H. OLKKONEN, J. T. OLKKONEN

Copyright © 2012 SciRes. OJDM

133

22

11

212 1

22

2

1

2

23 6

11 1

ss

sss

ss

(22)

Finally, we have (Lemma 2)

2121 21

11

211 21

1

22

2

11 2

ln 2

26 3

sss

sss

ss

(23)

6. Discussion

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

the computation of the Riemann zeta function

1

ln 2

rapidly

h corr

for

for (Lemma 3) is con-

erme sum yield the 9tect

odd s. The series

verging. First 120 t

3

s of th

decimals. The use of Euler’s transformation yields the

infinite sum for

3

3

0

1

4

331

n

nn

(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

42

R

54

3 3RR

Rs Rs

(25)

To obtain 9 correct decimals we need

summation for computation of

for ms for

30 terms of the

5

, 9 terms

Rs

and 7 ter

7

9

e recursiv

. Ang ob

vaht the comn method

w

interestin

putatio

ser-

tion is that thoug

2

as deduced for

21s

, it is valid also for computa-

tion of the

values. An alternatethod to ob-

he

2

i

tain

ve m

t

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

woul

vergence

dified

ursive comof putation

(Lemm

Th

a 4).

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

nificant value in testing the

values. Previously the

series

2

11

s

s

,

1

3

21

4

s

s

and

1

1

211 4

s

s

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

R

computed as

. The irrational number sequences can be

24 2sRsRs

eld an infinite number of irrat

(Lemma 4).

iional number

se

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

1!

s

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).

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et

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