Download Irrationality of the Zeta Constants

Irrationality of the Zeta Constants
arXiv:1212.4082v5 [math.GM] 24 Apr 2017
N. A. Carella
Abstract: A general technique for proving the irrationality of the zeta constants
ζ(2n + 1), n ≥ 1, from the known irrationality of the beta constants L(2n + 1, χ) is
developed in this note. The irrationality of the zeta constants ζ(2n), n ≥ 1, and ζ(3)
are well known, but the irrationality results for the zeta constants ζ(2n + 1), n ≥ 2,
are new, and seem to show that these are irrational numbers. By symmetry, the
irrationality of the beta constants L(2n, χ) are derived from the known irrationality
of the zeta constants ζ(2n).
Mathematics Subject Classifications: 11J72, 11J82.
Keywords: Irrational numbers, Transcendental numbers, Beta constants, Zeta constants.
1
Introduction
As the rationality or irrationality nature of a number is an arithmetic property, it is
not surprising to encounter important constants, whose rationality or irrationality
is linked to the properties of the integers
and the distribution of the prime numbers,
Q
2
for example, the number 6/π = p≥2 (1 − p−2 ). An estimate of the partial sum of
the Dedekind zeta function of quadratic numbers fields will be utilized to develop
a general technique for proving the irrationality of the zeta constants ζ(2n + 1)
from the known irrationality of the beta constants L(2n + 1, χ), 1 6= n ∈ N. This
technique provides another proof of the first odd case ζ(3), which have well known
proofs of irrationalities, see [1], [2], [16], et al, and an original proof for the other
odd cases ζ(2n+ 1), n ≥ 2, which seems to confirm the irrationality of these number.
Theorem 1.1. For each fixed odd integer s = 2k + 1 ≥ 3, the zeta constant ζ(s) is
an irrational number.
The current research literature on the zeta constants ζ(2n + 1) states the following:
1. The special zeta value ζ(3) is an irrational number, see [1], [2], [6], [16], et al.
2. At least one of the four numbers ζ(5), ζ(7), ζ(9), and ζ(11) is an irrational
number, see [19, p. 7] and [21].
3. The sequence ζ(5), ζ(7), ζ(9), ζ(11), . . . contains infinitely many irrational zeta
constants, see [3]. Various advanced techniques for studying the zeta constants
are surveyed in [11], and [17].
Zeta Constants — 2
By the symmetry of the factorization of the Dedekind zeta function ζK (s) = ζ(s)L(s, χ)
with respect to either ζ(s) or L(s, χ), almost the same analysis leads to a derivation
of the irrationality of the beta constants L(2n, χ) from the known irrationality of
the zeta constants ζ(2n) n ≥ 1.
Theorem 1.2. For each fixed even integer s = 2n ≥ P
2, and the nonprincipal
−s
quadratic character χ mod 4, the beta constant L(s, χ) = ∞
is an irran=1 χ(n)n
tional number.
Section 2 contains basic materials on the theory of irrationality, the transcendental properties of real numbers, and an estimate of the summatory function of the
Dedekind zeta function. The proof of Theorem 1.1 is given in section 6. Last but
not least, a sketch of the proof of Theorem 1.2 is given in the last Section.
2
Fundamental Concepts and Background
The basic notation, concepts and results employed throughout this work are stated
in this Section. All the materials covered in this subsection are standard definitions
and results in the literature, confer [7], [9], [12], [16], [19], et al.
2.1
Criteria for Rationality and for Irrationality
A real number α ∈ R is called rational if α = a/b, where a, b ∈ Z are integers.
Otherwise, the number is irrational. The irrational numbers are further classified
as algebraic if α is the root of an irreducible polynomial f (x) ∈ Z[x] of degree
deg(f ) > 1, otherwise it is transcendental.
Lemma 2.1. (Criterion for rationality) If a real number α ∈ Q is a rational number,
then there exists a constant c = c(α) such that
p c (1)
≤ α− q q
holds for any rational fraction p/q 6= α. Specifically, c ≥ 1/B if α = A/B.
