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Experimental Evaluation of Euler SumsDavid H. Bailey, Jonathan M. Borwein and Roland Girgensohn
CONTENTS
1. Introduction
2. Numerical Techniques
3. Experimental Setup and Optimizations
4. Integer Relation Detection Algorithms
5. Applications of the PSLQ Algorithm
6. Experimental Results
7. Conjectures
Acknowledgments
References
B o r w e i n w a s s u p p o r t e d b y N S E R C a n d t h e S h r u m E n d o w m e n t
a t S i m o n F r a s e r U n i v e r s i t y .
G i r g e n s o h n w a s s u p p o r t e d b y a D F G f e l l o w s h i p .
Euler expressed certain sums of the form
1
X
k = 1
1 +
1
2
m
+
+
1
k
m
( k + 1 )
n ,
where m and n are positive integers, in terms of the Riemann
zeta function. In [Borwein et al. 1993], Euler’s results were
extended to a significantly larger class of sums of this type,
including sums with alternating signs.
This research was facilitated by numerical computations us-
ing an algorithm that can determine, with high confidence,
whether or not a particular numerical value can be expressed
as a rational linear combination of several given constants. The
present paper presents the numerical techniques used in these
computations and lists many of the experimental results that
have been obtained.
1. INTRODUCTION
I n r e s p o n s e t o a l e t t e r f r o m G o l d b a c h , E u l e r c o n -
s i d e r e d s u m s o f t h e f o r m
1
X
k = 1
1 +
1
2
m
+
+
1
k
m
( k + 1 )
n
;
a n d w a s a b l e t o g i v e e x p l i c i t v a l u e s f o r c e r t a i n o f
t h e s e s u m s i n t e r m s o f t h e R i e m a n n z e t a f u n c t i o n
( t ) =
P
1
k = 1
k
t
. F o r e x a m p l e , E u l e r f o u n d a n
e x p l i c i t f o r m u l a f o r t h e c a s e m
= 1 , n
2 . L i t t l e
h a s b e e n d o n e o n t h i s p r o b l e m i n t h e i n t e r v e n i n g
y e a r s ( s e e B e r n d t 1 9 8 5 ] f o r s o m e r e f e r e n c e s ) .
I n A p r i l 1 9 9 3 , E n r i c o A u - Y e u n g , a n u n d e r g r a d -
u a t e a t t h e U n i v e r s i t y o f W a t e r l o o , b r o u g h t t o t h e
a t t e n t i o n o f o n e o f u s t h e c u r i o u s f a c t t h a t
1
X
k = 1
1 +
1
2
+
+
1
k
2
k
2
= 4 : 5 9 9 8 7
: : :
1 7
4
( 4 ) =
1 7
4
3 6 0
;
c
A K P e t e r s , L t d .
1 0 5 8 - 6 4 5 8 / 9 6 $ 0 . 5 0 p e r p a g e
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18 Experimental Mathematics, Vol. 3 (1994), No. 1
b a s e d o n a c o m p u t a t i o n t o 5 0 0 , 0 0 0 t e r m s . T h i s
a u t h o r ' s r e a c t i o n w a s t o c o m p u t e t h e v a l u e o f t h i s
c o n s t a n t t o a h i g h e r l e v e l o f p r e c i s i o n i n o r d e r t o
d i s p e l t h i s c o n j e c t u r e . S u r p r i s i n g l y , a c o m p u t a t i o n
t o 3 0 a n d l a t e r t o 1 0 0 d e c i m a l d i g i t s s t i l l a r m e d
i t . ( U n k n o w n t o u s a t t h a t t i m e , d e D o e l d e r 1 9 9 1 ]
h a d p r o v e d a r e l a t e d r e s u l t f r o m w h i c h t h e a b o v e
i d e n t i t y f o l l o w s . )
I n t r i g u e d b y t h i s e m p i r i c a l r e s u l t , w e c o m p u t e d
n u m e r i c a l v a l u e s f o r s e v e r a l o f t h e s e a n d s i m i l a r
s u m s , w h i c h w e h a v e t e r m e d E u l e r s u m s . W e t h e n
a n a l y z e d t h e s e v a l u e s b y a t e c h n i q u e , p r e s e n t e d b e -
l o w , t h a t p e r m i t s o n e t o d e t e r m i n e , w i t h a h i g h
l e v e l o f c o n d e n c e , w h e t h e r a n u m e r i c a l v a l u e c a n
b e e x p r e s s e d a s a r a t i o n a l l i n e a r c o m b i n a t i o n o f
s e v e r a l g i v e n c o n s t a n t s . T h e s e e o r t s p r o d u c e d
e v e n m o r e e m p i r i c a l e v a l u a t i o n s , s u g g e s t i n g b r o a d
p a t t e r n s a n d g e n e r a l c o n j e c t u r e s . U l t i m a t e l y w e
f o u n d p r o o f s f o r m a n y o f t h e s e e x p e r i m e n t a l r e -
s u l t s .
T h e c l a s s e s o f E u l e r s u m s t h a t w e w i l l c o n s i d e r
a r e l i s t e d i n T a b l e 1 . E x p l i c i t e v a l u a t i o n s o f s o m e
o f t h e c o n s t a n t s i n t h e s e c l a s s e s a r e p r e s e n t e d w i t h
p r o o f s i n B o r w e i n a n d B o r w e i n 1 9 9 4 ] a n d B o r -
w e i n e t a l . 1 9 9 4 ] . T a b l e 2 c o n t a i n s a s u m m a r y
o f t h e s e r e s u l t s , i n c l u d i n g s o m e a l r e a d y k n o w n t o
E u l e r . R e s u l t s f o r a l t e r n a t i n g s u m s a r e a l s o g i v e n
i n B o r w e i n e t a l . 1 9 9 4 ] .
