Research Article On Rational Approximations to Euler s Constant γ and to γ log a/b

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1 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 9, Article ID 66489, pages doi:.55/9/66489 Research Article On Rational Approximations to Euler s Constant γ and to γ loga/b Carsten Elsner Fachhochschule für die Wirtschaft Hannover, Freundallee 5, 373 Hannover, Germany Correspondence should be addressed to Carsten Elsner, carsten.elsner@fhdw.de Received 4 December 8; Accepted 3 April 9 Recommended by Stéphane Louboutin The author continues to study series transformations for the Euler-Mascheroni constant γ. Here, we discuss in detail recently published results of A. I. Aptekarev and T. Rivoal who found rational approximations to γ and γ log q q Q > defined by linear recurrence formulae. The main purpose of this paper is to adapt the concept of linear series transformations with integral coefficients such that rationals are given by explicit formulae which approximate γ and γ log q. It is shown that for every q Q > and every integer d 4 there are infinitely many rationals a m /b m for m,,...such that γ log q a m /b m /d d /d 4 d m and b m Z m with log Z m d m for m tending to infinity. Copyright q 9 Carsten Elsner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.. Introduction Let s n : log n n.. 3 n It is well known that the sequence s n n converges to Euler s constant γ, , where s n γ O n n.. Nothing is known on the algebraic background of such mathematical constants like Euler s constant γ. So we are interested in better diophantine approximations of these numbers, particularly in rational approximations. In 995 the author introduced a linear transformation for the series s n n with integer coefficients which improves the rate of convergence. Let τ be an additional positive integer parameter.

2 International Journal of Mathematics and Mathematical Sciences Proposition. see. For any integers n and τ one has n n k τ n nk s kτ γ n k τ! nn n n τ..3 Particularly, by choosing τ n, one gets the following result. Corollary.. For any integer n one has n n k n nk s nk γ n k n n n n 3/ 4 n..4 Some authors have generalized the result of Proposition. under various aspects. At first one cites a result due to Rivoal. Proposition.3 see. For n tending to infinity, one has γ n n k n n k s kn O n k n 7 n/..5 Kh. Hessami Pilehrood and T. Hessami Pilehrood have found some approximation formulas for the logarithms of some infinite products including Euler s constant γ. These results are obtained by using Euler-type integrals, hypergeometric series, and the Laplace method 3. Proposition.4 3. For n tending to infinity the following asymptotic formula holds: n n k n γ nk s kn..6 n k 4non Recently the author has found series transformations involving three parameters n, τ and τ, 4. InPropositions.5 and.6 certain integral representations of the discrete series transformations are given, which exhibit important analytical tools to estimate the error terms of the transformations. Proposition.5 see 4. Let n, τ, and τ be integers. Additionally one assumes that τ τ..7

3 International Journal of Mathematics and Mathematical Sciences 3 Then one has n n τ k n nk s kτ γ n k n u log u u τ τ n u n u nτ u n n! du..8 Proposition.6 see 4. Let n, τ and τ be integers. Additionally one assumes that τ τ n τ..9 Then one has n n τ k n nk s kτ γ n k nτ τ wt unτ u n t τ τ t nτ τ ut n du dt,. with wt :.. t t π log Setting n τ dm, τ d m, d,. one gets an explicit upper bound from Proposition.6 Corollary.7. For integers m, d 3, one has d m k dm dmk /d d s kdm γ dm k < C d d 4 d m,.3 where <C d /6π is some constant depending only on d. For d one gets m 3m k m k 6 s km γ m k < 7π m..4 64m

4 4 International Journal of Mathematics and Mathematical Sciences For an application of Corollary.7 let the integers B m and A m be defined by B m : l.c.m.,, 3,...,4m, m 3m k m A m : B m k m k k m..5 Λk denotes the von Mangoldt function. By [5, Theorem 434] one has ψm : k mλk m..6 Then, for ε : log 55/4 >.8, there is some integer m such that B m e ψ4m < e 4εm 55 m m m..7 Multiplying.4 by B m, we deduce the following corollary. Corollary.8. There is an integer m such that one has for all integers m m that B m m 3m k m k logk m γb m A m < m k 6 7π m..8. Results on Rational Approximations to γ In 7, Aptekarev and his collaborators 6 found rational approximations to γ, which are based on a linear third-order recurrence. For the sake of brevity, let Dn l.c.m.,,...,n. Proposition. see 6. Let p n n and q n n be two solutions of the linear recurrence 6n 5n u n 8n 3 4n 8n 45 u n u. n 56n 3 4n 64n 7 n nn 6n u n with p, p, p 3/ and q, q 3, q 5. Then, one has q n Z, Dnp n Z, and γ p n c e n, q n qn c n! e n, n /4 n!. with two positive constants c,c.