This is a statement about the lack of effective or good approximations of an arbitrary
rational number α ∈ Q by other rational numbers. On the other hand, irrational
numbers α ∈ R − Q have effective approximations by rational numbers.
If the complementary inequality |α − p/q| < c/q holds for infinitely many rational
approximations p/q, then it already shows that the real number α ∈ R is almost
irrational, so it is almost sufficient to prove the irrationality of real numbers.
Zeta Constants — 3
Lemma 2.2. (Criterion for irrationality) Let ψ(x) = o(1/x) be a monotonic decresing function, and let α ∈ Q be a real number. If
p
(2)
0 < α − < ψ(q)
q
holds for infinitely many rational fraction p/q ∈ Q, then α is irrational.
Proof. By Lemma 2.1 and the hypothesis, it follows that
c 1
p .
≤ α − < ψ(q) = o
q
q
q
But this is a contradiction since c/q 6= o(1/q).
(3)
More precise results for testing the irrationality of an arbitrary real number are
stated below.
Theorem 2.1. (Dirichlet) Suppose α ∈ R is an irrational number. Then there
exists an infinite sequence of rational numbers pn /qn satisfying
1
pn (4)
0 < α − < 2
qn
qn
for all integers n ∈ N.
For continued fractions α = [a0 , a1 , a2 , . . .] with sizable entries ai ≥ a > 1, where a
ia a constant, there is a slightly better inequality.
Theorem 2.2. Let α = [a0 , a1 , a2 , . . .] be the continued fraction of a real number,
and let {pn /qn : n ≥ 1} be the sequence of convergents. Then
p
1
n
0 < α − <
(5)
qn
an qn2
for all integers n ∈ N.
This is standard in the literature, the proof appears in [7, Theorem 171], [16, Corollary 3.7], and similar references. A more general result provides a family of inequalities for almost all real numbers.
Theorem 2.3. (Khinchin) Let ψ be a real and deceasing function, and let α ∈ R be
a real number. If there exists an infinite sequence of rational approximations pn /qn
such that pn /qn 6= α, and
α − pn < ψ(qn )
(6)
qn qn
P
and q ψ(q) < ∞, then real number α is ψ-approximatable.
Zeta Constants — 4
3
Estimate of An Arithmetic Function
Let q ≥ 2 be an integer, and let χ 6= 1 be the quadratic character
modulo q. The
√ Dedekind zeta function of a quadratic numbers field Q q is defined
K (s) =
P∞by ζ−s
ζ(s)L(s, χ). The factorization
consists of the zeta function ζ(s) = n=1 n , and
P
−s
the L-function L(s, χ) = ∞
n=1 χ(n)n , see [5, p. 219]. The product has the Dirichlet series expansion


! ∞
!
∞
∞
X
X
X
X
1
1
1

(7)
χ(n) s =
χ(d) s ,
ζ(s) · L(s, χ) =
s
n
n
n
n=1
n=1
n=1
d|n
and its summatory function is
P
n≤x
r(n) = 4
P
n≤x
P
d|n
χ(d), see [8, p. 17].
P
The finite sum rQ (n) = 4 d|n χ(d) is the number of respresentations of n by primitive forms Q(u, v) = au2 + buv + cv 2 of discriminant q. For q = 4, the
Pnonprincipal character χ 6= 1 has order 2, and the counting function r(n) = 4 d|n χ(d) =
# {(u, v) : n = u2 + v 2 } ≥ 0 tallies the number of representations of an integer n ≥ 1
as sums of two squares.
Lemma 3.1. The average order of the summatory function of the Dedekind function
is given by
X rQ (n)
c0
1
= ζ(s)L(s, χ) + s−1 + O
,
(8)
s
s−1/2
n
x
x
n≤x
where c0 > 0 is a constant, and x ≥ 1 is a large number.
