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The Spiral of Theodorus and Sums of Zeta-values at the Half-integers

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The total angular distance traversed by the spiral of Theodorus is governed by the Schneckenkonstante K introduced by Hlawka. The only published estimate of K is the bound K ≤ 0.75. We express K as a sum of Riemann zeta-values at the half-integers and compute it to 100 decimal places. We find similar formulas involving the Hurwitz zeta-function for the analytic Theodorus spiral and the Theodorus constant introduced by Davis.
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The spiral of Theodorus and sums of zeta-values at the half-integers
David Brink
July 2012
Abstract. The total angular distance traversed by the spiral of Theodorus
is governed by the Schneckenkonstante K introduced by Hlawka. The only
published estimate of K is the bound K 0.75. We express K as a sum
of Riemann zeta-values at the half-integers and compute it to 100 deci-
mal places. We find similar formulas involving the Hurwitz zeta-function for
the analytic Theodorus spiral and the Theodorus constant introduced by Davis.
1 Introduction
Theodorus of Cyrene (ca. 460–399 B.C.) taught Plato mathematics and was himself a pupil of
Protagoras. Plato’s dialogue Theaetetus tells that Theodorus was distinguished in the subjects of
the quadrivium and also contains the following intriguing passage on irrational square-roots, quoted
here from [12]:
[Theodorus] was proving to us a certain thing about square roots, I mean of three square feet and
of five square feet, namely that these roots are not commensurable in length with the foot-length,
and he went on in this way, taking all the separate cases up to the root of 17 square feet, at
which point, for some reason, he stopped.
It was discussed already in antiquity why Theodorus stopped at seventeen and what his method of
proof was. There are at least four fundamentally different theories—not including the suggestion of
Hardy and Wright that Theodorus simply became tired!—cf. [11, 12, 16].
One of these theories is due to the German amateur mathematician J. Anderhub, cf. [4, 14]. It
involves the so-called square-root spiral of Theodorus or Quadratwurzelschnecke. This spiral consists
of a sequence of points P
1
, P
2
, P
3
, . . . in the plane circulating anti-clockwise around a centre P
0
such
that |P
0
P
n
| =
n and |P
n
P
n+1
| = 1 for all n 1 (see Figure 1). Let θ
n
be the angle P
n
P
0
P
n+1
.
Then
θ
n
= arctan
1
n
1
since P
0
P
n
P
n+1
is a right angle. Further, let Θ
n
be the total angular distance traversed by the
spiral in n 1 steps, i.e.,
Θ
n
:=
n1
X
x=1
θ
x
. (1)
Then P
1
P
0
P
n
equals Θ
n
modulo 2π. The spiral of Theodorus can thus alternatively be defined in
the complex plane by P
0
= 0 and P
n
=
n · exp(Θ
n
i). It was Anderhub’s discovery that n = 17 is
the last value of n such that the spiral does not overlap, i.e., such that Θ
n
< 2π.
Figure 1.
2 An asymptotic formula and the Schneckenkonstante
Hlawka [14, eq. (13)] gives a formula for Θ
n
of the form Θ
n
= 2
n+K +(terms of lower order) with
a constant K which he terms Schneckenkonstante. However, the last coefficient in Hlawka’s formula,
which is also quoted in [4], seems to be incorrect. Hlawka moreover gives the bound K 0.75 which,
to the author’s best knowledge, is the only published estimate of K. Better estimates appear in
several unpublished manuscripts, see [5, 10] and the references in [7].
Theorem 1. The angular distance traversed by the spiral of Theodorus satisfies the asymptotic
formula
Θ
n
= 2
n + K +
1
6
n
1
120n
n
1
840n
2
n
+
5
8064n
3
n
+
1
4224n
4
n
+ O
1
n
5
n
, (2)
where K, Hlawka’s Schneckenkonstante, is given by
K =
X
k=0
(1)
k
ζ(k +
1
2
)
2k + 1
=
π
4
+
X
k=0
(1)
k
ζ(k +
1
2
) 1
2k + 1
(3)
2
or numerically
K = 2.1577829966 5944622092 9142786829 5777235041 3959860756
2455154895 5508588696 4679660648 1496694298 9463960898 . . .