V a r i a n t s o f t h e s u m s d e n e d i n T a b l e 1 c a n b e
e v a l u a t e d b y u s i n g t h e s e r e s u l t s . F o r e x a m p l e , f o r
a l l m
1 a n d n
2 o n e c a n w r i t e
1
X
k = 1
1 +
1
2
m
+
+
1
k
m
k
n
=
1
X
k = 0
1 +
1
2
m
+
+
1
( k + 1 )
m
( k + 1 )
n
=
1
X
k = 1
1 +
1
2
m
+
+
1
k
m
( k + 1 )
n
+
1
X
k = 1
k
m n
=
h
( m ; n ) +
( m + n ) :
s
h
( m ; n ) =
1
X
k = 1
1 +
1
2
+
+
1
k
m
( k + 1 )
n
f o r m 1
; n 2
s
a
( m ; n ) =
1
X
k = 1
1
1
2
+
+
( 1 )
k + 1
k
m
( k + 1 )
n
f o r m 1
; n 2
a
h
( m ; n ) =
1
X
k = 1
1 +
1
2
+
+
1
k
m
( 1 )
k + 1
( k + 1 )
n
f o r m 1
; n 1
a
a
( m ; n ) =
1
X
k = 1
1
1
2
+
+
( 1 )
k + 1
k
m
( 1 )
k + 1
( k + 1 )
n
f o r m 1
; n 1
h
( m ; n ) =
1
X
k = 1
1 +
1
2
m
+
+
1
k
m
( k + 1 )
n
f o r m 1
; n 2
a
( m ; n ) =
1
X
k = 1
1
1
2
m
+
+
( 1 )
k + 1
k
m
( k + 1 )
n
f o r m 1
; n 2
h
( m ; n ) =
1
X
k = 1
1 +
1
2
m
+
+
1
k
m
( 1 )
k + 1
( k + 1 )
n
f o r m 1
; n 1
a
( m ; n ) =
1
X
k = 1
1
1
2
m
+
+
( 1 )
k + 1
k
m
( 1 )
k + 1
( k + 1 )
n
f o r m 1
; n 1
TABLE 1. D e n i t i o n s o f t h e E u l e r s u m s c o n s i d e r e d .
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Bailey, Borwein and Girgensohn: Experimental Evaluation of Euler Sums 19
s
h
( 2 ;
2 ) =
3
2
( 4 ) +
1
2
2
( 2 ) =
1 1
3 6 0
4
s
h
( 2 ;
4 ) =
2
3
( 6 )
1
3
( 2 )
( 4 ) +
1
3
3
( 2 )
2
( 3 ) =
3 7
2 2 6 8 0
6
2
( 3 )
h
( 2 ;
2 ) =
1
2
2
( 2 )
1
2
( 4 ) =
1
1 2 0
4
h
( 2 ;
4 ) = 6
( 6 ) +
8
3
( 2 )
( 4 ) +
2
( 3 ) =
2
( 3 )
4
2 8 3 5
6
s
h
( 1 ; n ) =
h
( 1 ; n ) =
1
2
n ( n
+ 1 )
1
2
n 2
P
k = 1
( n k ) ( k + 1 )
s
h
( 2 ; n ) =
1
3
n ( n + 1 )
( n + 2 ) +
( 2 )
( n )
1
2
n
n 2
P
k = 0
( n k ) ( k + 2 )
+
1
3
n 2
P
k = 2
( n k )
k 1
P
j = 1
( j + 1 )
( k + 1
j ) +
h
( 2 ; n )
h
( 2 ; 2 n
1 ) =
1
2
2 n
2
+ n + 1
( 2
n + 1 ) +
( 2 )
( 2
n 1 ) +
n 1
P
k = 1
2 k ( 2
k + 1 )
( 2
n 2 k )
s
h
( 2 ; 2 n
1 ) =
1
6
( 2 n
2
7 n 3 )
( 2
n + 1 ) +
( 2 )
( 2
n 1 )
1
2
n 2
P
k = 1
( 2 k
1 )
( 2 n 1 2 k )
( 2 k
+ 2 )
+
1
3
n 2
P
k = 1
( 2
k + 1 )
n 2 k
P
j = 1
( 2
j + 1 )
( 2
n 1 2 k 2 j )
f o r m o d d , n e v e n :
h
( m ; n ) =
1
2
m + n
m
1
( m + n ) +
( m ) ( n )
m + n
P
j = 1
2 j 2
m 1
+
2 j 2
n 1
( 2
j 1 )
( m + n 2 j + 1 )
f o r m e v e n , n o d d :
h
( m ; n ) =
1
2
m + n
m
+ 1
( m + n ) +
m + n
P
j = 1
2 j 2
m 1
+
2 j 2
n 1
( 2
j 1 )
( m + n 2 j + 1 )
TABLE 2. S o m e e x p l i c i t e v a l u a t i o n s o f E u l e r s u m s
S i m i l a r l y , l e t h
k
=
P
k
j = 1
j
1
a n d h
0
= 0 . T h e n ,
f o r a l l n
2 ,
1
X
k = 1
1 +
1
2
+
+
1
k
2
k
n
=
1
X
k = 0
h
2
k + 1
( k + 1 )
n
=
1
X
k = 0
h
k
+
1
k + 1
2
( k + 1 )
n
=
1
X
k = 0
h
2
k
( k + 1 )
n
+ 2
1
X
k = 0
h
k
( k + 1 )
n 1
+
1
X
k = 0
( k + 1 )
n 2
= s
h
( 2 ; n ) + 2 s
h
( 1 ; n + 1 ) + ( n
+ 2 ) :
2. NUMERICAL TECHNIQUES
I t i s n o t e a s y t o n a v e l y c o m p u t e n u m e r i c a l v a l -
u e s o f a n y o f t h e s e E u l e r s u m s t o h i g h p r e c i s i o n .
S t r a i g h t f o r w a r d e v a l u a t i o n u s i n g t h e d e n i n g f o r -
m u l a s , t o s o m e u p p e r l i m i t f e a s i b l e o n p r e s e n t - d a y
c o m p u t e r s , y i e l d s o n l y a b o u t e i g h t d i g i t s a c c u r a c y .
B e c a u s e t h e i n t e g e r r e l a t i o n d e t e c t i o n a l g o r i t h m
d e s c r i b e d i n S e c t i o n 4 r e q u i r e s m u c h h i g h e r p r e c i -
s i o n t o o b t a i n r e l i a b l e r e s u l t s , m o r e a d v a n c e d t e c h -
n i q u e s m u s t b e e m p l o y e d .