A proof appears in [10, p. 369], slightly different version, based on the hyperbola
method, is computed in [15, p. 255].
4
The Irrationality of Some Constants
The different analytical techniques utilized to confirm the irrationality, transcendence, and irrationality measures of many constants are important in the development of other irrationality proofs. Some of these results will be used later on.
Theorem 4.1. The real numbers π, ζ(2), and ζ(3) are irrational numbers.
The various irrationality proofs of these numbers are widely available in the open
literature. These technique are valuable tools in the theory of irrational numbers,
refer to [1], [2], [6], [16], and others.
Theorem 4.2. For any fixed n ∈ N, and the nonprincipal character χ mod 4, the
followings statements are valid.
1. The real number ζ(2n) =
(−1)n+1 22n B2n 2n
π is a transcendental number,
(2n)!
Zeta Constants — 5
(−1)n E2n 2n+1
π
is a transcendental number,
22n+2 (2n)!
are the Bernoulli and Euler numbers respectively.
2. The real number L(2n + 1, χ) =
where B2n and E2n
Proof. Apply the Lindemann-Weierstrass theorem to the transcendental number
π.
The first few nonvanishing Bernoulli numbers and Euler numbers are as these.
1. B0 = 1, B1 =
−1
, B2
2
= 61 , B4 =
−1
, B6
30
=
1
,...,
42
2. E0 = 1, E2 = −1, E4 = 5, E6 = −161, . . . .
The generalization of these results to number fields is discussed in [20], and related
literature.
Theorem 4.3. (Klinger) Let K be a number field extension of degree k = [K : Q,
and discriminant D = disc(K). Then
π 2nk
1. If D > 0, the number field is totally real and ζK (2n) = rk √ , where n ≥ 1,
D
and rk ∈ Q.
2. If D < 0, the number field is totally complex and ζK (1−2n) = rk , where n ≥ 1,
and rk ∈ Q.
5
Irrationality of the Zeta Constants ζ(2n + 1)
For any integer 1 < s ∈ N, the zeta constant ζ(s) is a real number classified as a
period since it has a representation as an absolutely convergent integral of a rational
function:
Z
X 1
dxs
dx1 dx2
···
=
,
(9)
ζ(s) =
s
x1 x2
1 − xs
n
n≥1
1>x1 >x2 >···>xs
where s > 1. A few related integral representations are devised in [2] to prove the
irrationality ζ(2) and ζ(3). The general idea of a rational or nonrational integral
proof of the zeta constant ζ(s) for any integer s ≥ 2 is probably feasible.
6
The Main Result
A different technique using two independent infinite sequences of rational approximations of the two constants ζK (s), and 1/L(s, χ), which are linearly independent
over the rational numbers, will be used to construct an infinite sequence of rational
approximations for the zeta constant ζ(2n + 1), n ≥ 1. The properties of these sequences, such as sufficiently fast rates of convergence, are then used to derive the
irrationality of any zeta constant ζ(2n + 1), n ≥ 1.
Zeta Constants — 6
Proof. (Proof of Theorem 1.1.) Let χ 6= 1 be the quadratic character modulo q > 1,
and fix an odd integer s = 2k + 1 ≥ 3. By Lemma 3.1, the summatory function of
the Dedekind zeta function satisfies the expression
!−1
X r(n)
c0
1
1
(10)
= s−1 + O
−
ζ(s)
s
s−1/2
n
L(s,
χ)
x
x
n≤x
for every s > 1, and a constant c0 > 0. By Theorem 4.2, the real number number
L(2k + 1, χ) = aπ 2k+1 , where a ∈ Q, is a transcendental number, so there exists an
infinite sequence of rational approximations {pn /qn : n ∈ N } such that
1
p
n
< c1 ,
(11)
−
L(s, χ) qn an q 2
n
where [a0 , a1 , a2 , . . .] is the continued fraction of L(s, χ)−1 , and 0 < c1 < 1 is a
constant, see Theorem 2.1 or Theorem 2.2. The size of the constant c1 > 0 in
rational approximations is discussed in [19, p. 28]. Combining these data, taking
absolute value, and using the triangle inequality
||x − y|| ≥ ||x|| − ||y|| ,
(12)
lead to the followings.