5 International Journal of Mathematics and Mathematical Sciences 5 It seems interesting to replace the fraction p n /q n by A n B n : Dnp n Dnq n,.3 and to estimate the remainder in terms of B n. Corollary.. Let <ε<. Then there are two positive constants c,c 3, such that for all sufficiently large integers n one has c exp ε log B n / log log B n < γ A n < c 3 exp ε log B n / log log B n. B n.4 Recently, Rivoal 7 presented a related approach to the theory of rational approximations to Euler s constant γ, and, more generally, to rational approximations for values of derivatives of the Gamma function. He studied simultaneous Padé approximants to Euler s functions, from which he constructed a third-order recurrence formula that can be applied to construct a sequence in Qz that converges subexponentially to logzγ for any complex number z C \,. Here, log is defined by its principal branch. We cite a corollary from 7. Proposition.3 see 7. i The recurrence n 3 8n 8n 9U n3 4n 45n 5 8n U n 4n 3 5n 4n 5 8n 7U n.5 n 8n 98n 7U n, provides two sequences of rational numbers p n n and q n n with p, p 4, p 77/4 and q, q 7, q 65/ such that p n /q n n converges to γ. ii The recurrence n n n 3U n3 3n 9n 9 n U n 3n 3 6n 7n 3 U n n 3 U n,.6 provides two sequences of rational numbers p n n and q n n with p, p, p 7 and q, q 8, q 56 such that p n /q n n converges to logγ.

6 6 International Journal of Mathematics and Mathematical Sciences The goal of this paper is to construct rational approximations to γ loga/b without using recurrences by a new application of series transformations. The transformed sequences of rationals are constructed as simple as possible, only with few concessions to the rate of convergence see Theorems.4 and 6. below. In the following we denote by B n the Bernoulli numbers, that is, B /6, B 4 /3, B 6 /4, and so on In Sections 3 6 the Bernoulli numbers cannot be confused with the integers B n from Corollary.. In this paper we will prove the following result. Theorem.4. Let a, b, d 4 and m be positive integers, and S n : an bn j n j n j j B j j n j a j bj n n 4j, n..7 Then, d m k dm dmk S kdm γ log a /d d dm k b <c 4 d 4 d m,.8 where c 4 is some positive constant depending only on d. 3. Proof of Theorem.4 Lemma 3.. One has for positive integers d and m d m k dm gk : < 6 dm k dm. 3. dm k Proof. Applying the well known inequality g h g,weget d m k dm d mdm dm 4dm m < 6 dm. dm k 3. This proves the lemma. gk takes its maximum value for k k with k 5d 4d d m O, 3.3

7 International Journal of Mathematics and Mathematical Sciences 7 which leads to a better bound than 6 dm in Lemma 3.. But we are satisfied with Lemma 3.. A main tool in proving Theorem.4 is Euler s summation formula in the form n fi i n fx dx f fn r B j f j n f j R r, j! 3.4 where r N is a suitable chosen parameter, and the remainder R r is defined by a periodic Bernoulli polynomial P r x, namely R r r! n P r xf r xdx, 3.5 with P r x r sin πjx r! r. 3.6 πj Applying the summation formula to the function fx /x,wegetsee 8, equation 5 n i i log n r n B j j n j n P r x x r dx, n, r N. 3.7 It follows that n in i log n n n r B j j n j n 4j n n P r x x r dx, n, r N. 3.8 We prove Theorem.4 for a b. The case a<bis treated similarly. So we have again by the above summation formula that an ibn i log a b b a an bn r n B j jn j P r x x r dx, n, r N. b j a j 3.9