W e p r e s e n t h e r e a r e a s o n a b l y s t r a i g h t f o r w a r d
m e t h o d t h a t i s g e n e r a l l y a p p l i c a b l e t o a l l E u l e r
s u m s d i s c u s s e d i n t h i s p a p e r . I t i n v o l v e s t h e c o m -
p o u n d a p p l i c a t i o n o f t h e E u l e r { M a c l a u r i n s u m m a -
t i o n f o r m u l a A b r a m o w i t z a n d S t e g u n 1 9 7 2 , p . 8 0 6 ;
A t k i n s o n 1 9 8 9 , p . 2 8 9 ; K n u t h 1 9 7 3 , p . 1 0 8 ] , w h i c h
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20 Experimental Mathematics, Vol. 3 (1994), No. 1
c a n b e s t a t e d a s f o l l o w s . S u p p o s e f ( t
) h a s a t l e a s t
2 p + 2 c o n t i n u o u s d e r i v a t i v e s o n ( a ; b ) . L e t
D b e
t h e d i e r e n t i a t i o n o p e r a t o r , l e t B
k
d e n o t e t h e k
- t h
B e r n o u l l i n u m b e r , a n d l e t B
k
( ) d e n o t e t h e
k - t h
B e r n o u l l i p o l y n o m i a l . T h e n
b
X
j = a
f ( j ) =
Z
b
a
f ( t ) d t
+
1
2
( f ( a ) +
f ( b ) )
+
p
X
j = 1
B
2 j
( 2 j
) !
( D
2 j 1
f ( b ) D
2 j 1
f ( a ) )
+ R
p
( a ; b
) ; (2.1)
w h e r e t h e r e m a i n d e r R
p
( a ; b ) i s g i v e n A t k i n s o n
1 9 8 9 , p . 2 8 9 ] b y
R
p
( a ; b ) =
1
( 2 p
+ 2 ) !
Z
b
a
B
2 p + 2
( t t ] )
D
2 p + 2
f ( t ) d t :
W e w i l l s t a r t b y d e m o n s t r a t i n g t h e m e t h o d i n
t h e c o m p u t a t i o n o f s
h
( m ; n ) . L e t
h ( k ) =
P
k
j = 1
j
1
a n d f ( t
) = t
1
. N o t e t h a t l i m
k ! 1
( h ( k ) l n
k ) i s E u -
l e r ' s c o n s t a n t
. B y ( 2 . 1 ) ,
h ( k ) = l n
k +
1
2
+
1
2
k
1
+
p
P
j = 1
B
2 j
2 j k
2 j
p
P
j = 1
B
2 j
2 j
+ R
p
( 1 ; k ) :
S i n c e j B
2 k
( t ) j j
B
2 k
j f o r a l l
k a n d f o r
j t j
1
A b r a m o w i t z a n d S t e g u n 1 9 7 2 , p . 8 0 5 ] , t h e r e m a i n -
d e r R
p
( 1 ; k ) h a s a w e l l - d e n e d l i m i t R
p
( 1 ; 1
) a s k
a p p r o a c h e s i n n i t y . W e t h e r e f o r e h a v e =
1
2
p
P
j = 1
B
2 j
2 j
+ R
p
( 1 ; 1
) , w h i c h g i v e s
h ( k ) =
+ l n
k +
1
2 k
+
p
X
j = 1
B
2 j
2 j k
2 j
R
p
( k ;
1 ) :
N o w
j R
p
( k ;
1 ) j =
Z
1
k
B
2 p + 2
( t t ] )
t
2 p 3
d t
j B
2 p + 2
j
( 2 p
+ 2 ) k
2 p + 2
;
s o t h a t t h e r e m a i n d e r i n t h e e x p r e s s i o n f o r h ( k )
i s n o g r e a t e r t h a n t h e r s t t e r m o m i t t e d i n t h e
s u m m a t i o n . W e c a n t h e n w r i t e , f o r e x a m p l e ,
h ( k ) =
+ l n
k +
1
2 k
1
1 2 k
2
+
1
1 2 0 k
4
1
2 5 2 k
6
+
1
2 4 0 k
8
1
1 3 2 k
1 0
+
6 9 1
3 2 7 6 0 k
1 2
1
1 2 k
1 4
+
3 6 1 7
8 1 6 0 k
1 6
+ O ( k
1 8
) :
W e w i l l u s e
h ( k ) t o d e n o t e t h i s p a r t i c u l a r a p p r o x -
i m a t i o n ( w i t h o u t t h e e r r o r t e r m ) . I t i s a n u n f o r -
t u n a t e f a c t t h a t
h c a n n o t b e e x t e n d e d t o a v a l i d
i n n i t e s e r i e s . T h e d i c u l t y i s t h a t f o r a n y x e d
k , t h e B e r n o u l l i c o e c i e n t s e v e n t u a l l y b e c o m e v e r y
l a r g e a n d t h e s e r i e s d i v e r g e s . O n t h e o t h e r h a n d ,
i t i s c l e a r t h a t f o r a n y x e d n u m b e r o f t e r m s , a p -
p r o x i m a t i o n s s u c h a s
h b e c o m e e v e r m o r e a c c u r a t e
a s k
i n c r e a s e s t o i n n i t y .
N o w c o n s i d e r
s
h
( m ; n ) =
1
X
k = 1
1 +
1
2
+
+
1
k
m
( k + 1 )
n
:
L e t c
b e a l a r g e i n t e g e r , a n d l e t
g ( t ) =
h
m
( t ) (
t + 1 )
n
:
A p p l y i n g ( 2 . 1 ) a g a i n , w e c a n w r i t e
s
h
( m ; n ) =
c
X
k = 1
1 +
1
2
+
+
1
k
m
( k + 1 )
n
+
1
X
k = c + 1
1 +
1
2
+
+
1
k
m
( k + 1 )
n
=
c
X
k = 1
h
m
( k ) (
k + 1 )
n
+
Z
1
c + 1
g ( t ) d t
+
1
2
g ( c + 1 )
9
X
k = 1
B
2 k
( 2 k
) !
