Numerical approximation to ζ(2n+1)

Transcription

1 Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at:

2 Numeical Appoximation to ζ(n + 1) Michael J. Dancs and Tian-Xiao He Depatment of Mathematics and Compute Science Illinois Wesleyan Univesity Bloomington, IL 617-9, USA May 11th, 5 Abstact In this shot pape, we establish a family of apidly conveging seies expansions fo ζ(n + 1) by discetizing an integal epesentation given by D. Cvijović and J. Klinowski in []. The poofs ae elementay, using basic popeties of the Benoulli polynomials. AMS Subject Classification: 11M6, 11Y35, 41A1, 65D15. Key Wods and Phases: Riemann zeta function, Benoulli polynomial, Diichlet seies, Apéy s constant. 1 Intoduction In [1], Apéy poved the iationality of ζ(3) by using a apidly conveging infinite seies. No one has yet discoveed a simila expession fo ζ(n + 1) when n, but thee have appeaed seveal ecent papes (fo example, [3], [4], and [5]) devoted to finding othe types of seies expansions fo the zeta function at odd-intege aguments. The typical esult is an exponentially convegent seies, and thus insufficient to pove iationality. We establish a new seies expansion of ζ(n + 1) by discetizing the integal epesentations given in []. Ou esults similia in quality to the efeences cited peviously, but ou method has the advantage of being completely elementay. It may also be possible to use ou methods in 1

3 M. Dancs & T. X. He evaluating othe Diichlet seies, and we intend to etun to this poblem in the futue. Thoughout this pape, log t denotes the natual logaithm of t, and B m (t) is the mth Benoulli polynomial. We hencefoth assume that n is a fixed positive intege, and note that implied constants may depend on n. Ou main esult is: Theoem 1.1 Fo any positive intege n, thee is a ational numbe A n such that [ ζ(n + 1) = α n A n n + 1 ( log 3 + n log 1 ) ] + Z n, (1.1) whee α n = ( 1)n+1 n+ π n (1 n )(n + 1)! (1.) Z n = [ (ζ() )( 1 1 ) 1/ ] B n+1 (t)t 1 dt. (1.3) Specifically, if k denotes the kth iteate of the odinay fowad diffeence opeato, and then A n = 1/ 1/ f n (x) = 1 (3/)x+1 (x + 1) 1 x+1 n (x + 1), (1.4) B n+1 (t) 1 t dt + 1 3/ 1/ B n+1 (t) dt + n(n + 1) [ ] n 1 f n (). (1.5) t In light of this esult, we may appoximate ζ(n + 1) in tems of Z n. We have the following esult egading the ate of convegence: Theoem 1. Let Z n (k) denote the kth patial sum of the seies in (1.3). Then Z n = Z n (k) + O(k 1 k ).

4 Numeical Appoximation to ζ(n + 1) 3 Poofs of the Main Results Fom the identity tan x = cot x + cot x and the well-known fomula π cot(πz) = z 1 ζ()z 1, z / Z, (.1) one eadily obtains, fo < t < 1, tan(πt) = π ( n 1)ζ(n)t n 1, (.) n=1 and this emains valid when t =. Fom Section 4 of [1], we have ζ(n + 1) = ( 1) n+1 (π) n+1 (1 n )(n + 1)! and inseting (.) gives 1/ B n+1 (t) tan(πt)dt, (.3) ζ(n + 1) = α n lim δ 1/ δ ( 1)ζ()B n+1 (t)t 1 dt. (.4) Since δ < 1, the seies conveges unifomly on t δ. Thus ζ(n + 1) = α n lim δ 1/ ( 1)ζ() Let J (δ) be the integal in (.5), and wite δ B n+1 (t)t 1 dt. (.5) ( 1)ζ()J (δ) (.6) = J (δ) J (δ) + [ ] ζ() 1 ( 1)J (δ) (.7) = W n (δ) X n (δ) + Y n (δ). (.8)