c0
1
= ζ(s)
+
O
xs−1
xs−1/2 = ζ(s)
≥ ζ(s)
> ζ(s)
Rewrite it as
X r(n)
n≤x
ns
!−1
1 −
L(s, χ) pn pn 1
+
− −
s
n
L(s,
χ)
q
qn n
n≤x
!−1
X r(n)
pn pn 1
− − −
s
n
q
L(s,
χ)
qn
n
n≤x
!−1
X r(n)
c1
pn .
− −
s
n
qn an qn2
n≤x
X r(n)
!−1
X
r(n)
p
n
0 < ζ(s) −
s
qn n≤x n X
r(n)
c1 c0
1
+
<
+
O
.
2
s−1
s−1/2
qn
x
x
ns
n≤x
Now, taking x ≥ 1 to infinity yields
c2
pn X r(n) ,
0 < ζ(s) −
<
s
qn n≥1 n an qn2
(13)
(14)
(15)
Zeta Constants — 7
where 0 < c2 < 1 is a constant, and s > 2.
The rest of the proof is broken up into two cases.
P
−s
Case I. Assume that the constant ∞
∈ Q is a rational number, and
n=1 r(n)n
that the constant ζ(2k + 1) = A/B is a rational number. In this case Theorem 4.2
and (7) leads to a contradiction.
P
−s
∈ R − Q is an irrational number, and
Case II. Assume the constant ∞
n=1 r(n)n
that the constant ζ(2k + 1) = A/B is a rational number.
By Theorem 2.2, there exists an infinite sequence of rational approximations {um /vm :
m ∈ N} such that
X r(n) u c3
m
<
0<
,
(16)
−
s
2
bm vm
n
v
m
n≥1
P
where [b0 , b1 , , b2 , . . .] is the continued fraction of n≥1 r(n)n−s , and 0 < c3 < 1 is a
constant. This inequality is equivalent to
X r(n)
c3
um
c3
um
<
.
(17)
−
<
+
2
s
2
vm
bm vm
n
vm
bm vm
n≥1
Replacing this approximation into inequality (15) returns
p
u
n
m
≤ c4 p n + c2 ,
0 < ζ(s) −
2 q
qn vm bm vm
an qn2
n
where c4 pn < pn and 0 < c4 < 1 is a constant. By hypothesis, ζ(2k + 1) = A/B is a
rational number, thus, there exists a constant c5 ≥ 1/B such that
c5
pn um c4 p n
c2
≤
+
,
(18)
≤ ζ(s) −
2
qn vm
qn vm
bm qn vm an qn2
this follows from Lemma 2.1. Next, an infinite subsequence of rational approximations
pn u m
pnd umd
:d≥1 ⊂
: n, m ≥ 1
(19)
qnd vmd
qn vm
is generated by the following algorithm.
For each d ≥ 1, and ε > 0 is a small number.
1. If and > bmd , fix the convergent pnd /qnd and choose a convergent umd /vmd such
that
wd
c6 a1−ε
≤ vmd ≤ 2wd +1 ≤ c7 a1−ε
(20)
nd qnd ≤ 2
nd qnd ,
or
2. If and < bmd , fix the convergent pnd /qnd and and choose a convergent umd /vmd
such that
wd
c6 b1−ε
≤ qnd ≤ 2wd +1 ≤ c7 b1−ǫ
(21)
md vmd ≤ 2
md vmd ,
where 0 < c6 , c7 ≤ 2 are constants.