D
2 k 1
g ( c + 1 )
+ O ( c
1 8
) : (2.2)
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Bailey, Borwein and Girgensohn: Experimental Evaluation of Euler Sums 21
T h i s f o r m u l a s u g g e s t s t h e f o l l o w i n g c o m p u t a t i o n a l
s c h e m e . F i r s t e v a l u a t e e x p l i c i t l y t h e s u m
c
X
k = 1
h
m
( k ) (
k + 1 )
n
f o r c
= 1 0
8
, u s i n g a n u m e r i c w o r k i n g p r e c i s i o n o f
1 5 0 d i g i t s . T h e n p e r f o r m t h e s y m b o l i c i n t e g r a t i o n
a n d d i e r e n t i a t i o n s t e p s i n d i c a t e d i n ( 2 . 2 ) . F i -
n a l l y , e v a l u a t e t h e r e s u l t i n g e x p r e s s i o n , a g a i n u s i n g
a w o r k i n g p r e c i s i o n o f 1 5 0 d i g i t s . T h e n a l r e s u l t
s h o u l d b e e q u a l t o s
h
( m ; n ) t o a p p r o x i m a t e l y 1 3 5
s i g n i c a n t d i g i t s .
T h e d i c u l t y a n d c o s t o f p e r f o r m i n g t h e r e q u i -
s i t e s y m b o l i c i n t e g r a t i o n a n d d i e r e n t i a t i o n o p e r a -
t i o n s c a n b e g r e a t l y r e d u c e d b y a p p r o x i m a t i n g g ( t )
a s f o l l o w s : e x p a n d
h
m
( t ) , t h e n u m e r a t o r o f
g ( t ) ,
i n t o a s u m o f i n d i v i d u a l t e r m s ; w r i t e ( 1 + t )
n
a s
t
n
( 1 + t
1
)
n
; e x p a n d ( 1 + t
1
)
n
u s i n g t h e b i -
n o m i a l t h e o r e m t o 1 8 t e r m s ; m u l t i p l y t o g e t h e r t h e
r e s u l t i n g n u m e r a t o r a n d d e n o m i n a t o r e x p r e s s i o n s ;
a n d n a l l y o m i t a l l t e r m s i n w h i c h t h e e x p o n e n t o f
t
1
i s g r e a t e r t h a n 1 8 . T h e r e s u l t i s a l i n e a r s u m
o f t e r m s o f t h e f o r m t
p
l n
q
( t ) f o r m o d e s t - s i z e d i n -
t e g e r s p a n d
q .
O f c o u r s e , e v e n m o r e a c c u r a t e r e s u l t s c a n b e o b -
t a i n e d b y u t i l i z i n g m o r e t e r m s i n t h e E u l e r { M a c -
l a u r i n e x p a n s i o n s , a l t h o u g h t h e c o s t o f t h e r e q u i r e d
s y m b o l i c m a n i p u l a t i o n c o r r e s p o n d i n g l y i n c r e a s e s .
T h e d e t e r m i n a t i o n o f t h e o p t i m a l b a l a n c e b e t w e e n
n u m e r i c a n d s y m b o l i c c a l c u l a t i o n s , a n d o f t h e n u m -
b e r o f E u l e r { M a c l a u r i n t e r m s ( a t b o t h s t e p s ) r e -
q u i r e d f o r v a r i o u s l e v e l s o f p r e c i s i o n , i s a n i n t e r e s t -
i n g p r o b l e m i n i t s o w n r i g h t . H o w e v e r , w e f o u n d
t h a t o n l y m i n o r t u n i n g o f t h e a b o v e s c h e m e , b a s e d
o n s i m p l e t i m i n g a n d a c c u r a c y e x p e r i m e n t s , s u f -
c e d f o r t h e c a s e s w e s t u d i e d .
F o r a l t e r n a t i n g E u l e r s u m s , t h e s c h e m e i s a b i t
m o r e c o m p l i c a t e d . E u l e r { M a c l a u r i n s u m m a t i o n
w o r k s b a d l y f o r a l t e r n a t i n g s e r i e s , b e c a u s e o s c i l -
l a t i o n s l e a d t o l a r g e h i g h - o r d e r d e r i v a t i v e s . T h e
s o l u t i o n i s t o e e c t i v e l y s p l i t t h e s u m m a t i o n i n t o
t w o , o n e p o s i t i v e a n d o n e n e g a t i v e . C o n s i d e r , f o r
i n s t a n c e , t h e s u m s
a
( m ; n ) . W e c a n w r i t e
s
a
( m ; n ) =
1
X
k = 1
1
1
2
+
+
( 1 )
k + 1
k
m
( k + 1 )
n
=
1
X
k = 1
k
X
j = 1
1
2 j 1
k 1
X
j = 1
1
2 j
m
1
( 2 k )
n
+
1
X
k = 1
k
X
j = 1
1
2 j 1
k
X
j = 1
1
2 j
m
1
( 2 k
+ 1 )
n
=
1
X
k = 1
r
k
+
1
2 k
m
1
( 2 k )
n
+
1
X
k = 1
r
m
k
( 2 k
+ 1 )
n
;
w h e r e
r
k
=
k
X
j = 1
1
2 j 1
k
X
j = 1
1
2 j
=
k
X
j = 1
1
2 j ( 2
j 1 )
:
T h e E u l e r { M a c l a u r i n f o r m u l a ( 2 . 1 ) c a n t h e n b e a p -
p l i e d r s t t o o b t a i n a h i g h l y a c c u r a t e a p p r o x i m a -
t i o n t o r
k
, a n d t h e n t o e v a l u a t e t h e t w o r e m a i n i n g
o u t e r s u m m a t i o n s .
A n o t h e r a p p r o a c h f o r a l t e r n a t i n g E u l e r s u m s i s
t o a p p l y t h e B o o l e s u m m a t i o n f o r m u l a B o r w e i n e t
a l . 1 9 8 9 ] , w h i c h d e a l s s p e c i c a l l y w i t h a l t e r n a t i n g
s u m s .