8 8 International Journal of Mathematics and Mathematical Sciences First, we estimate the integral on the right-hand side of 3.8. We have n n P r x x r n dx n P r x dx x r r! r! π r n x r n P r x dx x r r dx πj [ r x r ] xn j r 3. r! 3r! ζr < π r nr π r n r, since ζr ζ3 < 3. Next, we assume that n a. Hence bn, an n, n,and therefore we estimate the integral on the right-hand side in 3.9 by an bn P r x x r an dx bn n n P r x dx x r P r x 3r! dx x r π r n. r 3. Inthesequelweputr dm. Moreover, in the above formula we now replace n by dm k with k dm. In order to estimate r! we use Stirling s formula πm m e m m <m! < πm e m, m >. 3. Then, it follows that dmk dmk P r x x r dx 3r! π r dm k r 3dm! π dm dm dm 3 πdm dm πdm dm e 3r! π r dm r dm πdm πdmπe dm,

9 International Journal of Mathematics and Mathematical Sciences 9 and similarly we have admk bdmk P r x x r dx 3 3πdm, dm a. 3.4 dm πdmπe By using the definition of S n in Theorem.4, the formula. for s n, and the identities 3.8, 3.9, it follows that S n γ log a b s n γ s n s n s an s bn r n B j j s n γ n b a n n P r x x r dx n j an bn P r x x r a j bj dx, n 4j 3.5 where r is specified to r dm and n to n dm k. Moreover, we know from 4, Lemma that dmk gk, m. 3.6 By setting n dm k, the above formula for the series transformation of S dmk simplifies to dmk gks dmk γ log a b dmk gk s dmk γ b a dmk gk dm k dmk gk dmk gk dmk dmk admk bdmk P r x x r P r x x r dmk gk s dmk γ gk admk bdmk P r x x r dx dx dx gk dmk dmk P r x x r dx 3.7 <C d /d d d 4 d m 3 3πdm gk πdmπe, dm

10 International Journal of Mathematics and Mathematical Sciences where dm a, m, and d 3. Here, we have used the results from Corollary.7, 3.3, and 3.4. Thesum dmk gk dm k 3.8 vanishes, since for every real number x> dm we have dmk d mk dm dm k dm k x d m x m x dm x dm, 3.9 where on the right-hand side for an integer x with m x d m one term in the numerator equals to zero. The inequality 64 πe d < /d d d 3. holds for all integers d 4. Now, using Lemma 3., we estimate the right-hand side in 3.7 for dm a and d 4 as follows: dmk gks dmk γ log a b <C d /d d d 4 d m 3 3πdm πdm 6 dm πe dm C d /d d d 4 d m 3dm 3dm dm π 4 dm 64 πe dm 3. /d d <C d d 4 d C d /d d d 4 d m 3dm 3dm m 85 3d 8 π dm m π /d d d 4 d /d d d 4 d m m c 4 /d d d 4 d m. 3. The last but one estimate holds for all integers m, d 4, and c 4 is a suitable positive real constant depending on d. This completes the proof of Theorem.4.

11 International Journal of Mathematics and Mathematical Sciences 4. On the Denominators of S n In this section we will investigate the size of the denominators b m transformations a m b m dmk gks kdm, of our series 4. for m tending to infinity, where a m Z and b m N are coprime integers. Theorem 4.. For every m there is an integer Z m with Z m >, b m Z m, and log Z m d m, m. 4. Proof. We will need some basic facts on the arithmetical functions ϑx and ψx. Let ϑx log p, p x x >, ψx [ log x p x log p ] log p, x >, 4.3 where p is restricted on primes. Moreover, let D n : l.c.m,,...,n for positive integers n. Then, ψn log D n, n, 4.4 ψx ϑx πx log x x, x, 4.5 where 4.5 follows from 5, Theorem 4 and the prime number theorem. By 5, Theorem 8 von Staudt s theorem we know how to obtain the prime divisors of the denominators of Bernoulli numbers B k : The denominators of B k are squarefree, and they are divisible exactly by those primes p with p k. Hence, B k p Z, k,, p k Next, let max{a, b} dm n dm n k dm are the subscripts of S kdm in Theorem.4. First, we consider the following terms from the series transformation in S m : an bn j n j n j j : n e j j, 4.7