5 4 M. Dancs & T. X. He When δ 1, J (δ) = O( 1 ), and hence the seies fo X n conveges unifomly on this inteval. The th tem of Y n is dominated by ζ() 1, hence this seies conveges unifomly on the same inteval. As we emaked peviously, the infinite seies Z n = Y n (1/) is the chief ingedient in ou appoximation of ζ(n + 1). The next two lemmas ae devoted to evaluating W n and X n. Lemma.1 With W n and X n defined as above, lim W n(δ) = n + 1 δ 1/ X n (1/) = n + 1 1/ 1/ Poof. Integating J (δ) by pats, we have W n (δ) = 1 B n+1(δ) (δ) n + 1 B n (t) log(1 4t )dt (.9) B n (t) log(1 t )dt. (.1) δ = 1 B n+1(δ) log(1 4δ ) n + 1 (t) B n (t)dt (.11) δ (t) B n (t)dt. (.1) The odd Benoulli polynomials have 1/ as a oot, hence the fist tem tends to zeo as δ 1/. The seies (t) B n (t) conveges unifomly when t δ < 1/. We thus have lim W n(δ) = n + 1 δ lim B n (t) δ 1/ δ 1/ (t) dt, and this establishes (.9). The poof of (.1) is simila.

6 Numeical Appoximation to ζ(n + 1) 5 at 1 The domain of W n can thus be extended so that W n is left-continuous. With this in mind, we now simplify the above integals. Lemma. Thee exist ational numbes p n and q n such that W n (1/) = 1/ 1/ X n (1/) = 1 Poof. Fom (.9), we have W n (1/) = n + 1 1/ 3/ 1/ B n+1 (t) 1 t dt p n n + 1 n+1 log 1. (.13) B n+1 (t) dt q n + n + 1 log 3 t. (.14) B n (t) log(1 t)dt + n + 1 Again integating by pats, the fist tem becomes and the second tem is Hence 1/ W n (1/) = 1/ B n+1 (t) 1 + t dt = 1/ 1/ = 1/ B n (t) log(1 + t)dt. (.15) B n+1 (t) dt, (.16) 1 t 1/ 1/ B n+1 (t) dt + (n + 1) 1 t B n+1 ( x) dx (.17) 1 x B n+1 (x) + (n + 1)x n dx. (.18) 1 x 1/ x n dx. (.19) 1 x

7 6 M. Dancs & T. X. He (.13) follows upon letting x = 1 y and evaluating the esulting integal. We can wite n(n + 1) n 1 1 x+1 p n = n (x + 1), (.) x= whee is the fowad diffeence opeato: ( f)(x) = f(x + 1) f(x). The poof of (.14) is simila, with n 1 1 (3/)x+1 q n = n(n + 1) (x + 1). (.1) x= Theoem 1.1 follows immediately upon inseting the esults of this lemma, along with the given expessions fo p n and q n, into equation (.8). Tuning to Theoem 1., we have aleady seen that the integal appeaing in (1.3) is O( 1 ). To estimate ζ() 1, we use the epesenation ζ(s) = s ( 1) n 1 n=1 n s, (.) which is valid when R(s) >. The altenating seies is 1 + O( ), and fom this we can deduce that ζ() 1 = O( ). Hence ( Z n = Z n (k) + O =k+1 1 ) (.3) and this establishes Theoem 1. Refeences [1] R. Apéy Iationalité de ζ() et ζ(3). Astéisque 41(1979), [] D. Cvijović and J. Klinowski Integal epesentations of the Riemann zeta function fo odd-intege aguments. J. Comp. and Appl. Math. 14(), [3] D. Cvijović and J. Klinowski New apidly convegent seies epesentations fo ζ(n + 1). Poc. Ame. Math. Soc. 13(1995), no. 5.,

8 Numeical Appoximation to ζ(n + 1) 7 [4] J. A. Ewell On values of the Riemann zeta function at integal aguments. Canad. Math. Bull. 34(1991), no. 1, [5] H. M. Sivastava Some apidly conveging seies fo ζ(n+1). Poc. Ame. Math. Soc. 17 (1999), no..,