Limit Representations of Riemann's Zeta Function

0.11 MB·English

by  Djurdje Cvijovic

Limit Representations of Riemann’s Zeta
3 Function
1
0
2 Djurdje Cvijovi´c and Hari M. Srivastava

n
a
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Abstract

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1 InthispaperitisshownthatRiemann’szetafunctionζ(s)admitstwo
limit representations when ℜ(s) > 1. Each of these limit representations

] isdeducedbyusingsimpleargumentsbasedupontheclassical Tannery’s
A
(limiting) theorem for series.
C
.
h
1 Introduction.
t
a
m
Riemann’szetafunctionζ(s)isacomplex-valuedfunctionofacomplexvariable
[
s and is holomorphic everywhere in the complex s-plane except at the point

1 s = 1 where a first-order pole exists with residue equal to 1. It is, as usual,
v defined as the analytic continuation of the function given by the sum of the
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following series:
5
6
3
∞ ∞
1 1 1
.
1 = ℜ(s)>1
ns 1−2−s (2n−1)s
0  n=1 n=1
13 ζ(s):= X1 ∞ (−1)nX−1 (cid:0)ℜ(s)>0(cid:1); s6=1 . (1)
: 1−21−s ns
v n=1
Xi Moreover,anumberofotXherinfiniteseries,infinitepro(cid:0)ducts,improper(cid:1)integrals,
r complex contour integrals and closed-form expressions based upon the Euler-
a
Maclaurin summation formula may be used to represent ζ(s) in certain regions
of the complex s-plane (see, for details, [3] and [9]). Some illustrative examples
are given below.
Euler’s product formula for the zeta function:
∞
1
ζ(s)= ℜ(s)>1 , (2)
1−p−s
m=1 m
Y (cid:0) (cid:1)
where p is mth prime number;
m
1 ∞ xs−1
ζ(s)= dx ℜ(s)>1 ; (3)
Γ(s) ex−1
Z0
(cid:0) (cid:1)
1
∞
1 γ
ζ(s)= + (−1)n n (s−1)n (s6=1), (4)
s−1 n!
n=0
X
where
m (logk)n (logm)n+1
γ = lim − n∈N:={1,2,3,···} ; (5)
n
m→∞ k n+1 !
k=1
X (cid:0) (cid:1)
n 1 n1−s ∞ x−⌊x⌋
ζ(s)= + −s dx ℜ(s)>0;n∈N , (6)
ms s−1 xs+1
m=1 Zn
X (cid:0) (cid:1)
where ⌊x⌋ stands for the floor function which gives the largestinteger less than
or equal to x∈R;
Γ(1−s) (0+) zs−1
ζ(s)= dz (s∈C\N), (7)
2πı e−z−1
Z−∞
wherethecontourofintegrationisalooparoundthenegativerealaxis;itstarts
at −∞, encircles the origin once in the positive (counter-clockwise) direction
without enclosing any of the points
z =±2πı,±4πı,··· ,
and returns to −∞.
All of the above and many other representations of ζ(s) are known for a
considerable time. For an exhaustive list of such and other representations of
ζ(s), the interested reader is referred (for instance) to [3] and [9]. This is not
surprising since there is a long and rich history of research on Riemann’s zeta
functionζ(s)thatgoesbacktoEulerin1735(see,fordetails,[4]and[10]). What
is surprising, however,is that it has not been noticed hitherto that ζ(s) admits
two limit representations which are asserted here by the following theorem.
Theorem. Suppose that s is a complex number and let m, n, p and q be
nonnegative integers. Then, for ℜ(s)>1, the values of Riemann’s zeta function
ζ(s) are given by
s⌊(2q+n−1)/2⌋
π pπ
(a) ζ(s)= lim cots (8)
q→∞ 2q+m 2q+n
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
and
s⌊(2q+n−1)/2⌋
π pπ
(b) ζ(s)= lim cscs (9)
q→∞ 2q+m 2q+n
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
(p∈N,m∈N :=N∪{0}; n=0 and q ∈N\{1}; n∈N and q ∈N).
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Remark 1. Severalspecialcasesofthelimitrelationships(8)and(9)involving
ζ(2n) and ζ(2n+1) when n ∈ N can be found in the literature. For example,
by making use of elementary arguments and complex function theory, respec-
tively,thefollowingtwolimitformulaswereestablishedbyWilliams[12,p. 273,
Lemma 1; p. 275, Lemma 2]:
π 2n q pπ
ζ(2n)= lim cot2n (10)
q→∞ 2q 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
and
2n q
π pπ
ζ(2n)= lim cot2n . (11)
q→∞ 2q+1 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