12 International Journal of Mathematics and Mathematical Sciences with, if j n, if n j bn e j :, if bn j an, if an j n, if j n, if n j an e j :, if an j bn, if bn j n. a b, a <b. 4.8 For every m there is a rational x m /y m defined by x m y m dmk gk kdm e j j, 4.9 where x m Z, y m N, x m,y m, and y m Y m : D 4d m, dm max{a, b}. 4. Similarly, we define rationals u m /v m by u m v m dmk gk k dm B j j k dm j a j bj k dm 4j, 4. where u m Z, v m N and u m,v m. We have k dm j k dm 4dm, k dm, j dm. 4. Therefore, using the conclusion 4.6 from von Staudt s theorem, we get v m V m : ab dm D dm p dm p Ddm 4dm, dm max{a, b}. 4.3 Note that D dm l.c.m. dm,...,dm, since every integer n with n <dmdivides at least one integer n with dm n dm.

13 International Journal of Mathematics and Mathematical Sciences 3 From 4. and 4.3 we conclude on b m Z m : ab dm D dm D 4d m D dm 4dm p. 4.4 p dm Hence we have from 4.4 and 4.5 that log Z m log dm log ab ψdm ψ 4d m 4dmψdm ϑdm log dm log ab dm 4d m 8d m dm log 3 logab dm d m 4.5 d m m. The theorem is proved. Remark 4.. On the one side we have shown that log Y m 4d m and log V m 8d m.onthe other side, every prime p dividing V m satisfies p max{a, b, dm, dm, dm} dmand therefore p divides Y m D 4d m. Conversely, all primes p with dm <p<4d m divide Y m,butnotv m. That means: V m is much bigger than Y m,butv m is formed by powers of small primes, whereas Y m is divisible by many big primes. 5. Simplification of the Transformed Series Let R n : n B j j n j a j bj, 5. n 4j such that S n an bn j n j n j j R n. 5. In Theorem.4 the sequence S n is transformed. In view of a simplified process we now investigate the transformation of the series S n R n. Therefore we have to estimate the contribution of R kdm to the series transformation in Theorem.4. For this purpose, we define E m : d m k dm dmk R kdm dmk gk dm k dm k B j j a dmk gk dmk gk. j bj dm k j dm k 4j A major step in estimating E m is to express the sums on the right-hand side by integrals. 5.3

14 4 International Journal of Mathematics and Mathematical Sciences Lemma 5.. For positive integers d, j and m one has dmk gk dm dm k j dm! j! u m log u j dm u d m u dm du. u dm 5.4 Proof. For integers k, r and a real number ρ with k ρ> the identity r k ρ r! e kρt t r dt 5.5 holds, which we apply with r j and ρ dm to substitute the fraction /dm k j. Introducing the new variable u : e t, we then get m dmk gk dm k gk k dm j j! u kdm log u j du dm k gku u kd m m log u j du. j! 5.6 The sum inside the brackets of the integrand can be expressed by using the equation n n τ k n k n k u τk n u n u nτ u n n!, n, τ N {}, 5.7 in which we put n dm and τ d m. This gives the identity stated in the lemma. The following result deals with the case j, in which we express the finite sum by a double integral on a rational function. Corollary 5.. For every positive integer m one has dmk gk d m dm k u dm w m u d m w d m du dw. 5.8 uw dm Proof. Set j inlemma 5., and note that dw log u u uw. 5.9

15 International Journal of Mathematics and Mathematical Sciences 5 Hence, dmk gk dm k dm! dm dm! dm u m log u dm u d m u dm du u dm uu m uw dm u d m u dm du dw. u dm 5. Let s be any positive integer. Then we have the following decomposition of a rational function, in which u is considered as variable and w as parameter: u s s uw w ν ν w ν w u s ν w s uw. 5. We additionally assume that s < dm. Then, differentiating this identity dm-times with respect to u, the polynomial in u on the right-hand side vanishes identically: dm u dm u s uw w w s dm dm! w dm uw dm. 5. Therefore, we get from 5. by iterated integrations by parts: dmk gk dm k dm! dm! u d m u dm dm u dm u m u m w m u d m u dm w uw du dw w m w 5.3 dm dm! w dm du dw uw dm. The corollary is proved by noting that w w m w w m m w m w m. 5.4