Proof. The series
arctan
1
x
=
X
k=0
(1)
k
(2k + 1)x
k+1/2
(4)
converges for x 1. Hence, for any N 1, one has
Θ
n
=
n1
X
x=1
arctan
1
x
=
n1
X
x=1
X
k=0
(1)
k
(2k + 1)x
k+1/2
=
X
k=0
(1)
k
2k + 1
n1
X
x=1
1
x
k+1/2
=
N1
X
k=0
(1)
k
2k + 1
n1
X
x=1
1
x
k+1/2
+
X
k=N
(1)
k
ζ(k +
1
2
)
2k + 1
+ O
1
n
N1/2
(5)
for n , where ζ is Riemann’s zeta-function. For any complex exponent s and any positive
integers m and n, Euler’s summation formula [8, p. 469] gives
n1
X
x=1
x
s
=
Z
n
1
x
s
dx
1
2
(n
s
1) +
m
X
t=2
B
t
t!
s
(t1)
(n
s+1t
1)
+ (1)
m+1
Z
n
1
B
m
({x})
m!
s
(m)
x
sm
dx. (6)
Here, B
t
and B
m
(x) are Bernoulli numbers and polynomials, {x} is the fractional part of x, and s
(t)
is the falling factorial s(s 1) ···(s t + 1). It is only necessary to sum over the even values of t
since B
t
= 0 for odd t > 1. For s 6= 1 and m > <(s) + 1, (6) can be rewritten as
n1
X
x=1
x
s
= C(s) +
1
s + 1
n
s+1
1
2
n
s
+
m
X
t=2
B
t
t!
s
(t1)
n
s+1t
+ O(n
sm
) (7)
for n , where all terms independent of n have been collected in the constant
C(s) =
1
s + 1
+
1
2
m
X
t=2
B
t
t!
s
(t1)
+ (1)
m+1
Z
1
B
m
({x})
m!
s
(m)
x
sm
dx. (8)
It follows from (7) that C(s) is independent of m, and also that
C(s) =
X
x=1
x
s
= ζ(s) for <(s) < 1.
It follows from (8) by Leibniz’s integral rule that C(s) is an analytic function of s. Consequently,
C(s) and ζ(s) agree for all complex s 6= 1 by analytic continuation. Thus, for example,
n1
X
x=1
1
x
= 2
n+ ζ(
1
2
)
1
2
n
1
24n
n
+
1
384n
2
n
1
1024n
3
n
+
143
163840n
4
n
+O
1
n
5
n
, (9)
3
and similarly for s =
3
2
,
5
2
, etc. Inserting these formulas into (5) with N = 4 gives (2) and the
first equality of (3); the second equality follows from Leibniz’s formula
π
4
=
1
1
1
3
+
1
5
···
For computational purposes, the second series in (3) is much superior to the first since ζ(x)1 2
x
for real x . Thus, the first 322 terms give the 100 decimal places stated.
Figure 2 shows the spiral of Theodorus together with the curve with polar coordinates r(t) = t and
ϕ(t) = 2t + K +
1
6
t
1
, t > 0, and the points on that curve corresponding to t =
1,
2,
3, . . . As
it appears, the first three terms of (2) approximate Θ
n
very well.
Figure 2.
Comparing (2) with the equivalent expression
Θ
n+1
= 2
n + K +
7
6
n
41
120n
n
+
167
840n
2
n
1147
8064n
3
n
+
1411
12672n
4
n
+ O
1
n
5
n
(10)
suggests that, for reasons unknown to the author, (1) is the “right” definition of Θ
n
rather than
P
n
x=1
θ
x
.
It is worth noting that letting s be a nonnegative integer in (6) and adding n
s
to each side gives
n
X
x=1
x
s
=
1
s + 1
n
s+1
+
1
2
n
s
+
s+1
X
t=2
B
t
t!
s
(t1)
n
s+1t
+ ζ(s).
4
These are the power sum polynomials known already to Jakob Bernoulli (who was, incidentally, the
inventor of another spiral, the spira mirabilis). A slightly modified application of Euler’s summation
formula shows that these polynomials have no constant terms. Hence, comparison with the constant
term of the above formula gives the zeta-values at the nonpositive integers:
ζ(0) =
1
2
and ζ(s) =
B
s+1
s + 1
for s > 0.
