NOTES
327
Acknowledgement
Thanks are due to the anonymous referee for very helpful advice.
References
1.
E. C. Titchmarsh, The theory off unctions (corrected 2nd edn.) Oxford
University Press (1958), misc. ex. 16. pp. 135-136.
2.
M. R. Spiegel et aI, Complex variables (2nd edn.), Schaum's Outline
Series, McGraw-Hill (2009).
J. A. SCOTT
1 Shiptons Lane, Great Somerford, Chippenham SN15 5EJ
98.13 Some double series related to '(3)
Recall that for integers k
~ (k)
;ll
1
Lco
=
n:
Ink
2, ~ (k) is defined by
1
1
2
3
= 1 + -k + -k +
It is very well known that the value of ~ (2) is n2 / 6, and for all even k, ~ (k)
can be expressed in terms of the Bernoulli numbers. However, no closed
expressions are known for odd k, and in particular for ~ (3), though of
course it can be calculated numerically to any desired degree of accuracy:
~ (3) = 1.202057 to six decimal places. It was shown by Apery [1] that
~ (3) is irrational, and it is sometimes known as Apery's constant. Apery's
proof is quite difficult; a simpler one was given by Beukers [2].
Here we describe a number of double series whose sums can be
expressed in terms of ~ (3), demonstrating that it really is of some interest.
First, let
ce
1
ce
LL--'
m=ln~lmn(m+n)
~
and
S)
=
1
ee
L L ----.
mn max(m,
m=1
n~l
n)
We start with Sl. Substituting m + n = r, and then reversing the order of
summation, we can rewrite it in two ways as follows:
1
1
1 r-l 1
Sl =
"2 =
(1)
m~lmr=m+lr
r_2r m~lm
ee
ee
L- L
co
L"2 L -.
Also, by writing Sl with m and n interchanged
expression, we have
and adding the original
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328
TIIE MATHEMATICAL
- 1 (1
I- I--+m
1)
m=ln=l(m+n)2
-
-
II-m=1
mn(m
GAZETIE
n
1
+
n~1
n)
(2)
S2.
Now
1
mn(m
1
+
n)
12 (1
-----
m
m2 n(m
+
n)
m
n
1)
m+n
and by cancellation it is easily seen that
1) I- (1--n
m+n
1
1
+ -.
+ - +
2
n=l
m
So
i~
(1 +.!. +
S2
m=l
m
-
C(3) +
+-m1)
2
1
m-I
1
I2 I-n
m=2m
n-I
C (3) +
SI>
(3)
from (1). From (2) and (3), we arrive at the rather pleasing conclusion
SI
=
C(3),
=
S2
2C(3).
(4)
Now consider S3' The terms with m = n contribute C (3). The terms
with n > m contribute the same as those with m > n, so we have
C (3) +
2SI>
from (1) again. So, from (4), we have
S3
=
3C(3).
The identity Sl = C (3) was already known to Euler. It is discussed
extensively in [3] (thanks to Nick Lord for providing this reference), where
the quick proof given above is attributed to Steinberg [4]. Numerous other
proofs are given in [3]. For example, one can equate both Sl and C (3) to the
integral
f o Inx In x(1 l
x) dx
or to the double integral
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NOTES
329
-f f
I
I _In_{_l ---xy-) dx dy.
ool-xy
The proof using the first of these integrals was given in [5]. However, it
must surely be conceded that the method above is simpler and more direct.
The article [3] also describes a number of generalisations and related
results. In what is now the established notation for 'Euler sums", one
defines
--
~ (j, k)
=
1
L L
-1-1
L k ,am+lr'
L "'
Y =
k
m=ln=lm(m+n
m=lm
so that our SI is ~ (2,1) {warning: some writers interchange the} and k). By
a fairly straightforward extension of the method above (which readers might
care to attempt for themselves), one can prove
n-2
~(n)
L ~(n
=
- },})
j= I
for n ~ 3. Euler's equality SI = ~ (2,1) = ~ (3) is the case n
Another variation, also known to Euler, introduces alternating signs:
i.!. i
(-1)'
=
3.
g (3) ;
,-2
m=lm,_m+]
see [2, p. 8-9].
Several much more exotic series expressions that equate to ~ (3) are
listed on page 6 of [3]. One that converges very rapidly, so is good for
calculation, is
5 ~ (3)
2:
=
(_1)n-1
L
n
n=l
3
en) .
n
This identity was used by Apery in his proof, and has sometimes been
attributed to him, but in fact it was established by Hjortnaes in 1953[6]. A
proof can be seen in my website notes [7].