16 6 International Journal of Mathematics and Mathematical Sciences 6. Estimating E m In this section we estimate E m defined in 5.3. Substituting u for u into the integral in Lemma 5. and applying iterated integration by parts, we get dmk gk dm k j dm dm! j! dm u dm u m log u j u d m u dm du. 6. Set fu : u m log u j, 6. where m and j are kept fixed. We have f. For an integer k> we use Cauchy s formula f k a k! πi C fz dz k z a 6.3 to estimate f k. LetC denote the circle in the complex plane centered around with radius R : /k.witha andfz defined above, Cauchy s formula yields the identity f k k! π e ikφ Re iφ m log j Re iφ dφ. πr k π 6.4 For the complex logarithm function occurring in 6.4 we cut the complex plane along the negative real axis and exclude the origin by a small circle. All arguments φ of a complex number z /, are taken from the interval π, π. Therefore, using Re iφ R cos φ ir sin φ,weget Re iφ R R cos φ : A R, φ, arg Re iφ R sin φ arctan R cos φ. 6.5 Hence, log Re iφ R sin φ ln R R cos φ i arctan R cos φ ln A R, φ R sin φ i arctan R cos φ. 6.6

17 International Journal of Mathematics and Mathematical Sciences 7 Thus, it follows from 6.4 that f k k! πr k π π Re iφ m log Re iφ j dφ k! π m/ A R, φ πr k π 4 ln AR, φ R sin φ arctan R cos φ k! πr k π A R, φ m/ 4 ln AR, φ arctan R sin φ R cos φ j / dφ j / dφ. 6.7 From <R< we conclude on < R R R R R cos φ A R, φ < 4, φ π, R sin φ R cos φ sin φ cos φ, 6.8 <φ π. Since arctan is a strictly increasing function, we get R sin φ arctan R cos φ sin φ arctan cos φ π φ arctan tan π φ, <φ π. φ arctan cot 6.9 For <R<, this upper bound also holds for φ. Finally, we note that R k /k k /. Altogether, we conclude from 6.7 on f k k! πr k π m k! π m k! π ln 4 m/ 4 arctan 4 π φ ln π π ln π 4 j / dφ sin φ cos φ j / dφ j / dφ 6. m k! π π 3 j / dφ m 3 j k!.

18 8 International Journal of Mathematics and Mathematical Sciences It follows that the Taylor series expansion of fu, fu f k k! u k, 6. converges at least for <u<. Then, f dm u f k f kdm k dm! uk dm k! kdm u k, 6. and the estimate given by 6. implies for <u<that f dm u f kdm u k m 3 j k! m 3 j dm! k dm k k dm! k! u k u k m 3 j dm! u dm. 6.3 Combining 6.3 with the result from 6., wegetform> dmk gk dm k j m 3 j u j! d m u dm du m 3 j j! Γdm Γd m Γd m 6.4 d d m 3 j j!d m d m dm We estimate the binomial coefficient by Stirling s formula 3.. For this purpose we additionally assume that m d :. d m d m d d dm πdm d m d d d d d d d m. πd m d d d d m 6.5 We now assume m d and substitute the above inequality into 6.4: dmk gk dm k j dd m 3 j πm d d d d m. 6.6 j!d d m d d d

19 International Journal of Mathematics and Mathematical Sciences 9 For all integers m andd we have d πm d < πdm, d d m d d m. 6.7 Thus we have proven the following result. Lemma 6.. For all integers d, m with d 3 and m d one has dmk gk dm k j < m 3 j πd 3 d d d d m. 6.8 j!d d m d d Next, we need an upper bound for the Bernoulli numbers B j cf. 9, 3..5: j! Bj π j j 4 j! π, j. 6.9 j Let d 3andm max{d,a/}. UsingthisandLemma 6., we estimate E m in 5.3: πd3 m d d d d m πd3 m d d d d m E m < 3 d d m B j j d d a 3 j 3 j j bj j! 4j! d d m d d m πd3 m d d d d d d m d d 3 4 j! 3 j 3 j π j j! 4j! < 4 πd 3 d d m d d d d m 3 d d 8 3 π j 3 π j < 9 πd 3 d d m d d d d m. d d 6.

