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 Open Journal of Discrete Mathematics, 2012, 2, 131-133 http://dx.doi.org/10.4236/ojdm.2012.24025 Published Online October 2012 (http://www.SciRP.org/journal/ojdm) Fast Converging Series for Riemann Zeta Function Hannu Olkkonen1, Juuso T. Olkkonen2 1Department of Applied Physics, University of Eastern Finland, Kuopio, Finland 2VTT Technical Research Centre of Finland, Espoo, Finland Email: hannu.olkkonen@uef.fi, juuso.olkkonen@vtt.fi Received August 29, 2012; revised September 13, 2012; accepted September 25, 2012 ABSTRACT Riemann zeta function s has a key role in number theory and in its applications. In this paper we present a new fast converging series for s. Applications of the series include the computation of the and recursive com- putation of , 321s2s and generally ,ssR. We discuss on the production of irrational number se- quences e.g. for encryption coding and zeta function maps for analysis and synthesis of log-time sampled signals. Keywords: Riemann Zeta Function; Converging Series; Number Theory; Cryptography; Signal Processing 1. Introduction Riemann zeta function plays an important role in modern number theory and its applications, especially in infor- mation theory and signal processing [1]. The Riemann zeta function s is defined for complex numbers s with by Rs111snsn (1) For the Riemann zeta function is of the gen- eral form sN 2π2ssNs (2) where is an integer. For example Ns22π6, 4π904 and 66π945. Closed form solution for is not known. Especially the infinite sums converging to 21,ssN3 have been extensively studied. Apery has proved that is irrational [2,3]. However, his proof does not concern on other values of 321s. Also, it is not known if 3 is transcendental or not. The series (1) converges very slowly. Some acceleration on this series is achieved by asymptotic expansion with Bernoulli numbers [4]. In this context the Euler’s transformation is also efficient. Integral representations of the Riemann zeta function at odd integers has been described in [5, 6]. In this work we describe some new results on the con- verging series of the zeta function. Our primary aim is to main result is as follows. Theorem 1. Suppose ths e folldevelop fast converging series for . The at 21,ssNis the Riemann zeta function defined by (1). Thowing infinite series converges as 2ln 22sss. 2 we provide a proof of Theorem 1. In Sec- tioIn Section n 3 we present some applications of the Theorem 1, which include the converging series for even and odd Riemann function (Lemmas 1 and 2) and fastly converg-ing series for 3 (Lemma 3). In Section 4 we apply the results of heorem 1 to develop fast recursive method for computing the the Ts values (Lemma 4). In Section 5, we describe a md version of the Theo-rem 1. odifie2. Proof of Theorem 1 We may write 22112210111 12221122122 1ssssssnnssnsnsnnnnnn (3) Then we apply the well known logarithmic series 11111n1ln 21 234nn   (4) By grouping the terms into pairs we obtain Copyright © 2012 SciRes. OJDM H. OLKKONEN, J. T. OLKKONEN 132 111ln 21   1123411 1212 221nnnn nn   (5) Due to (3) we have 2ln 22sss (6) which completes the proof. rem 1 3. Applications of TheoLemma 1. 212122sss (7) Proof: We may write 22211122 210 12s112211 144 41sssssnsns nnnn n    (8) By dividing the last series into two parts we have 2110111 1 12212141111112212212nnnnnnnnn (9) Lemma 2. 21121 1ln 222sss (10) Proof: We may write 2121 21211102121 11 122 4124 1ssssssnsnnnnn(11) By separating the last series into two parts and due to Thseorem 1 and (9) we have 2211111122 1412411ln 22nnnnn nnn(12) Lemma 3. 321138ln24 41nnn  (13) Proof: We may write (10) as  2123212138ln248 218ln2 441ssnsnn  (14) which equals (13). 4. Recursive Computation of s Lemma 4. 42sRs Rs, where 211sRs  4nnn (15) Proof: First we prove the Lemma 4 fo1r 5. Based terms as on Lemma 2 we can write the first two 3213511 183232 241nnnln 2 (16) By substituting the equation for (Lemmahave 3 3) we  32 241541nn n51141nnn (17) 7 we obtain Correspondingly, for  52 721141 4nnn nn4171n (18) and generally  22 2114141 41ssnnsnn nn (19) By denoting 21141snRs nn (20) we obtain Lemma 4. 