3. EXPERIMENTAL SETUP AND OPTIMIZATIONS
W e h a v e p e r f o r m e d m a n y c o m p u t a t i o n s o f t h e t y p e
d e s c r i b e d . T h e i n t e g r a t i o n s a n d d i e r e n t i a t i o n s r e -
q u i r e d i n ( 2 . 2 ) c a n b e h a n d l e d b y a s y m b o l i c m a t h -
e m a t i c s p a c k a g e s u c h a s M a p l e C h a r e t a l . 1 9 9 1 ]
o r M a t h e m a t i c a W o l f r a m 1 9 9 1 ] . T h e e x p l i c i t s u m -
m a t i o n o f t h e r s t c
t e r m s , a s i n d i c a t e d i n ( 2 . 2 ) ,
c o u l d b e p e r f o r m e d b y u t i l i z i n g t h e m u l t i p l e p r e c i -
s i o n f a c i l i t y i n t h e M a p l e o r M a t h e m a t i c a p a c k a g e s .
H o w e v e r , w e f o u n d t h a t t h e M P F U N m u l t i p l e p r e -
c i s i o n p a c k a g e a n d t r a n s l a t o r d e v e l o p e d b y o n e o f
u s B a i l e y ] w a s s i g n i c a n t l y f a s t e r f o r t h i s p u r p o s e .
W h a t e v e r s o f t w a r e i s u s e d , t h i s e x p l i c i t s u m m a -
t i o n i s a v e r y e x p e n s i v e o p e r a t i o n . F o r e x a m p l e ,
t h e e v a l u a t i o n o f s
h
( 3 ;
4 ) t o 1 0
8
t e r m s , u s i n g M P -
F U N w i t h 1 5 0 - d i g i t p r e c i s i o n a r i t h m e t i c , r e q u i r e s
t w e n t y h o u r s o n a C r i m s o n S i l i c o n G r a p h i c s w o r k -
s t a t i o n . S u c h r u n s c a n b e m a d e , b u t c l e a r l y t h i s i s
p r e s s i n g t h e c a p a b i l i t i e s o f c u r r e n t w o r k s t a t i o n s .
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22 Experimental Mathematics, Vol. 3 (1994), No. 1
M P F U N i s a v a i l a b l e o n v e c t o r s u p e r c o m p u t e r s
s u c h a s t h o s e m a n u f a c t u r e d b y C r a y R e s e a r c h , I n c .
H o w e v e r , f o r t h e m o d e s t p r e c i s i o n l e v e l s t y p i c a l o f
t h e s e p r o b l e m s ( 1 0 0 t o 2 0 0 d e c i m a l d i g i t s ) , t h e r e -
s u l t i n g v e c t o r l e n g t h s a r e t o o s h o r t t o y i e l d t h e
h i g h p e r f o r m a n c e t h e s e s y s t e m s a r e c a p a b l e o f . O n
t h e o t h e r h a n d , m u l t i p l e p r e c i s i o n c a l c u l a t i o n s o f
t h i s t y p e a r e w e l l s u i t e d f o r R I S C p r o c e s s o r s , b e -
c a u s e t h e y a r e w e l l b e h a v e d i n c a c h e m e m o r y s y s -
t e m s .
T h e s e c o n s i d e r a t i o n s s u g g e s t t h a t , i n p r i n c i p l e ,
a h i g h l y p a r a l l e l c o m p u t e r b a s e d o n R I S C p r o c e s -
s o r s c o u l d b e e e c t i v e l y e m p l o y e d f o r c o m p u t i n g
t h e s e e x p l i c i t s u m s . H o w e v e r , a t r s t g l a n c e t h e s e
c o m p u t a t i o n s a p p e a r n o t t o p o s s e s s a n y s i g n i c a n t
o p p o r t u n i t y f o r p a r a l l e l i s m , s i n c e e v i d e n t l y b o t h
t h e i n n e r a n d o u t e r s u m s m u s t b e s i m u l t a n e o u s l y
a c c u m u l a t e d .
F o r t u n a t e l y , t h e r e a r e a l g o r i t h m s t h a t e c i e n t l y
e x p l o i t p a r a l l e l i s m . T h e b a s i c i d e a i s s i m p l e : s u p -
p o s e , s a y , t h a t w e w i s h t o c o m p u t e s
h
( m ; n ) u s i n g
( 2 . 2 ) , a n d t h u s n e e d t o s u m h
m
( k ) (
k + 1 )
n
u p t o
k = c . I f t h e p a r a l l e l s y s t e m h a s
P p r o c e s s o r n o d e s ,
a n d c =
J P , e a c h p r o c e s s o r s t o r e s a l e n g t h - J
r u n
o f v a l u e s o f k
1
, t h e n a d d s t h e s e v a l u e s t o g e t h e r
a n d c o m b i n e s t h e s u m w i t h t h e a c c u m u l a t e d s u m
f r o m t h e p r e c e d i n g p r o c e s s o r s . M o r e p r e c i s e l y :
Algorithm 1 (Parallel Summation).S u p p o s e g i v e n
P
p a r a l l e l p r o c e s s o r n o d e s , e a c h w i t h a n a s s o c i a t e d
a r r a y r
p
o f l e n g t h J
+ 1 a n d s t o r a g e c e l l s h
p
a n d
s
p
, f o r 1 p P
. A l l o a t i n g - p o i n t s t o r a g e a n d
a r i t h m e t i c i s a s s u m e d t o b e d o n e i n m u l t i p l e p r e -
c i s i o n . W e c o m p u t e
S =
J P
X
k = 1
h
m
( k ) (
k + 1 )
n
a s f o l l o w s :
I n i t i a l i z e b y s e t t i n g H
0 a n d S
0 .
F o r p
1 t o P
, i n p a r a l l e l :
S e t h
p
0 a n d
s
p
0 .
F o r j
1 t o J
, s e t r
p ; j
1 = k ( w h e r e
k =
j + (
p 1 )
J ) , a n d
h
p
h
p
+ r
p ; j
.
S e t r
p ; J + 1
1 = ( 1 + p J ) .
F o r p
1 t o P
, s e q u e n t i a l l y :
S e t H
0
H , H H + h
p
, a n d h
p
H
0
.
F o r p
1 t o P
, i n p a r a l l e l :
F o r j
1 t o J :
S e t h
p
h
p
+ r
p ; j
a n d s
p
s
p
+ h
m
p
r
n
p ; j + 1
.
