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
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  

2
s
 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

1
1
s
n
sn 




(1)
For the Riemann zeta function is of the gen-
eral form
sN
 
2
π
2
s
sNs   (2)
where is an integer. For example

Ns

2
2π6  ,

4
π904  and

6
6π945  .
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 foll
develop fast converging series for . The
at
21,ss  N
is the Riemann zeta
function defined by (1). Thowing infinite series
converges as


2
ln 2
2s
s
s 

.

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


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.
odifie
2. Proof of Theorem 1
We may write




22112
2
10
1
11 1
2 22
11
2
2
1
22 1
s
sss
ssnns
s
ns
n
s
n
n
n
n
nn


















(3)
Then we apply the well known logarithmic series

1
1111
n

1
ln 21 234
nn 

  
 (4)
By grouping the terms into pairs we obtain
C
opyright © 2012 SciRes. OJDM

H. OLKKONEN, J. T. OLKKONEN 132

111
ln 21
 
  

11
23 4
11 1
212 221
nn
nn nn


  
 






(5)
Due to (3) we have

2
ln 22
s
s
s




(6)
which completes the proof.
rem 1 3. Applications of Theo
Lemma 1.

2
1
21
2 2s
s
s 



 (7)
Proof: We may write

222
111
22 2
10 1
2s 

11
22
11 1
44 41
sss
ssn
s
ns n
n
nn n


 
 






 
(8)
By dividing the last series into two parts we have

2
11
01
11 1 1




22121 41
11111
2212212
nn
nn
nn
n
nn




 



(9)
Lemma 2.

21
1
21 1
ln 22
2s
s
s 




(10)
Proof: We may write


2121 212
1110
2
1
21 11 1
22 4
1
24 1
s
sss
ssns
n
nn
nn












(11)
By separating the last series into two parts and due to
Th
s 


eorem 1 and (9) we have


2
2
111
1


11
22 141
241
1
ln 22
nnn
nn n
nn







(12)
Lemma 3.



32
1
1
38ln24 41
nnn


 
 (13)
Proof: We may write (10) as

 

21
2
32
1
21
38ln248 2
1
8ln2 441
s
s

n
s
nn







 

 (14)
which equals (13).
4. Recursive Computation of

s 
Lemma 4.




42
s
Rs Rs, where  


2
1
1
s
Rs  
4 nnn  (15)
Proof: First we prove the Lemma 4 fo
1
r


5  . Based
terms as on Lemma 2 we can write the first two





32
1
35
11 1
83232 2
41
nnn



ln 2



 (16) 
By substituting the equation for (Lemma
have

3  3) we

 
32 2
41
541nn n 


5
11
41
nn
n




(17)


7  we obtain Correspondingly, for

 
52 72
11
41 4
nn
n nn

41
71n 





(18)
and generally

 
22 2
11
41
41 41
ss
nn
snn nn 






(19)
By denoting


2
1
1
41
s
n
Rs nn



 (20)
we obtain Lemma 4.
5. Modified Version of Theorem 1
series Let us consider the zeta


22 s

1
s
s 


Due to the limit value the nominator ap-
proaches zero and the seriesrated convergence.
We directly have
1  
has accele


222
111
ln 22
222
sss
sss
ss 





 (21)
Due to Lemma 1 we obtain, correspondingly

H. OLKKONEN, J. T. OLKKONEN 133



22
11
212 1
22
2
1
2
23 6
11 1
s
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


s

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
s
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

s
 values. An alternatethod to ob-
he

2
i
tain
ve m
t
s
 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
s
 (Lemm
Th
a 4).
e fast convergence of the zeta series (21 - 23) has a sig-
nificant value in testing the


s
 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


s


1! s
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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2  et

3 
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
3  ,” Bul-
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