On the other hand, an elementary proof for the following limit formula was
given by Apostol [2, p. 430, Eq. (16)]:
2n+1 q
π pπ
ζ(2n+1)= lim cot2n+1 , (12)
q→∞ 2q 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
who also found an asymptotic expansion of the finite sum in (10) [2, p. 428,
Eq. (7)], which readily leads to Euler’s celebrated relation:
(2π)2n
ζ(2n)=(−1)n+1 B (n∈N ) (13)
2n 0
2·(2n)!
betweentheeven-indexedBernoullinumbersB andthevaluesoftheζ(2n).It
2n
shouldbe notedthat(12)canbe provenbyfollowingWilliams’argumentsused
in the case of (10) with necessary changes. Cvijovi´c et al. [6] resorted to the
calculusofresiduesinorderto derivethe cotangentfinite sumin(10)andsome
other related sums in a closed form. Furthermore, as immediate consequences
of their results, Williams’ limit formulas (10) and (11) as well as the following
three related limit formulas for ζ(2n) were obtained by Cvijovi´c et al. [6, p.
206, Theorem 2]:
2nq−1
π pπ
ζ(2n)= lim cot2n , (14)
q→∞ 2q 2q
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
2nq−1
π pπ
ζ(2n)= lim csc2n (15)
q→∞ 2q 2q
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
and
2n q
π pπ
ζ(2n)= lim csc2n . (16)
q→∞ 2q 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
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2 Demonstration of the Theorem.
After numerous unsuccessful attempts to generalize the limit formulas (10) to
(12)and(14)to (16),wehaveencounteredanoldandalmostelementaryresult
of the classical analysis, known as Tannery’s (limiting) theorem for series (see
[11,p. 292],[5,pp,123and124],[8,pp. 371and372]and[7,pp. 199and200]),
which indeed providesa simple and direct proofof our Theorem. We firststate
Tannery’stheoremherewithoutproof,notingthatitsstandardapplicationisto
show that the following two usual definitions of ex are the same:
x n n n x k ∞ xk
lim 1+ = lim = =:ex. (17)
n→∞ n n→∞ k n k!
(cid:16) (cid:17) kX=0(cid:18) (cid:19)(cid:16) (cid:17) kX=0
Tannery’s Theorem (cf. [8, pp. 371 and 372]). For a given double sequence
{fm(n)}m,n∈N0,suppose that each of the following two conditions is satisfied for
any fixed m∈N :
0
(i) lim f (n)=f ;
m m
n→∞
(ii) |f (n)| ≦ M , where M > 0 is independent of n and the infinite
m m m
series:
∞
M
m
m=0
X
is convergent.
Then the following limit relationship holds true:
α(n) ∞
lim f (n)= f , (18)
m m
n→∞
m=0 m=0
X X
where {α(n)}n∈N0 is a monotonically increasing integer-valued sequence which
tends to infinity as n→∞.
In what follows, it is assumed that m, n, p and q are nonnegative integers
and we also suppose, for a moment, that s is a real number.
To prove Part (a) when s > 1, we consider the following double sequence
which appears in (8):
s
π pπ
Φ (q|m,n,s):= cots , (19)
p
2q+m 2q+n
(cid:18) (cid:19) (cid:18) (cid:19)
where p and q are indices and m, n and s are parameters with
2q+n−1
p=1,··· , (20)
2
j k
(p∈N,m∈N ; n=0 and q ∈N\{1}; n∈N and q ∈N).
0
We now show that the double sequence defined by (19) satisfies the conditions
(i) and (ii) of Tannery’s theorem.
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Condition (i) of Tannery’s Theorem. For a fixed p and fixed m, n and s,
in view of the well-known facts that
pπ 2q+n
tanx∼x (x→0), →0 and →1 (q →∞),
2q+n 2q+m
it follows without difficulty that
s s
π pπ π 2q+n 1
lim cot = lim = . (21)
q→∞ 2q+m 2q+n q→∞ 2q+m pπ ps
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)(cid:18) (cid:19)(cid:21)
Condition(ii)ofTamnnery’sTheorem. Inthiscase,werecallthefollowing
elementary inequality [1, p. 75, Entry 4.3.80]:
π
sinx<x<tanx 0<x< ,
2
(cid:16) (cid:17)
which yields
1 π
0<cotx< 0<x< ,
x 2
(cid:16) (cid:17)
so that, since [cf. Equation (20)]
pπ π
0< < ,
2q+n 2
we have
pπ 2q+n
0<cot < ,
2q+n pπ
(cid:18) (cid:19)
which, upon noticing that π/(2q+m)>0, becomes
π pπ 2q+n 1