F o r p
1 t o P
, s e q u e n t i a l l y :
S e t S S + s
p
.
N o t e t h a t , a s s t a t e d , t h e a l g o r i t h m m a i n t a i n s a l l
t h e c =
J P i n v e r s e s a t o n c e i n t h e a r r a y r
. I n a
t y p i c a l c o m p u t a t i o n o n e m a y h a v e c
= 1 0
8
, a n d t h e
r e q u i r e d a m o u n t o f m e m o r y p e r n o d e m a y n o t b e
a v a i l a b l e o n s o m e h i g h l y p a r a l l e l c o m p u t e r s . T h i s
d i c u l t y c a n b e r e m e d i e d b y s e t t i n g u p a n o u t e r
l o o p : w e w r i t e c =
I J P a n d r u n t h e a l g o r i t h m s u c -
c e s s i v e l y f o r e a c h b l o c k k 2
( i
1 ) J P + 1 ; i J P ] ,
w h e r e 1 i I
. ( N a t u r a l l y , H
a n d S
a r e m a i n -
t a i n e d f r o m o n e i t e r a t i o n t o t h e n e x t i n s t e a d o f
b e i n g i n i t i a l i z e d t o 0 , a n d t h e v a l u e a s s i g n e d t o
r
p ; j
i s 1 = k w h e r e
k = j + (
p 1 )
J + (
i 1 ) J P :
)
U p o n c o m p l e t i o n o f I
i t e r a t i o n s , S
i s t h e d e s i r e d
o v e r a l l s u m .
T h i s a l g o r i t h m h a s b e e n i m p l e m e n t e d o n a n I n -
t e l P a r a g o n p a r a l l e l c o m p u t e r a t N A S A A m e s R e -
s e a r c h C e n t e r , u s i n g M P F U N . F o r m
= 3 a n d n =
4 i n 1 5 0 - d i g i t p r e c i s i o n a r i t h m e t i c , w i t h P
= 1 2 8 ,
J = 2
1 3
a n d I
= 1 0 0 ( s o t h a t c
= 1 0 4 ;
8 5 7 ;
6 0 0 ) ,
t h e p r o g r a m t o o k o n l y 9 7 1 s e c o n d s . T h i s i s 1 1 0
t i m e s f a s t e r t h a n t h e t i m e r e q u i r e d b y a s t r a i g h t -
f o r w a r d s e r i a l a l g o r i t h m t h a t e x e r c i s e s o n l y o n e
n o d e o f t h e P a r a g o n , a n d 4 0 t i m e s f a s t e r t h a n t h e
s a m e a l g o r i t h m r u n n i n g o n o n e p r o c e s s o r o f a C r a y
Y M P , u s i n g M P F U N t u n e d f o r t h e C r a y . I t m a y
b e p o s s i b l e , b y r e o r g a n i z i n g t h e c o m p u t a t i o n , t o
a c h i e v e h i g h e r p e r f o r m a n c e o n t h e C r a y ; t h u s c a u -
t i o n s h o u l d b e e x e r c i s e d w h e n i n t e r p r e t i n g t h i s l a s t
g u r e . B u t t h e s e r e s u l t s n o n e t h e l e s s c o n r m t h a t
A l g o r i t h m 1 , r u n n i n g o n a h i g h l y p a r a l l e l R I S C s u -
p e r c o m p u t e r , i s a h i g h l y e c i e n t a n d c o s t - e e c t i v e
s o l u t i o n t o t h e p r o b l e m o f c o m p u t i n g t h e e x p l i c i t
s u m s r e q u i r e d i n ( 2 . 2 ) .
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Bailey, Borwein and Girgensohn: Experimental Evaluation of Euler Sums 23
4. INTEGER RELATION DETECTION ALGORITHMS
L e t x
= ( x
1
; x
2
; : : : ; x
n
) b e a v e c t o r o f r e a l n u m -
b e r s . W e s a y t h a t x
p o s s e s s e s a n i n t e g e r r e l a t i o n
i f t h e r e e x i s t i n t e g e r s a
i
n o t a l l z e r o s u c h t h a t
a
1
x
1
+ a
2
x
2
+
+ a
n
x
n
= 0 :
B y a n i n t e g e r r e l a t i o n a l g o r i t h m w e m e a n a n a l g o -
r i t h m t h a t i s g u a r a n t e e d ( p r o v i d e d t h e c o m p u t e r
i m p l e m e n t a t i o n h a s s u c i e n t n u m e r i c p r e c i s i o n ) t o
r e c o v e r t h e v e c t o r o f i n t e g e r s a
i
, i f i t e x i s t s , o r t o
p r o d u c e b o u n d s w i t h i n w h i c h n o i n t e g e r r e l a t i o n
c a n e x i s t .
T h e p r o b l e m o f n d i n g i n t e g e r r e l a t i o n s a m o n g a
s e t o f r e a l n u m b e r s w a s r s t s t u d i e d b y E u c l i d , w h o
g a v e a n i t e r a t i v e a l g o r i t h m w h i c h , w h e n a p p l i e d t o
t w o r e a l n u m b e r s , e i t h e r t e r m i n a t e s , y i e l d i n g a n
e x a c t r e l a t i o n , o r p r o d u c e s a n i n n i t e s e q u e n c e o f
a p p r o x i m a t e r e l a t i o n s . T h e g e n e r a l i z a t i o n o f t h i s
p r o b l e m f o r n > 2 h a s b e e n a t t e m p t e d b y E u l e r ,
J a c o b i , P o i n c a r e , M i n k o w s k i , P e r r o n , B r u n , a n d
B e r n s t e i n , a m o n g o t h e r s . H o w e v e r , n o n e o f t h e i r
a l g o r i t h m s h a s b e e n p r o v e d t o w o r k f o r n > 3 , a n d
n u m e r o u s c o u n t e r e x a m p l e s h a v e b e e n f o u n d .