20 International Journal of Mathematics and Mathematical Sciences Now, let T n : an bn j n j n j j n e j, j n >, 6. with the numbers e j introduced in the proof of Theorem 4.. By definition of R n and S n we then have T n S n R n, and therefore we can estimate the series transformation of T n by applying the results from Theorem.4 and 6.. Again, let m max{d,a/} and d 4. d m k dmk dm T kdm γ log a dm k b dmk gks kdm γ log a b dmk gks kdm γ log a b E m dmk gkr kdm 6. /d d m 9 πd <c 4 3 d 4 d d d m d d d d m. d d By similar arguments we get the same bound when b>a. For d 3 it can easily be seen that d d d d d d d d /d d /d d 4 < d 4d Thus, we finally have proven the following theorem. Theorem 6.. Let T n : an bn j n j n j, n >, 6.4 j where a, b are positive integers. Let d 4 be an integer. Then, there is a positive constant c 5 depending at most on a, b and d such that d m k dm k T kdm γ log a dm k b < c 5 8 m 4 d m, m. 6.5

21 International Journal of Mathematics and Mathematical Sciences 7. Concluding Remarks It seems that in Theorem 6. a smaller bound holds. Conjecture 7.. Let a, b be positive integers. Let d be an integer. Then there is a positive constant c 6 depending at most on a, b and d such that for all integers m one has d m k dm dmk T kdm γ log a dm k b /d d m <c 6. d 4 d 7. A proof of this conjecture would be implied by suitable bounds for the integral stated in Lemma 5.. For j such a bound follows from the double integral given in Corollary 5.: dmk gk dm k u dm w m u d m w d m du dw uw dm u d wu uu w uw 3 uw u d wu d w d uw d m du dw 7. /d d 4 d d 4 d d 3 /d d m u wu uw 3 du dw /d d m, m, d 4 d where the double integral in the last but one line equals to /4. Note that the rational functions uu w uw, u d wu d w d, 7.3 uw d take their maximum values 4 d and /d d /d 4 d inside the unit square,, at u, w /, and u, w /, d /d, respectively. Finally, we compare the bound for the series transformation given by Theorem.4 with the bound proven for Theorem 6..InTheorem.4 the bound is T d, m : c 4 /d d m, d 4, m, 7.4 d 4 d

22 International Journal of Mathematics and Mathematical Sciences whereas we have in Theorem 6. that T d, m : c 5 8 m, d 4, m. 7.5 m 4 d For fixed d 4 and sufficiently large m it is clear on the one hand that T d, m <T d, m, but on the other hand we have lim lim m d log T d, m lim log T d, m lim d m log T d, m log T d, m. 7.6 Conversely, for d tending to infinity, one gets log T d, m dm log 4, log T d, m dm log 4, 7.7 with implicit constants depending at most on m. For the denominators b m of the transformed series S kdm in Theorem.4 we have the bound log b m d m from Theorem 4., anda similar inequality holds for the denominators of the transformed series T kdm in Theorem 6.. References C. Elsner, On a sequence transformation with integral coefficients for Euler s constant, Proceedings of the American Mathematical Society, vol. 3, no. 5, pp , 995. T. Rivoal, Polynômes de type Legendre et approximations de la constante d Euler, notes, 5, rivoal/articles/euler.pdf. 3 K. H. Pilehrood and T. H. Pilehrood, Arithmetical properties of some series with logarithmic coefficients, Mathematische Zeitschrift, vol. 55, no., pp. 7 3, 7. 4 C. Elsner, On a sequence transformation with integral coefficients for Euler s constant. II, Journal of Number Theory, vol. 4, no., pp , 7. 5 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, UK, A. I. Aptekarev, Ed., Rational Approximation of Euler s Constant and Recurrence Relations, Sovremennye Problemy Matematiki, Vol. 9, MIAN Steklov Institute, Moscow, Russia, 7. 7 T. Rivoal, Rational approximations for values of derivatives of the Gamma function, rivoal/articles/eulerconstant.pdf. 8 D. E. Knuth, Euler s constant to 7 places, Mathematics of Computation, vol. 6, no. 79, pp. 75 8, M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 97.