PII. S0161171204307180 http://ijmms.hindawi.com
© Hindawi Publishing Corp.
q-RIEMANN ZETA FUNCTION
TAEKYUN KIM Received 19 July 2003
We consider the modifiedq-analogue of Riemann zeta function which is defined byζq(s)=
∞
n=1(qn(s−1)/[n]s), 0< q <1,s∈C. In this paper, we giveq-Bernoulli numbers which can be viewed as interpolation of the aboveq-analogue of Riemann zeta function at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Also, we will treat some identities of q-Bernoulli numbers using non- Archimedeanq-integration.
2000 Mathematics Subject Classification: 11S80, 11B68.
1. Introduction. Throughout this paper,Zp,Qp,C, andCpwill respectively denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, the complex number field, and the completion of algebraic closure ofQp.
Thep-adic absolute value inCpis normalized so that|p|p=1/p. When one talks of q-extension,qis considered in many ways such as an indeterminate, a complex number q∈C, or ap-adic numberq∈Cp. Ifq∈C, we normally assume|q|<1. Ifq∈Cp, then we normally assume|q−1|p< p−1/(p−1)so thatqx=exp(xlogq)for|x|p≤1. We use the notation
[x]=[x:q]=1−qx
1−q =1+q+q2+···+qx−1. (1.1)
Note that limq→1[x]=xforx∈Zpin thep-adic case.
LetUD(Zp)be denoted by the set of uniformly differentiable functions onZp. Forf∈UD(Zp), we start with the expression
1
pN
0≤j<pN
qjf (j)=
0≤j<pN
f (j)µq
j+pNZp
(1.2)
representing the analogue of Riemann’s sums forf(cf. [4]).
The integral off onZp will be defined as the limit(N→ ∞)of these sums, which exists. Thep-adicq-integral of a functionf∈UD(Zp)is defined by (see [4])
Zp
f (x)dµq(x)= lim
N→∞
1
pN
0≤j<pN
f (j)qj. (1.3)
Fordthat is a fixed positive integer with(p,d)=1, let X=Xd=lim←N Z
dpNZ, X1=Zp, X∗=
0<a<dp (a,p)=1
a+dpZp, a+dpNZp= x∈X|x≡a
moddpN ,
(1.4)
wherea∈Zlies in 0≤a < dpN.
Let N be the set of positive integers. For m,k ∈N, theq-Bernoulli polynomials, β(−m,k)m (x,q), of higher order for the variablexinCpare defined usingp-adicq-integral by (cf. [4])
β(−m,k)m (x,q)
=
Zp
Zp
···
Zp
ktimes
x+x1+x2+···+xk
m
·q−x1(m+1)−x2(m+2)−···−xk(m+k)dµq
x1
dµq
x2
···dµq
xk
. (1.5)
Now, we define theq-Bernoulli numbers of higher order as follows (cf. [2,4,7]):
β(−m,k)m
=β(−m,k)m (q)
=β(−m,k)m (0,q). (1.6)
By (1.5), it is known that (cf. [4])
β(−m,k)m =lim
N→∞
1 pNk
pN−1 x1=0
···
pN−1 xk=0
x1+···+xkm
q−x1m−x2(m+1)+···−xk(m+k−1)
= 1 (1−q)m
m i=0
m i
(−1)i(i−m)(i−m−1)···(i−m−k+1) [i−m][i−m−1]···[i−m−k+1],
(1.7)
where
m i
are the binomial coefficients.
Note that limq→1β(−m,k)m =Bm(k), whereBm(k)are ordinary Bernoulli numbers of orderk (cf. [2,3,5,7,9]). By (1.5) and (1.7), it is easy to see that
β(−m,1)m (x,q)= m i=0
m i
qxiβ(−m,1)i [x]m−i
= 1 (1−q)m
m j=0
qjx m
j
(−1)j j−m [j−m].
(1.8)
We modify theq-analogue of Riemann zeta function which is defined in [1] as follows:
forq∈Cwith 0< q <1,s∈C, define
ζq(s)= ∞ n=1
q(s−1)n
[n]s . (1.9)
q-RIEMANN ZETA FUNCTION
The numerator ensures the analytic continuation for(s) >1. In (1.9), we can consider the following problem.
“Are there q-Bernoulli numbers which can be viewed as interpolation ofζq(s)at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers?”
In this paper, we give the valueζq(−m)form∈N, which is the answer of the above problem, and construct a new complexq-analogue of Hurwitz’s zeta function andq-L- series. Also, we will treat some interesting identities ofq-Bernoulli numbers.
2. Some identities ofq-Bernoulli numbersβ(−m,1)m . In this section, we assumeq∈Cp
with|1−q|p< p−1/(p−1). By (1.5), we have β(−n,1)n (x,q)=
Xq−(n+1)t[x+t]ndµq(t)
=[d]n−1
d−1
i=0
q−ni
Zp
q−(n+1)dx x+i
d :qd n
dµqd(x).
(2.1)
Thus, we have
β(−n,1)n (x,q)=[d]n−1
d−1
i=0
q−niβ(−n,1)n x+i
d ,qd
, (2.2)
whered,nare positive integers.