T h e r s t i n t e g e r r e l a t i o n a l g o r i t h m w i t h t h e d e -
s i r e d p r o p e r t i e s m e n t i o n e d a b o v e w a s d i s c o v e r e d
b y F e r g u s o n a n d F o r c a d e 1 9 7 9 ] . I n t h e i n t e r v e n -
i n g y e a r s a n u m b e r o f o t h e r i n t e g e r r e l a t i o n a l g o -
r i t h m s h a v e b e e n d i s c o v e r e d , i n c l u d i n g a v a r i a n t o f
t h e o r i g i n a l a l g o r i t h m F e r g u s o n 1 9 8 7 ] , t h e L L L a l -
g o r i t h m L e n s t r a e t a l . 1 9 8 2 ] , t h e H J L S a l g o r i t h m
H a s t a d e t a l . 1 9 8 8 ] , w h i c h i s b a s e d o n L L L , a n d
t h e P S O S a l g o r i t h m B a i l e y a n d F e r g u s o n 1 9 8 9 ] .
R e c e n t l y a n e w a l g o r i t h m , k n o w n a s t h e P S L Q
a l g o r i t h m , w a s d e v e l o p e d b y F e r g u s o n a n d o n e o f
u s F e r g u s o n a n d B a i l e y 1 9 9 1 ] . I t a p p e a r s t o c o m -
b i n e s o m e o f t h e b e s t f e a t u r e s s e p a r a t e l y p o s s e s s e d
b y p r e v i o u s a l g o r i t h m s , i n c l u d i n g f a s t r u n t i m e s ,
n u m e r i c a l s t a b i l i t y , n u m e r i c a l e c i e n c y ( t h a t i s , i t
s u c c e s s f u l l y r e c o v e r s a r e l a t i o n w h e n t h e i n p u t i s
k n o w n t o o n l y l i m i t e d p r e c i s i o n ) , a n d a g u a r a n t e e d
c o m p l e t i o n i n a p o l y n o m i a l l y b o u n d e d n u m b e r o f
i t e r a t i o n s . W e p r e s e n t h e r e a s i m p l i e d b u t e q u i v -
a l e n t v e r s i o n o f P S L Q . T h e p r o o f o f t h e a l g o r i t h m
a n d n o t e s f o r e c i e n t i m p l e m e n t a t i o n s a r e g i v e n
i n F e r g u s o n a n d B a i l e y 1 9 9 1 ] .
Algorithm 2 (PSLQ).L e t
x b e t h e t h e i n p u t r e a l v e c -
t o r o f l e n g t h n
, a n d l e t n i n t d e n o t e t h e n e a r e s t i n -
t e g e r f u n c t i o n ( f o r e x a c t h a l f - i n t e g e r v a l u e s , d e n e
n i n t t o b e t h e i n t e g e r w i t h g r e a t e r a b s o l u t e v a l u e ) .
L e t =
p
4 = 3 . L e t
A a n d
B b e
n n m a t r i c e s ,
a n d H
a n n ( n
1 ) m a t r i x .
Initialization
S e t A
a n d B
t o t h e i d e n t i t y .
F o r k
1 t o n
, c o m p u t e s
k
q
P
n
j = k
x
2
j
.
F o r k
1 t o n
, s e t y
k
x
k
= s
1
a n d s
k
s
k
= s
1
.
(Initialize H ) F o r i
1 t o n :
f o r j i
+ 1 t o n
1 , s e t H
i j
0 ;
i f i n
1 , s e t H
i i
s
i + 1
= s
i
;
f o r j
1 t o i
1 , s e t H
i j
y
i
y
j
= ( s
j
s
j + 1
) .
(Perform full reduction onH
while updatingy
,A
,B
)
F o r i
2 t o n :
f o r j i
1 t o 1 b y
1 :
s e t t
n i n t ( H
i j
= H
j j
) a n d y
j
y
j
+ t y
i
;
f o r k
1 t o j
, s e t H
i k
H
i k
t H
j k
;
f o r k
1 t o n
, s e t A
i k
A
i k
t A
j k
a n d
B
k j
B
k j
+ t B
k i
.
Main loop
R e p e a t u n t i l p r e c i s i o n i s e x h a u s t e d o r a r e l a t i o n
h a s b e e n d e t e c t e d ( s e e t e r m i n a t i o n t e s t a t t h e
e n d ) :
S e l e c t m
s u c h t h a t
i
j H
i i
j i s m a x i m a l w h e n
i =
m .
(Perform block reduction onH
while updatingy
,A
,
B )
F o r i m
+ 1 t o n :
f o r j
m i n ( i 1
; m + 1 ) t o 1 b y
1 :
s e t t
n i n t ( H
i j
= H
j j
) a n d y
j
y
j
+ t y
i
;
f o r k
1 t o j
, s e t H
i k
H
i k
t H
j k
;
f o r k
1 t o n
, s e t A
i k
A
i k
t A
j k
a n d
B
k j
B
k j
+ t B
k i
.
E x c h a n g e e n t r i e s m
a n d m
+ 1 o f y
, c o r r e -
s p o n d i n g r o w s o f A
a n d H
, a n d c o r r e s p o n d i n g
c o l u m n s o f B .
I f m n
2 , u p d a t e H
a s f o l l o w s :
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24 Experimental Mathematics, Vol. 3 (1994), No. 1
S e t t
0
p
H
2
m m
+ H
2
m ; m + 1
, t
1
H
m m
= t
0
a n d t
2
H
m ; m + 1
= t
0
.
F o r i m
t o n
, s e t t
3
H
i m
, t
4
H
i ; m + 1
,
H
i m
t
1
t
3
+ t
2
t
4
, H
i ; m + 1
t
2
t
3
+ t
1
t
4
.
(Norm bound)S e t
M 1 = m a x
j
j H
j
j , w h e r e
H
j
d e n o t e s t h e j
- t h r o w o f H
. T h e r e c a n e x i s t n o
r e l a t i o n v e c t o r w h o s e E u c l i d e a n n o r m i s l e s s
t h a n M .
(Termination test)I f t h e l a r g e s t e n t r y o f
A e x c e e d s
t h e l e v e l o f n u m e r i c p r e c i s i o n u s e d , t h e n p r e c i -
s i o n i s e x h a u s t e d . I f t h e s m a l l e s t e n t r y o f t h e
y v e c t o r i s l e s s t h a n t h e d e t e c t i o n t h r e s h o l d , a
r e l a t i o n h a s b e e n d e t e c t e d a n d i s g i v e n i n t h e
c o r r e s p o n d i n g c o l u m n o f B .