We mention some alternative approaches to the numerical computation of K. Using Euler’s
summation formula on θ
x
directly gives
K = 1
3π
8
Z
1
{x}
1
2
2
x(x + 1)
dx, (11)
but this integral representation hardly gives K to more than a few places. Another way to calculate
K is to isolate it in (2) as
K Θ
n
2
n
1
6
n
+ ··· (12)
Taking, say, n = 10
4
and including 400 terms on the right gives K to more than 1000 places. Finally,
subtracting (9) from (2) gives
K = ζ
1
2
+
X
x=1
arctan
1
x
1
x
. (13)
Although this sum converges rather slowly compared to the second sum of (3), it can be computed
very efficiently using convergence acceleration techniques such as Levin’s u-transform [15].
3 Uniform distribution
Figure 2 above suggests that the sequence
n
) is uniformly distributed modulo 2π. Using Θ
n
=
2
n+K +o(1) and a theorem of Fej´er, Hlawka [13, 14] gives a short proof of this fact (originally due
to W. Neiss). We note that there is an even shorter proof using a generalization of Fej´er’s theorem
due to van der Corput [3]. This result states that a real sequence (x
n
) is uniformly distributed to
any modulus if the sequence of differences x
n
:= x
n+1
x
n
satisfies x
n
0 monotonically and
n ·x
n
±∞. In our case, ∆Θ
n
= arctan n
1/2
n
1/2
clearly satisfies the conditions of van der
Corput’s theorem.
4 The analytic Theodorus spiral and the Theodorus constant
Davis [4] defines an analytic Theodorus spiral with polar coordinates
r(t) =
t , ϕ(t) =
X
x=1
arctan
1
x
arctan
1
x + t 1
(14)
5
for real t > 0. If n is a positive integer, then clearly ϕ(n) =
P
n1
x=1
arctan x
1/2
= Θ
n
. Gronau [9]
shows a uniqueness result for ϕ(t) similar to the Bohr-Mollerup theorem on the Gamma-function.
The angular velocity is obtained by term-wise differentiation of (14) as ϕ
0
(t) =
1
2
U(t) with
U(t) =
X
x=1
1
(x + t)
x + t 1
.
Davis defines the Theodorus constant as the attractive sum
T = U (1) =
X
x=1
1
(x + 1)
x
= 1.8600250792 . . .
and computes it to 10 places. In an interesting analysis, Gautschi [4, 7] shows how T can be
computed to higher precision using Gaussian quadrature.
Theorem 2. The angular coordinate of Davis’s analytic Theodorus spiral satisfies
ϕ(t) = K arctan
1
t
X
k=0
(1)
k
ζ(k +
1
2
, t + 1)
2k + 1
(15)
for real t > 0, where ζ(s, q) is the Hurwitz zeta-function, and K is the Schneckenkonstante. The
Theodorus constant is given by the quickly converging series
T =
1
2
+
X
k=1
(1)
k+1
ζ(k +
1
2
) 1
. (16)
Proof. For real q > 0 and complex s 6= 1, the Hurwitz zeta-function ζ(s, q) is defined by
ζ(s, q) =
X
x=0
1
(q + x)
s
for <(s) > 1 and for all s 6= 1 by analytic continuation. Thus, ζ(s, 1) equals the usual Riemann
zeta-function ζ(s). Similarly, ζ(s, 2) equals ζ(s) 1, and
X
x=2
1
x
s
1
(x + t 1)
s
= ζ(s, 2) ζ(s, t + 1) (17)
for all s 6= 1, again by analytic continuation. Recall that (4) converges for x 1. Now,
ϕ(t) =
X
x=1
arctan
1
x
arctan
1
x + t 1
=
π
4
arctan
1
t
+
X
x=2
X
k=0
(1)
k
2k + 1
1
x
k+1/2
1
(x + t 1)
k+1/2
=
π
4
arctan
1
t
+
X
k=0
(1)
k
2k + 1
X
x=2
1
x
k+1/2
1
(x + t 1)
k+1/2
=
π
4
arctan
1
t
+
X
k=0
(1)
k
2k + 1
ζ(k +
1
2
, 2) ζ(k +
1
2
, t + 1)
= K arctan
1
t
X
k=0
(1)
k
2k + 1
ζ(k +
1
2
, t + 1),
6
where we have used (3), (17), the identity ζ(s, 2) = ζ(s) 1, and the (easily seen) fact that the
double sum in the second line converges absolutely.