If we takex=0, then we have
[n]β(−m,1)m −n[n]mβ(−m,1)m
qn
=
m−1 k=0
m k
[n]kβ(−m,1)k qnn−1
j=1
q−(m−j)k[j]m−k. (2.3) It is easy to see that limq→1β(−m,1)m =Bm, whereBm are ordinary Bernoulli numbers (cf. [7]).
Remark2.1. By (2.3), note that
n 1−nm
Bm=
m−1
k=0
m k
nkBk
n−1
j=1
jm−k. (2.4)
LetFq(t)be the generating function ofβ(−n,1)n as follows:
Fq(t)= ∞ k=0
β(−k,1)k tk
k!. (2.5)
By (1.7) and (2.5), we easily see that
Fq(t)= − ∞ m=0
m
∞ n=0
q−mn[n]m−1 tm
m!. (2.6)
Through differentiating both sides with respect totin (2.5) and (2.6), and comparing coefficients, we obtain the following proposition.
Proposition2.2. Form >0, there exists
−β(−m,1)m
m =
∞ n=1
q−nm[n]m−1. (2.7)
Moreover,β(0,1)0 =(q−1)/logq.
Remark2.3. Note thatProposition 2.2is aq-analogue ofζ(1−2m)for any positive integerm.
Letχbe a primitive Dirichlet character with conductorf∈N. Form∈N, we define
β(−m,1)m,χ =
Xq−(m+1)xχ(x)[x]mdµq(x), form≥0. (2.8) Note that
β(−m,1)m,χ =[d]m−1
d−1 i=0
χ(i)q−miβ(−m,1)m
i d,qd
. (2.9)
3. q-analogs of zeta functions. In this section, we assumeq∈Cwith|q|<1. In [1], theq-analogue of Riemann zeta function was defined by (cf. [1])
ζq∗(s)= ∞ n=1
qns
[n]s, (s) >0. (3.1)
Now, we modify the aboveq-analogue of Riemann zeta function as follows: forq∈C with 0<|q|<1,s∈C, define
ζq(s)= ∞ n=1
q(s−1)n
[n]s . (3.2)
By (2.5), (2.6), and (2.7), we obtain the following proposition.
Proposition3.1. Form∈N, there exists (i) ζq(1−m)= −β(−m,1)m /m, form≥1;
(ii) ζq(s)having simple pole ats=1with residue(q−1)/logq. By (1.7) and (1.8), we see that
β(−n,1)n (x,q)= −n ∞ k=0
[k]qx+[x]n−1
q−n(k+x), where 0≤x <1. (3.3)
q-RIEMANN ZETA FUNCTION
Hence, we can defineq-analogue of Hurwitzζ-function as follows: fors∈C, define
ζq(s,x)= ∞ n=0
q(s−1)(n+x)
[n]qx+[x]s. (3.4)
Note thatζq(s,x)has an analytic continuation inCwith only one simple pole ats=1.
By (3.3) and (3.4), we have the following theorem.
Theorem3.2. For any positive integerk, there exists
ζq(1−k,x)= −β(−k,1)k (x,q)
k . (3.5)
Letχbe Dirichlet character with conductord∈N. By (2.9), the generalizedq-Bernoulli numbers withχcan be defined by
β(−m,1)m,χ =[d]m−1
d−1
i=0
χ(i)q−miβ(−m,1)m i
d,qd
. (3.6)
Fors∈C, we define
Lq(s,χ)=∞
n=1
χ(n)q(s−1)n
[n]s . (3.7)
It is easy to see that
Lq(χ,s)=[d]−s d a=1
χ(a)q(s−1)aζqd
s,a
d
. (3.8)
By (3.6), (3.7), and (3.8), we obtain the following theorem.
Theorem3.3. Letkbe a positive integer. Then there exists
Lq(1−k,χ)= −β(−k,1)k,χ
k . (3.9)
LetaandFbe integers with 0< a < F. Fors∈C, we consider the functionsHq(s,a,F) as follows:
Hq(s,a,F)=
m≡a(F),m>0
qm(s−1)
[m]s =[F]−sζqF
s,a
F
. (3.10)
Then we have
Hq(1−n,a,F)= −[F]n−1 n β(−n,1)n
a F,qF
, (3.11)
wherenis any positive integer.
Therefore, we obtain the following theorem.
Theorem3.4. LetaandF be integers with0< a < F. Fors∈C, there exists (i) Hq(1−n,a,F)= −([F]n−1/n)β(−n,1)n (a/F,qF);
(ii) Hq(s,a,F)having a simple pole ats=1with residue(1/[F]F)((qF−1)/logq). In a recent paper, theq-analogue of Riemann zeta function was studied by Cherednik (see [1]). In [1], we can consider theq-Bernoulli numbers which can be viewed as an interpolation of theq-analogue of Riemann zeta function at negative integers. In this paper, we have shown that the q-analogue of zeta function interpolates q-Bernoulli numbers at negative integers in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers (cf. [2,5,7]).
