Fast Converging Series for Riemann Zeta Function

doi:10.4236/ojdm.2012.24025

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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, [email protected]

Received August 29, 2012; revised September 13, 2012; accepted September 25, 2012

ABSTRACT

Riemann zeta function







has a key role in number theory and in its applications. In this paper we present a new

fast converging series for







. Applications of the series include the computation of the and recursive com-

putation of ,







21s









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





is defined for complex numbers

with by



Rs1











(1)

For the Riemann zeta function is of the gen-

eral form

sN

 

sNs



 (2)

where is an integer. For example



2π6



,



π904



and



6π945



.

Closed form solution for is not

known. Especially the infinite sums converging to



21,ss



N







have been extensively studied. Apery has proved that

is irrational [2,3]. However, his proof does not

concern on other values of









21s



. Also, it is not

known if







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







e foll

develop fast converging series for . The

21,ss



N

is the Riemann zeta

function defined by (1). Thowing infinite series

converges as





ln 2





.



2 we provide a proof of Theorem 1. In Sec-

tio

In 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







(Lemma 3). In Section 4 we apply

the results of heorem 1 to develop fast recursive

method for computing the

the T







values (Lemma 4). In

Section 5, we describe a md version of the Theo-

rem 1.

odifie

2. Proof of Theorem 1

We may write







22112

11 1

22 1

sss

ssnns





































(3)

Then we apply the well known logarithmic series



1111

n



ln 21 234



  

 (4)

By grouping the terms into pairs we obtain

H. OLKKONEN, J. T. OLKKONEN

132



111

ln 21

 

  



11 1

212 221

nn nn







 

 













(5)

Due to (3) we have



ln 22









(6)

which completes the proof.

rem 1 3. Applications of Theo

Lemma 1.









 (7)

Proof: We may write



222

111

22 2

10 1





11

11 1

44 41

sss

ssn

ns n

nn n





 

 













 

(8)

By dividing the last series into two parts we have



11 1 1









22121

11111

2212212



















(9)

Lemma 2.



21 1

ln 22











(10)

Proof: We may write



2121 212

1110

21 11 1

22 4

24 1

sss

ssns

























(11)

By separating the last series into two parts and due to







eorem 1 and (9) we have



111





22 141

241

ln 22

nnn

nn n











(12)

Lemma 3.





38ln24 41

nnn





 

 (13)

Proof: We may write (10) as



 



38ln248 2

8ln2 441

s















 



 (14)

which equals (13).

4. Recursive Computation of





Lemma 4.









Rs Rs, where







Rs 



nnn

 (15)

Proof: First we prove the Lemma 4 fo







. Based

terms as on Lemma 2 we can write the first two











11 1

83232 2

nnn







ln 2







 (16) 

By substituting the equation for (Lemma

have





3) we



 

32 2

541nn n













(17)







we obtain Correspondingly, for



 

52 72

41 4

n nn



71n











(18)

and generally



 

22 2

41 41

snn nn















(19)

By denoting



Rs nn







 (20)

we obtain Lemma 4.

5. Modified Version of Theorem 1

series Let us consider the zeta





s









Due to the limit value the nominator ap-

proaches zero and the seriesrated convergence.

We directly have





has accele





222

111

ln 22

222

sss













 (21)

Due to Lemma 1 we obtain, correspondingly

H. OLKKONEN, J. T. OLKKONEN

133







212 1

23 6

11 1

sss











(22)

Finally, we have (Lemma 2)



 

2121 21

211 21

11 2

ln 2

26 3

sss









 







(23)

6. Discussion

One application of the present main result (Theorem 1) is

the computation of the Riemann zeta function



ln 2







rapidly

h corr

for

for (Lemma 3) is con-

erme sum yield the 9tect

odd s. The series

verging. First 120 t





s of th

decimals. The use of Euler’s transformation yields the

infinite sum for





 



331











 (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 3

  

3 3RR

Rs Rs









 (25)

To obtain 9 correct decimals we need

summation for computation of

for ms for

30 terms of the





, 9 terms



and 7 ter











e recursiv

. Ang ob

vaht the comn method

interestin

putatio

ser-

tion is that thoug



as deduced for



21s



, it is valid also for computa-

tion of the





values. An alternatethod to ob-



tain

ve m



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





(Lemma 3) and the algo-

rithm for rec



woul

vergence

dified

ursive comof putation



(Lemm

a 4).

e fast convergence of the zeta series (21 - 23) has a sig-

nificant value in testing the







values. Previously the

series













,













 and



211 4











 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 app

compunces use





,ss





computed as

. The irrational number sequences can be





24 2sRsRs









eld an infinite number of irrat

(Lemma 4).

iional number

applied in co

nd the corresponding appli

This would y

quences 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











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)00336-8

offers 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. Brad

putational Strategies for the Riemann Zeta Function,”

Journal of Computational a

121, No. 1-2, 2000, pp. 247-296

[2] R. Apéry, “Irr,” Asté risque, ationalité de









Vol. 61,1979, pp. 11-13.

[3] F. Beukers, “A Note on the Irrationality of







,” Bul-

letin London Mathematical Society

pp. 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, V

doi:10.1016/0022-314X(72)90049-2

[5] D. Cvijovic and J. Klinowski, “Integral Representations

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doi:10.1016/S0377-0427(02)00358-8

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doi:10.4236/ojdm.2011.12008