The series
1
(x + 1)
x
=
X
k=1
(1)
k+1
x
k+1/2
converges for x > 1. As before we get
U(t) =
X
x=1
1
(x + t)
x + t 1
=
1
(t + 1)
t
+
X
x=2
X
k=1
(1)
k+1
(x + t 1)
k+1/2
=
1
(t + 1)
t
+
X
k=1
(1)
k+1
X
x=2
1
(x + t 1)
k+1/2
=
1
(t + 1)
t
+
X
k=1
(1)
k+1
ζ(k +
1
2
, t + 1)
for t > 0. Letting t = 1 gives T = U(1).
5 Sums of zeta-values at the integers and half-integers
There is a vast literature on sums of zeta-values at the integers, cf. [1, 2]. To give some elementary
examples, one has
X
k=2
{ζ(k) 1} = 1,
X
k=2
(1)
k
{ζ(k) 1} =
1
2
,
X
k=2
ζ(k) 1
k
= 1 γ,
X
k=2
(1)
k
ζ(k) 1
k
= γ 1 + log 2,
where γ is Euler’s constant. Other classical constants such as π, Mertens’s constant, Hardy-
Littlewood’s twin prime constant, Khinchin’s constant, and Landau-Ramanujan’s constant are ex-
pressed as sums or products of zeta-values at the integers in [6].
In contrast, the author has been unable to find a closed-form expression for any sum of zeta-
values at the half-integers such as (3) and (16). To conclude, we mention without proof
X
x=2
1
(x 1)
x
=
X
k=1
ζ
k +
1
2
1
= 2.1840094702 . . . (18)
and
n
X
x=2
artanh
1
x
= 2
n + K
0
+ O
1
n
(19)
with
K
0
=
X
k=0
ζ
k +
1
2
1
2k + 1
= 1.8265078108 . . . (20)
7
References
[1] V. S. Adamchik, H. M. Srivastava, Some series of the zeta and related functions, Analysis
(Munich) 18 (1998) 131–144.
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function, J. Comput. Appl. Math. 121 (2000) 247–296.
[3] J. G. van der Corput, Diophantische Ungleichungen. I. Zur Gleichverteilung modulo Eins, Acta.
Math. 56 (1931) 373–456.
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A K Peters, Wellesley, MA, 1993.
[5] S. Finch, Constant of Theodorus (2005), available at http://algo.inria.fr/csolve/th.pdf.
[6] P. Flajolet, I. Vardi, Zeta function expansions of classical constants (1996), available at
http://algo.inria.fr/flajolet/Publications/.
[7] W. Gautschi, The spiral of Theodorus, numerical analysis, and special functions, J. Comput.
Appl. Math. 235 (2010) 1042–1052.
[8] R. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, second edition. Addison Wesley,
Reading, MA, 1994.
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[10] H. K. Hahn, K. Schoenberger, The ordered distribution of natural numbers on the square root
spiral (2007), available at http://arxiv.org/abs/0712.2184.
[11] G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, fourth edition. Claren-
don Press, Oxford, 1960.
[12] T. Heath, A History of Greek Mathematics, Vol. I. Clarendon Press, Oxford, 1921.
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The Theory of Uniform Distribution. AB Academic Publishers, Berkhamsted, 1984.
[14] ———, Gleichverteilung und Quadratwurzelschnecke, Monatsh. Math. 89 (1980) 19–44.
[15] D. Levin, Development of non-linear transformations of improving convergence of sequences,
Internat. J. Comput. Math. 3 (1973) 371–388.
[16] R. L. McCabe, Theodorus’ irrationality proofs, Math. Mag. 49 (1976) 201–203.
8
... making the most of a small number of given integer and half-integer zetavalues, ζ(n) and ζ(n + 1/2). The constant K is known as the Schneckenkonstante and arises in connection with the Spiral of Theodorus [6]. ...
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