Zeta Constants — 8
Replacing the appropiate convergent (20) or (21) into (18) yields
c9 p n d
c2
c8
1
p
u
n
m
d
d
≤
+
≤ ζ(s) −
,
2(1−ǫ)
1−ǫ
2
and qnd
qnd vmd
qnd and bmd qn2
and qn2 d
(22)
d
or
c9 p n d
1
p
u
c2
c8
n
m
d
d
≤
≤ ζ(s) −
+
,
2(1−ǫ)
1−ǫ
2
2
2
bmd vmd
qnd vmd
qnd bmd vmd and bmd vm
d
where 0 < c8 , c9 , c10 < 1 are constants.
Since the terms
1
c8 p n d
c10
≤
,
2(1−ǫ)
qnd an2(1−ǫ)
qn2 d
and qn2 d
d
and
c10
1
c8 p n d
≤
,
2
2
qnd bmd vmd
bmd vm
d
(23)
(24)
(25)
for all large integers nd ≥ 1, see (11) and (16) respectively, the relation (22) or (23)
reduces to
c8
1
c9 nd
c2
≤
+
1−ǫ
2
2(1−ǫ)
and qnd
qnd and qn2
and qn2 d
d
c10
c2
,
(26)
≤ 2(1−ǫ)
+
and qn2 d
and qn2
d
or
c8
1−ǫ
2
bmd vm
d
1
c9 p n d
c2
+
2
2
qnd bmd vmd bmd vm
d
c2
c10
+ 2(1−ǫ)
.
≤
2
2
bmd vm
bmd vm
d
≤
(27)
d
Clearly, the inequality (26) or (27) leads to a contradiction for infinitely many arge
pairs of convergents pnd /qnd and umd /vmd as nd , md → ∞. Ergo, the constant
ζ(2k + 1) is not a rational number.
The infinite subsequence of rational approximations
pnd umd
−→ ζ(s)
as nd , md −→ ∞,
qnd vmd
(28)
is probably sparse depending on the magnitute of quotiens. However, iterative process in algorithm (20) shows that it is infinite, and it suffices to prove the irrationality
of ζ(2k + 1), k ≥ 1.
7
Irrationality of the Beta Constants L(2n, χ)
For q = 4 the quadratic symbol is defined by χ(n) = (−1)(n−1)/2 if n ∈ N is odd,
else χ(n) = 0. The corresponding Dedekind zeta function is given by
ζK (s) = ζ(s)L(s, χ) =
1 X r(n)
,
4 n≥1 ns
(29)
Zeta Constants — 9
P
where the counting function r(n) = 4 d|n χ(d) = # {(a, b) : n = a2 + b2 } ≥ 0 tallies
the number of representations of an integer n ≥ 1 as sums of two squares, and s ∈ C
is a complex
√ number. This is the zeta function of the Gaussian quadratic numbers
field Q −1 . The corresponding L-series is
L(s, χ) =
X χ(n)
n≥1
ns
=1−
1
1
1
+ s − s +··· .
s
3
5
7
(30)
The evaluation at s = 2 is known as Catalan constant
L(2, χ) = 1 −
1
1
1
+ 2 − 2 + · · · = .915965594177 . . . .
2
3
5
7
(31)
Proof. (Proof of Theorem 1.2.) Use the symmetry of the factorization of the Dedekind
zeta function ζK (s) = ζ(s)L(s, χ) with respect to the zeta function ζ(s) and the Lfunction L(s, χ) to arrive at the asymptotic formula
L(s, χ)
X r(n)
n≤x
ns
!−1
c0
1
= s−1 + O
−
ζ(s)
x
1
xs−1/2
(32)
compare this to (10). Now, proceed as before in the proof of Theorem 1.1 for the
verification of the irrationality of the zeta constant ζ(2k + 1), mutatis mutandis. 8
Products of Irrationals
A basic extension of the Dirichlet inequality in Theorem 2.2 privided here uses a
pair of irrational numbers and the corresponding parameters.
Lemma 8.1. Let α = [a0 , a1 , a2 , . . .] and β = [b0 , b1 , b2 , . . .] be the continued fractions
of two distinct irrational numbers α and β ∈ R such that αβ 6= ±1, respectively.