5. Modified Version of Theorem 1 series Let us consider the zeta22s1ss Due to the limit value the nominator ap- proaches zero and the seriesrated convergence. We directly have 1 has accele222111ln 22222ssssssss (21) Due to Lemma 1 we obtain, correspondingly Copyright © 2012 SciRes. OJDM H. OLKKONEN, J. T. OLKKONEN Copyright © 2012 SciRes. OJDM 1332211212 12221223 611 1ssssssss (22) Finally, we have (Lemma 2)  2121 2111211 21122211 2ln 226 3ssssssss  (23) 6. Discussion One application of the present main result (Theorem 1) is the computation of the Riemann zeta function 1ln 2srapidlyh corr for for (Lemma 3) is con-erme sum yield the 9tect odd s. The seriesverging. First 120 t3 s of thdecimals. The use of Euler’s transformation yields the infinite sum for 3  3014331nnn (24) However, first0 term 160s is needed for the 9 correct decimals. Due to Lemma 4 the convergence of the series of can be accelerated by recursive computation:   38ln24 342R  543 3RRsRs Rs (25) To obtain 9 correct decimals we need summation for computation offor ms for 30 terms of the 5, 9 terms Rsand 7 ter 7 9e recursiv. Ang obvaht the comn methodw interestinputatioser- tion is that thoug2as deduced for 21s, it is valid also for computa- tion of the s values. An alternatethod to ob- he 2itainve m ts values d be the deduction of the recursive method similar to Lemma 4 based on Lemma 1. However, the conof the series is not so acceler- ated. The mo zeta series (21 - 23) have accelerated convergence. However, their application yields the same results for the series for 3 (Lemma 3) and the algo- rithm for recwoulvergence difiedursive comof putation s (LemmTha 4). e fast convergence of the zeta series (21 - 23) has a sig- nificant value in testing the svalues. Previously the series 211ss,  13214ss and 11211 4ss have used for that purpose been [1]. lication of the fast converging series is the tation of the irrational number sequed in encryption coding. The seed number would be any One appcompunces use,ssRcomputed as. The irrational number sequences can be 24 2sRsRseld an infinite number of irrat (Lemma 4). iional number seapplied in cond the corresponding appliThis would yquences for encryption keys. The fast recursion for computation of the zeta function maps is also useful in the analysis and synthesis of the log-sampled signals mpressive sampling scheme [7]. Theorem 1 acations (Lem- mas 1 - 3) give a new converging series for the Riemann zeta function. To the best of authors’ knowledge no previous studies concern on the convergence of the simi- lar series. Theoretically, the series of the form s1!sswhere s! denotes the factorial function, would guarantee accelerated convergence. However, to conduct the cor- responding results (Lemmas 1 - 4) the convergence of the series (26) or some of its variant should be proved. This , NCES ley and R. E. Crandall, “Com- nd Applied Mathematics, Vol. . doi:10.1016/S0377-0427(00)00336-8offers an interesting subject for future work. 7. Acknowledgements This work was supported by the National Technology Agency of Finland (TEKES). REFERE[1] J. M. Borwein, D. M. Bradputational Strategies for the Riemann Zeta Function,” Journal of Computational a121, No. 1-2, 2000, pp. 247-296 [2] R. Apéry, “Irr,” Asté risque, ationalité de 2 et 3Vol. 61,1979, pp. 11-13. [3] F. Beukers, “A Note on the Irrationality of 3,” Bul-letin London Mathematical Societypp. 268-272. , Vol. 11, No. 3, 1979, ol. 4, No. 3, 1972, pp. 225-235. [4] E. Grosswald, “Remarks Concerning the Values of the Riemann Zeta Function at Integral, Odd Arguments,” Jour- nal of Number Theory, Vdoi:10.1016/0022-314X(72)90049-2 [5] D. Cvijovic and J. Klinowski, “Integral Representations of the Riemann Zeta Function for Odd-Integer Argu- ments,” Journal of Computational and Applied M athe mat- ics, Vol. 142, No. 2, 2002, pp. 435-439. doi:10.1016/S0377-0427(02)00358-8 [6] T. Ito, “On an Integral Representation of Special Values of the Zeta Function at Odd Integers,” Journal of the Mathematical Society of Japan, Vol. 58, No. 3, 2006, pp. 681-691. doi:10.2969/jmsj/1156342033 [7] H. Olkkonen and J. T. Olkkonen, “Log-Time Sampling of Signals: Zeta Transform,” Open Journal of Discrete Ma- thematics, Vol. 1, No. 2, 2011, pp. 62-65. doi:10.4236/ojdm.2011.12008