0< cot < · . (22)
2q+m 2q+n 2q+m p
(cid:18) (cid:19)
Moreover,from the graph of the function defined by
2x+n
f(x):= ,
2x+m
it is easily concluded that
1 (n≦m)
2q+n
≦C := (m,n∈N ; q ∈N). (23)
m,n  0
2q+m
1+n (n>m)
1+m
Thus,bymakinguseoftheselasttwoequations(22)and(23),wefindfors>0
that
s s
π pπ π pπ 1
cots = cot ≦Cs . (24)
2q+m 2q+n 2q+m 2q+n m,nps
(cid:12)(cid:18) (cid:19) (cid:18) (cid:19)(cid:12) (cid:20) (cid:18) (cid:19)(cid:21)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
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Clearly, therefore,the condition (ii) of Tannery’s theorem is fulfilled only when
s>1, because the infinite series
∞
1
ps
p=1
X
is then convergent.
In conclusion, we may apply Tannery’s theorem to the double sequence
Φ (q|m,n,s) (s>1),
p
given by the equations (19) and (20), since the needed conditions are satisfied
and ⌊(2q+n−1)/2⌋ is evidently an increasing integer-valued function which
tends to infinity as q → ∞. Thus, for s > 1, the desired limit formula in (8)
follows in view of the limit relationship:
⌊(2q+n−1)/2⌋ s ∞
π pπ 1
lim cots = =ζ(s) (s>1). (25)
q→∞ 2q+m 2q+n ps
p=1 (cid:18) (cid:19) (cid:18) (cid:19) p=1
X X
To prove Part (b) when s>1, we consider the double sequence defined by
s
π pπ
Ψ (q|m,n,s)= cscs , (26)
p
2q+m 2q+n
(cid:18) (cid:19) (cid:18) (cid:19)
together with the restrictions on the integers p, q, m and n given in (20) and
proceed along the same lines as in the proof of Theorem (a). In the process, it
is necessary to employ the following well-known asymptotic relation:
sinx∼x (x→0)
as well as the inequality [1, p. 75, Entry 4.3.79]:
π π
0<cscx< 0<x< .
2x 2
(cid:16) (cid:17)
In this way, by Tannery’s theorem for series, we obtain the limit relationship:
⌊(2q+n−1)/2⌋ s ∞
π pπ 1
lim cscs = =ζ(s) (s>1), (27)
q→∞ 2q+m 2q+n ps
p=1 (cid:18) (cid:19) (cid:18) (cid:19) p=1
X X
which obviously implies the limit formula (9) which holds true for s>1.
Remark 2. Observethat the limit formulas (25)and (27)arededuced hereon
the suppositionthats isa realnumber. However,these limit formulasarevalid
in the entire half-plane ℜ(s) > 1, since they may be extended by applying the
principle of analytic continuation on s as far as possible.
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3 Concluding Remarks and Observations.
By suitably applying the Theorem, the above-presented limit formulas in (10)
to (12) and (14) to (16) could be generalizedso as to be valid for ℜ(s)>1. We
thus have
s q
π pπ
ζ(s)= lim cots (ℜ(s)>1), (28)
q→∞ 2q 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
s q
π pπ
ζ(s)= lim cots (ℜ(s)>1), (29)
q→∞ 2q+1 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
sq−1
π pπ
ζ(s)= lim cots (ℜ(s)>1), (30)
q→∞ 2q 2q
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
π s q pπ
ζ(s)= lim cscs (ℜ(s)>1) (31)
q→∞ 2q 2q+1
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
and
sq−1
π pπ
ζ(s)= lim cscs (ℜ(s)>1). (32)
q→∞ 2q 2q
(cid:18) (cid:19) p=1 (cid:18) (cid:19)
X
Remark 3. We remark that many elementary and special functions possess
limitrepresentationsandtheratherwell-knownonesarethoseoftheexponential
function in (17) and Euler’s limit formula for the gamma function [1, p. 255,
Entry 6.1.2]:
n!nz
Γ(z)= lim (33)
n→∞z(z+1)(z+2)···(z+n)
z ∈C\Z− (Z− :={0,−1,−2,−3,···}) .
0 0
(cid:0) (cid:1)
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k=1
P
Djurdje U. Cvijovi´c graduated from the University of Belgrade, Serbia, and
received his Ph.D. from the University of Cambridge, England, in 1994. His
main research interests include global optimization, theory of special functions
and elementary number theory.
AtomicPhysics Laboratory, Vinˇca Instituteof Nuclear Sciences, P. O.Box 522,
Belgrade, Republic of Serbia
[e-mail]
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