Then
1 pn
1 um
pn um +
,
(33)
≤
0 < αβ −
2
2 q
qn vm
an qn vm bm vm
n
where {pn /qn : n ≥ 1} and {um/vm : m ≥ 1} are the sequences of convergents
respectively.
Proof. By Theorem 2.2 or Theorem 2.3, there exists a sequence of convergents
{pn /qn : n ∈ N} such that
p
n
α − < 1 .
(34)
qn an qn2
Similarly, there exists a sequence of convergents {um /vm : m ∈ N} such that
u
m
β −
< 1 .
2
vm bm vm
(35)
Zeta Constants — 10
The product of the last two inequalities returns
um
pn
um
1
1
1
1
pn
< αβ <
.
−
−
+
+
2
2
qn an qn2
vm
bm vm
qn an qn2
vm
bm vm
This is rewritten as a standard inequality
1 um
1 pn
pn um ≤
+
.
0 < αβ −
2
2 q
qn vm
an qn vm bm vm
n
(36)
(37)
Often, the simpler version
p
u
n
m
≤ 2β + 2α
0 < αβ −
2
qn vm an qn2 bm vm
(38)
can be used to stream line the proof of a result such as Theorem 9.1, see (52) and
(52).
Theorem 8.1. (Euler) The continued fraction e = [2, 12, 1, 1, 4, 1, 1, 6, 1, 1, 8, . . .] of
the natural base has unbounded quotients and the subsequence of convergents pn /qn
satisfies the inequality
p
n
e − < 1 .
(39)
qn 2nqn2
The quotients have the precise form
a0 = 2,
a3n = a3n−2 = 1,
a3n−1 = 2n,
(40)
for n ≥ 1. The (3n + 1)th convergent have the recurrent formulas
pn = 2np3n−2 + p3n−5 ,
qn = 2nq3n−2 + q3n−5 .
(41)
The derivation appears in [13], [9, Theorem 2], [16, Theorem 3.10], [4], and other.
Futhermore, since
c
e − p >
,
(42)
2
q
q log q
for all p/q ∈ Q, and some constant c > 0, the irrationality measure of the number e
is µ(e) = 2, see [16, p. 51].
9
The Natural Base e And The Circle Number π
The natural base e and the circle number π are irrational numbers, but the arithmetic properties of the sum e + π and the product eπ seem to be unknonwn. The
known information on the continued fractions and the convergents of the two irrational numbers e and π are used here to construct an infinite subsequense of rational
approximations for the product eπ.
Zeta Constants — 11
Theorem 9.1. The product eπ is an irrational number.
Proof. Let e = [a0 , a1 , a2 , . . .] be the continued fraction of the irrational number
e, see Theorem 8.1. By Theorem 2.2, there exists a subsequence of convergents
{pn /qn : n ∈ N} such that
p
n
e − < 1 .
(43)
qn an qn2
Similarly, let π = [b0 , b1 , b2 , . . .] be the continued fraction of the irrationl number π,
and let {um /vm : m ∈ N} be the sequence of convergents such that
π − um < 1 .
(44)
2
vm bm vm
Now, suppose that the product eπ ∈ Q is a rational number. Then, there exists a
constant c > 0 such that
p
u
c
n
m
≤ eπ −
(45)
qn vm
qn vm 1 pn
1 um
+
,
<
2 q
an qn2 vm bm vm
n
this follows from Lemma 8.1 and Lemma 2.1. Next, an infinite subsequence of
rational approximations
pn u m
pn u m
: m, n ≥ 1 ⊂
: n, m ≥ 1
(46)
qn vm
qn vm
for which
pn u m
qn vm
−→
eπ
as n, m −→ ∞,
(47)
is generated by the following algorithm.
For each n ≥ 1, and a small number ε > 0. (The subscripts nk , mk are omitted to
simplifiy the notation).
