r +! 3.6 7

8 Proof. We proceed by iductio as promised earlier. The base cases of are just Theorems 3.4 ad 3.6. Assumig that the statemets hold for, we must show that they hold for +. Sice Theorem 3.3 guaratees that + it follows that cos ζ + + Ivoig the iductio hypothesis, oe has cos + + ζ + + r r0 r+ + r0 cos π + cos +, +. π tr+ r ζ r r +! + r Similar reasoig applies for the sie couterpart. Corollary 3.8. Let Z +. The, the followig hold: ζ + r. ζ+ r r π ζ r r r 0.. r0 r r ζ r π r r+! 0. Proof.. Put 0 i the cosie series of Theorem Put 0 i the sie series of Theorem 3.7. r! π r. r! From Corollary 3.8, substitutig values of,,... oe obtais the liear system A b, where A a ij is defied by { i+j π i j a ij j 3 i! if i < j; 0 otherwise. ad ζ0 ζ ζ T ad b 0 0T. Via forward substitutio, oe easily obtais a epressio for ζ as a ratioal multiple of π without the use of Beroulli umbers. 8

9 Theorem 3.9. Let N. The, oe has ζ π α, where α 0 ad α r+ α r r +!, Z+. r Proof. The proof proceeds by iductio o. For the base case of 0, because 0 π 0 0 α 0 ζ0, the statemet holds. For the iductive step, we assume that ζ π α, <. The proof at the iductive step ow proceeds as follows: ζ r+ r p π r r ζ r π r r +! r+ r r r π r r α π r r r +! r+ α r r +!. r π α. Remar 3.0. A similar result epressig ζ as a π where a are give recursively by m j a j j m+ j! m m+! ca be foud i []. Our recurrece relatio i Theorem 3.9 used to calculate the α s is easier to apply, compared to Che s recurrece formula. As is well ow that ζ! B π, where B is the th Beroulli umber, it must be that! B π π α, which implies that Corollary 3.. For ay positive iteger, B! α. Compared to the form ivolvig Beroulli umbers, the formula for computig the eact values of ζ give i Theorem 3.9 is easier to use. For istace, a direct computatio yields α 6, α 7 360, α , α , α 5 This the returs: ζ π α 6 π, ζ4 π 4 α 90 π4, ζ6 3 π 6 3 α π6, ζ8 4 π 8 4 α π8, ζ0 5 π 0 5 α π0. 9

10 4 Cocludig remars I this paper, we have preseted a uified approach to evaluate ζ based o elemetary calculus techiques applied to associated Clause fuctios. I the course of provig the famous formula. for ζ, we produced two recurrece formulae for ζ c.f. Corollary 3.8. Our Fourier-series methodology has bee further developed to yield represetatios of ζ + as a ifiite series ivolvig ζj s, fastcoverget series represetatios for ζ +, log-sie itegral formulae ad so o. Refereces [] M.P. Che. A elemetary evaluatio of ζ. Chiese Joural of Mathematics, 3: 5, 975. [] J. Choi, Y.J. Cho, ad H.M. Srivastava. Log-sie itegrals ivolvig series associated with the zeta fuctio ad polylogarithms. Math. Scad., 05:99 7, 009. [3] E. de Amo, M.D. Carrillo, ad J. Ferádez-Sáchez. Aother proof of Euler s formula for ζ. Proc. Amer. Math. Soc., 394:44 449, April 00. [4] L. Debath. The Legacy of Leohard Euler A Triceteial Tribute. Imperial College Press, 00. [5] L. Euler. De summis serierum reciprocarum. I Opera Omia, volume 4, pages Commet. Acad. Sci. Petropolit., 740. [6] W.K. Ho, F.H. Ho, ad T.Y. Lee. A elemetary proof of the idetity cot θ θ + θ. IJMEST, 0. To appear. θ π [7] R.A. Kortram. Simple Proofs for π 6 ad si. Mathematics Magazie, 69: 5, April π 996. [8] T.Y. Lee. Hestoc-Kurzweil Itegratio o Euclidea Spaces, volume of Series i Real Aalysis. World Scietific, 0. [9] N. Robbis. Revisitig a Old Favourite: ζm. Mathematics Magazie, 74:37 39, October 999. [0] E.E. Scheufes. From Fourier Series to Rapidly Coverget Series for Zeta3. Math. Mag., 84:6 3, 0. 0