Instead of repeating any of these results and proofs here, we will now go
in a more number-theoretic direction and consider some double sums
involving coprime pairs or lowest common multiples. Write (a, b) for the
greatest common divisor of a and b and for r, s ~ 2, let
r.,
1
L- -.
o'b'
=
a,b = I
=I
(a,b)
Clearly, we have
--1
L L -m'n:
m~
Now write (m, n)
-1-1
=
I n_ I
=
g and m
L -m' L -n'
m~
=
1
ag, n
= ~ (r)
~ (s).
n=1
=
bg, so that (a, b)
=
1. Then
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330
THE MATHEMATICAL
~
1
~
g,a,b- I
(a,b) = I
~
1
£oJ -s' + S
~
~
£oJ
~
g'+sarb'
a,bK\
(a,b) = I
gml
1
arb' = '(r
-
GAZETTE
+ s)Tr, ••
and hence
'(r)'
= '(r +
Tr,s
In particular, using the known values'
have
(s)
s)'
(5)
= :rc21 6 and' (4)
(2)
:rc4190, we
= '(2)2 = :rc4/36 = ~
T
:rc4190
'(4)
2,2
(6)
2'
Now write [m, n] for the lowest common multiple of m and n. Let
U
ii
=
m~l
n=l
=
With g, a, b as above, we have [m, n]
~
U
=
~
g,a,b-
I
.
mn[m, n]
abg, hence
1
I
(a.b)~ 1
~ (3) T2,2,
g3a2b2
so by (6), we have
U
= ~
~(3).
By obvious variations of this reasoning, one obtains, for example,
~
~
m~l
n~l
~
~
5:rc2
1
[m,
nF =
'(2)
T2•2
12
=
and
1
L L -[m,n ]3
m=l
n=l
~ (3)3
= '(3)T3,3
'(6) .
More generally, one can apply (6) to express
~
u.;
=
~
L L mn
r
m~1
n=l
1
S [
m, n
j1
in terms of zeta values: we leave this to the reader.
References
1.
R. Apery, Interpolation de fractions continues et irrationalite de
certaines constantes, Math. CTHS Bull. Sec. Sci. Ill, Bib!. Nat. Paris
(1981), pp. 37-53.
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NOTES
331
5.
F. Beukers, A note on the irrationality of S (2) and C (3), Bull. London
Math. Soc. 11 (1979), pp. 268-272.
Jonathan M. Borwein and David M. Bradley, Thirty-two Goldbach
variations, Int. J. Number Theory 2 (2006), pp. 65-103; also at:
arxiv.org/abs/math/0502034, pp. 1-41.
R. Steinberg, Solution to problem 4431, Amer. Math. Monthly 59
(1952), pp. 471-472.
Solution to Problem 89D, Math. Gaz: 89 (2005), p. 541.
6.
M. M. Hjortnaes,
2.
3.
4.
Overfering
I.
av rekken
k-I
1/ k3 til et bestemt
integral, Proc. 12th Congo Scand. Math. (Lund, 1953), Lund (1954).
Tim Jameson, Polylogarithms, multiple zeta values and the series of
Hjortnaes and Comtet, at: www.maths.Iancs.ac.ukz-jameson
TIM JAMESON
Sadly, Tim Jameson died in September 2013. This note was submitted
after his death by his father, Graham Jameson.
7.
98.14 An unexpected closed form
According to a dictionary of mathematics, a closed form for a series or
integral is an expression in terms of well-known quantities [1]. In this note
we will show that the power series
y}
S(x)
= x
~
J?
x4
+ -2 + -2 + -2 + - +
2
3
4
52
(clearly summable for x = 0, ± 1) is closed in the special case x
We start with the logarithmic series
y}
10g(1 - x)
-x
- -
2
~
- -
J?
x4
- -
345
- -
(-I
-
= ~.
E;; x
< 1),
whence
I
f
1
= t
log (1 - x) d
x
I
x
[-S(x)]t
= S
(
1)
:rr
2
2 - 6'
(1)
Next, integration by parts leads to
I
=
[log(1 - x) logxh
I
fl
logx
+ t~
lim (log (1 - x) log x] - (Iog2/
x~1
+
dx
ft logO Y -
y) dy
0
(substituting x
The first term vanishes because x log x ~
0 as x ~
I = - (log 2)2 - S
- y).
0, so
m,
(2)
Finally, eliminating I from (I) and (2) yields the closed form
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