Information Sciences and Computing
Volume 2014, Number 1, Article ID ISC251013, 03 pages
Available online at http://www.infoscicomp.com/
Research Article
On the Riemann Zeta Function at Even Integers
Z. Ahsana, P. Lam-Estradab and J. López-Bonillac
a
Department of Mathematics, Aligarh Muslim University, Aligarh, India
Departamento de Matemáticas, ESFM, Edif. 9, Instituto Politécnico Nacional (IPN)
c
ESIME-Zacatenco, IPN, Col. Lindavista, CP 07738, México DF
b
Corresponding author: J. López-Bonilla; E-mail: jlopezb@ipn.mx
Received 12 October 2013; Accepted 19 December 2013
Abstract: Without the explicit participation of Bernoulli polynomials ( ), Balanzario [1]
exhibits an elementary method to determine (2 ), n = 1, 2,… Here we show the connection
between his polynomials and the ( ).
Keywords: Riemann zeta function; Bernoulli polynomials.
1. Introduction
In [1] the principal aim is to calculate the Riemann zeta function [2, 3] at even
integers, with the following result:
(2 ) = −
(
)
(2 )
,
(1)
where
( )=
,
( )=
( )=
−
!
,
#( ) =
%
! !
−
!
( )=
,
( )=
+
#
"
# !
−
'
( )=(
( )=(
'
without constants of integration.
,
+
!
,
(2)
,…
!
=
These polynomials can be generated with
'
#
−
'
)* −
and the recurrence relations:
)* ,
'
,
(3)
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2
(2) =
Thus with (1) and (2) is easy to obtain that
+"
+
,
(4) =
+
-!
,
(6) =
, … , in harmony with the values reported in the literature [2, 3]. It would be
interesting to have a closed expression for (2 + 1) 041. In the next Section, we
show how to write (1) and (2) in terms of the polynomials and numbers of Bernoulli.
-
2. Balanzario’s polynomials
We have the Bernoulli polynomials [5]:
( )=
,
( )=
( )=
−2
+
( )=
−3
+
−
,
−
+
,
−
+
−
,
( )=
,
5
2
( )=
−
(4)
, ….
2
with the properties:
=
=0
7
4 (1)
,
=0 ,
6 = 2, 3, … ..
' (0)
, 8 = 1, 2, …. ;
=0
∀ ;.
(5)
Then the polynomials (2) can be written in terms of (4), for example,
( )=
0
( )=
# !
(2 ) − 4
03
( )+
(2 ) − 48
( )1 ,
( )1 , etc.
( )−7
(6)
therefore:
=
thus (1) gives
,
(2) =
+
#
=−
and
(4) =
+
-!
,
!
(7)
.
Besides, it is possible to prove that:
= − >?
'
where >?
>? =
@A
= (−1)
(
)!
>?
,
(8)
are the Faulhaber [6]–Bernoulli [7] numbers:
,
>? = >? =
!
,
with the Bernoulli-like polynomials [5]:
>? =
, … ..
(9)
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>? ( ) =
>? ( ) =
>? ( ) =
, >? ( ) =
!
(15
!
Then
(21
>?
(3
− 30
>? ( ) =
− 1) ,
>? ( ) =
+ 7) ,
− 105
=−
3
+ 147
,
!
(
(3
− ),
− 10
+7 ) ,
(10)
− 31) , ….
>?
=
#
# !
,
(11)
in accordance with (7) and (8). If into (1) we employ (8) appears the known relation
[2, 8]:
@A
(2 ) =
>? .
(
)!
The expressions (6) and (8) show the relationship between the polynomials of
Balanzario and Bernoulli.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
E.P. Balanzario, Elementary method to evaluate the Riemann zeta function at even
integers, Miscelánea Matemática, 33(2001), 31-41.
E.C. Titchmarsh, The Theory of the Riemann Zeta Function, Clarendon Press,
Oxford, 1986.
C. Calderón, The Riemann zeta function, Rev. Real Acad. Ciencias, Zaragoza,
57(2002), 67-87.
E.L. Stark, The series ∑∞
'F ' D , E = 2, 3, 4, … . , once more, Math Mag., 47(1974),
197-202.
C. Lanczos, Discourse on Fourier Series, Oliver & Boyd, London, 1966.
J. Faulhaber, Academiae Algebrae, Ulm, Germany, 1631.
J. Bernoulli, Ars Conjectandi, Thurneysen, Basel, 1713 (The Art of Conjecturing,
Johns Hopkins University Press, Baltimore, 2005.)
Z. Ahsan, P. Lam-Estrada, J. López-Bonilla and R. López-Vázquez, On the LaforgiaNatalini’s inequality for the Riemann zeta function, Transnational J. of Mathematical
Analysis and Applications, 1(1) (2013), 1-4.
Copyright © 2014 Z. Ahsan, P. Lam-Estrada and J. López-Bonilla. 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.