1. If an = 2n > bm , fix the convergent pn /qn and choose a convergent um /vm
such that
2n1−ε qn ≤ 2wn ≤ vm ≤ 2wn +1 ≤ 2(2n)1−ε qn .
(48)
2. If an = 2n < bm , fix the convergent um /vm and choose a convergent pn /qn
such that
wn
b1−ε
≤ qn ≤ 2wn +1 ≤ 2b1−ε
(49)
m vm ≤ 2
m vm .
Various versions of this algorithm are possible, for example, by modifying the intervals in (48) and (49).
Zeta Constants — 12
To simplify the notation and the proof, take c = 1. Replacing the subsequence of
rational approximations constructed in (48) into (45) yields
p
u
1
n
m
eπ −
≤
(50)
2(2n)1−ε q 2
qn vm n
1 um
1
pn
+
2
2(1−ε)
2
2nqn vm
2(2n)
bm qn qn
2π
2e
≤
+
,
2
2(1−ε)
2nqn (2n)
bm qn2
≤
or replacing the subsequence of rational approximations constructed in (49) into
(45) yields
p
u
1
n
m
eπ −
≤
2b1−ε v 2
qn vm m
m
1 pn
um
+
2 q
vm bm vm
n
2π
2e
.
≤ 2(1−ε)
+
2
2
bm
vm bm vm
≤
1
2(1−ε) 2
bm
vm
(51)
The last lines in (50) and (51) use the fact that the terms
1 pn
2e
,
≤
2
2
bm vm qn
bm vm
and
(52)
1
2π
um
≤ 2(1−ε) ,
2
vm
bm
vm
for all large integers n, m ≥ 1, confer (43) and (44).
(53)
2(1−ε) 2
bm
vm
Clearly, both of these inequalities (50) and (51) are contradictions for infinitely many
large pairs of convergents pn /qn and um /vm as n, m → ∞. Ergo, the product eπ is
not a rational number.
10
Numerical Data
For each fixed pair of index (n, m), the basic product inequality
pn um 1 um
1 pn
0 < eπ −
<
+
2
qn vm
an q vm bm v 2 qn
n
(54)
m
is used to test each rational approximation. It is quite easy to find the pairs (n, m)
to construct a subsequence of rational approximations
pn u m
→ eπ
(55)
qn vm
as n, m → ∞. The parameters for two small but very accurate approximations are
listed in the Tables 1 and 2. These examples demostrate the practicality of the
algorithmic proof of Theorem (9.1).
Zeta Constants — 13
(n, m) =
(an , bm ) =
pn =
qn =
um =
vm =
p n um eπ − qn vm =
1 um
+ bm1v2 pqnn =
an q 2 vm
n
m
(19,10)
(12, 1)
13580623
4996032
5419351
1725033
0.00000000000012256862192
0.000000000000136378880
Table 1: A Rational Approximation Of The Product eπ.
(n, m) =
(an , bm ) =
pn =
qn =
um =
vm =
p n um eπ − qn vm =
1 um
2 v
an qn
m
+
1 pn
2 q
bm vm
n
(31,21)
(20, 2)
22526049624551
8286870547680
3587785776203
1142027682075
8.32849575322710174432272 × 10−25
= 1.04439176914510045201022 × 10−24
Table 2: A Rational Approximation Of The Product eπ.
Zeta Constants — 14
References
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Zeta Constants — 15
[18] Waldschmidt, Michel. Report on some recent advances in Diophantine approximation, 2011, http://www.math.jussieu.fr/∼miw/.
[19] Waldschmidt,
Michel.
An
Introduction
tionality
and
Transcendence
Methods,
www.math.jussieu.fr/∼miw/articles/pdf/AWSLecture1.pdf.
to
Lecture
Irra1.,
[20] Zagier, Don Hyperbolic manifolds and special values of Dedekind zetafunctions. Invent. Math. 83 (1986), no. 2, 285-301.
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