Remark3.5. Letq∈Cpwith|1−q|p< p−1/(p−1). Then thep-adicq-gamma function was defined as (see [8])
Γp,q(n)=(−1)n
1≤j<n,(j,p)=1
[j]. (3.12)
For allx∈Zp, we have
Γp,q(x+1)=p,q(x)Γp,q(x), (3.13) wherep,q(x)= −[x]for|x|p=1, andp,q(x)= −1 for|x|p<1, (see [8]). By (3.13), we easily see that (cf. [6])
logΓp,q(x+1)=logp,q(x)+logΓp,q(x). (3.14) By the differentiation of both sides in (3.14), we have (cf. [6])
Γp,q (x+1)
Γp,q(x+1)=Γp,q (x)
Γp,q(x)+p,q(x)
p,q(x). (3.15)
By (3.15), we easily see that (cf. [6]) Γp,q (x) Γp,q(x)=
x−1
j=1
qj [j]
logq
q−1+Γp,q (1)
Γp,q(1). (3.16)
Define
Lp,q(x)=
x−1 j=0
p,q(j)
p,q(j). (3.17)
It is easy to check thatLp,q(1)=0. By (3.15), we also see that Γp,q (x)
Γp,q(x)=Lp,q(x)+Γp,q (1)
Γp,q(1), forx∈Zp, (3.18)
q-RIEMANN ZETA FUNCTION
whereLp,q(x)denotes the indefinite sum ofp,q(x)/p,q(x). By using (3.18) after sub- stitutingx=1, we obtainLp,q(1)=0. The classical Euler constant was known asγ=
−Γ(1)/Γ(1). In [8], Koblitz defined thep-adicq-Euler constantγp,q= −Γp,q (1)/Γp,q(1) (cf. [6,8]). By using (3.16) and the congruence of Andrews (cf. [3]), we obtain the follow- ing congruence:
q−1 logq
Γp,q (p) Γp,q(p)−γp,q
=
p−1 j=1
qj
[j]≡p−1 2 (q−1)
mod[p]
. (3.19)
Acknowledgments. The author expresses his gratitude to the referees for their valuable suggestions and comments. This work was supported by the Korea Research Foundation Grant (KRF-2002-050-C00001).
References
[1] I. Cherednik,Onq-analogues of Riemann’s zeta function, Selecta Math. (N.S.)7(2001), no. 4, 447–491.
[2] T. Kim,On explicit formulas ofp-adicq-L-functions, Kyushu J. Math.48(1994), no. 1, 73–86.
[3] ,On p-adicq-L-functions and sums of powers, Discrete Math.252(2002), no. 1–3, 179–187.
[4] ,q-Volkenborn integration, Russ. J. Math. Phys.9(2002), no. 3, 288–299.
[5] ,Non-archimedeanq-integrals associated with multiple Changheeq-Bernoulli polyno- mials, Russ. J. Math. Phys.10(2003), 91–98.
[6] T. Kim, L. C. Jang, K-H. Koh, and I.-S. Pyung,A note on analogue ofΓ-functions, Proceedings of the Conference on 5th Transcendental Number Theory, vol. 5, no. 1, Gakushuin University, Tokyo, 1997, pp. 37–44.
[7] T. Kim and S.-H. Rim,Generalized Carlitz’sq-Bernoulli numbers in thep-adic number field, Adv. Stud. Contemp. Math. (Pusan)2(2000), 9–19.
[8] N. Koblitz,q-extension of thep-adic gamma function, Trans. Amer. Math. Soc.260(1980), no. 2, 449–457.
[9] Q.-M. Luo, Z.-L. Wei, and F. Qi,Lower and upper bounds ofζ(3), Adv. Stud. Contemp. Math.
(Kyungshang)6(2003), no. 1, 47–51.
Taekyun Kim: Institute of Science Education, Kongju National University, Kongju 314-701, Korea
E-mail address:[email protected]
Mathematical Problems in Engineering
Special Issue on Space Dynamics
Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities. This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics. It is possible to make a division in two main categories: astronomy and astrodynamics. By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth. Many important topics of research nowadays are related to those subjects.
By astrodynamics, we mean topics related to spaceflight dynamics.
It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the grav- itational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects. Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts. Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines. The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research. All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
Before submission authors should carefully read over the journal’s Author Guidelines, which are located at http://www .hindawi.com/journals/mpe/guidelines.html. Prospective au- thors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sy- stem at http://mts.hindawi.com/ according to the following timetable:
Manuscript Due July 1, 2009 First Round of Reviews October 1, 2009 Publication Date January 1, 2010
Lead Guest Editor
Antonio F. Bertachini A. Prado, Instituto Nacional de Pesquisas Espaciais (INPE), São José dos Campos, 12227-010 São Paulo, Brazil; [email protected]
Guest Editors
Maria Cecilia Zanardi, São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;
[email protected]
Tadashi Yokoyama, Universidade Estadual Paulista (UNESP), Rio Claro, 13506-900 São Paulo, Brazil;
[email protected]
Silvia Maria Giuliatti Winter, São Paulo State University (UNESP), Guaratinguetá, 12516-410 São Paulo, Brazil;
[email protected]
Hindawi Publishing Corporation http://www.hindawi.com