R e g a r d i n g t h e t e r m i n a t i o n t e s t , i t s o m e t i m e s h a p -
p e n s t h a t a r e l a t i o n i s m i s s e d a t t h e p o i n t o f p o -
t e n t i a l d e t e c t i o n b e c a u s e t h e y
e n t r y i s n o t q u i t e
a s s m a l l a s t h e d e t e c t i o n t h r e s h o l d b e i n g u s e d ( t h e
t h r e s h o l d i s t y p i c a l l y s e t t o t h e \ e p s i l o n " o f t h e
p r e c i s i o n l e v e l ) . W h e n t h i s h a p p e n s , h o w e v e r , o n e
w i l l n o t e t h a t t h e r a t i o o f t h e s m a l l e s t a n d l a r g e s t
y v e c t o r e n t r i e s i s s u d d e n l y v e r y s m a l l , p r o v i d e d
s u c i e n t n u m e r i c p r e c i s i o n i s b e i n g u s e d .
T h e a c t u a l p r o b a b i l i t y d i s t r i b u t i o n o f t h i s r a t i o
i s n o t k n o w n f o r t h e P S L Q a l g o r i t h m . M o s t l i k e l y ,
h o w e v e r , t h e p r o b a b i l i t y o f t h i s r a t i o b e i n g l e s s
t h a n "
i s c l o s e l y a p p r o x i m a t e d b y a m o d e s t - s i z e d
c o n s t a n t t i m e s "
. T h i s i s b e c a u s e t h e e n t r i e s o f t h e
v e c t o r y a r e r e l a t e d t o t h e i t e r a t e s o f t h e c o n t i n u e d
f r a c t i o n a l g o r i t h m , w h i c h a r e d i s t r i b u t e d a c c o r d i n g
t o t h e K u z m i n d i s t r i b u t i o n K n u t h 1 9 8 1 , p . 3 4 6 ] .
I n a n o r m a l r u n o f P S L Q , p r i o r t o t h e d e t e c t i o n o f
a r e l a t i o n , t h i s r a t i o i s s e l d o m s m a l l e r t h a n 0 :
0 1 .
T h u s i f t h i s r a t i o s u d d e n l y d e c r e a s e s t o a v e r y s m a l l
v a l u e , s u c h a s 1 0
2 0
, i t i s a l m o s t c e r t a i n t h a t a r e -
l a t i o n h a s b e e n d e t e c t e d | o n e n e e d o n l y a d j u s t t h e
d e t e c t i o n t h r e s h o l d f o r t h e a l g o r i t h m t o t e r m i n a t e
p r o p e r l y a n d o u t p u t t h e r e l a t i o n . W h e n d e t e c t i o n
d o e s o c c u r , t h i s r a t i o m a y b e t h o u g h t o f a s a \ c o n -
d e n c e l e v e l " o f t h e d e t e c t i o n .
I n p r a c t i c e , t h e P S L Q a l g o r i t h m i s v e r y e e c -
t i v e i n n d i n g r e l a t i o n s . F o r e x a m p l e , i n t e s t s
d e s c r i b e d i n F e r g u s o n a n d B a i l e y 1 9 9 1 ] , r e l a t i o n s
w e r e d e t e c t e d f o r i n p u t v e c t o r s i n d i m e n s i o n s u p t o
n = 8 2 , t h e c o e c i e n t s o f t h e r e l a t i o n h a v i n g s i z e
u p t o 1 0
1 4
. A s a g e n e r a l r u l e , o n e c a n e x p e c t t o
d e t e c t a r e l a t i o n w i t h c o e c i e n t s o f s i z e 1 0
m
f o r a n
i n p u t v e c t o r o f d i m e n s i o n n
i f t h e i n p u t i s k n o w n
t o s o m e w h a t g r e a t e r t h a n m n d i g i t p r e c i s i o n , a n d
p r o v i d e d t h a t c o m p u t a t i o n s a r e p e r f o r m e d u s i n g
a t l e a s t t h i s l e v e l o f n u m e r i c p r e c i s i o n .
5. APPLICATIONS OF THE PSLQ ALGORITHM
I n t e g e r r e l a t i o n d e t e c t i o n a l g o r i t h m s h a v e a n u m -
b e r o f a p p l i c a t i o n s i n c o m p u t a t i o n a l m a t h e m a t i c s .
O n e i s t o n d o u t w h e t h e r o r n o t a g i v e n c o n s t a n t
, w h o s e v a l u e c a n b e c o m p u t e d t o h i g h p r e c i s i o n ,
i s a l g e b r a i c o f s o m e d e g r e e n
o r l e s s . T h i s i s d o n e
b y c o m p u t i n g t h e v e c t o r x
= ( 1 ; ;
2
; : : : ;
n
) t o
h i g h p r e c i s i o n a n d t h e n a p p l y i n g a n i n t e g e r r e l a -
t i o n a l g o r i t h m t o t h e v e c t o r x
. I f a r e l a t i o n i s
f o u n d , t h i s i n t e g e r v e c t o r i s p r e c i s e l y t h e s e t o f c o -
e c i e n t s o f a p o l y n o m i a l s a t i s e d b y
. W h e n a
r e l a t i o n i s n o t f o u n d , t h e r e s u l t i n g b o u n d m e a n s
t h a t
c a n n o t p o s s i b l y b e t h e r o o t o f a p o l y n o m i a l
o f d e g r e e n
, w i t h c o e c i e n t s o f s i z e l e s s t h a n t h e
e s t a b l i s h e d b o u n d . E v e n n e g a t i v e r e s u l t s o f t h i s
s o r t a r e o f t e n o f i n t e r e s t .
O n e o f u s h a s p e r f o r m e d s e v e r a l c o m p u t a t i o n s
o f t h i s t y p e B a i l e y a n d F e r g u s o n 1 9 8 9 ] , e s t a b l i s h -
i n g , f o r e x a m p l e , t h a t i f E u l e r ' s c o n s t a n t s a t -
i s e s a n i n t e g e r p o l y n o m i a l o f d e g r e e 5 0 o r l e s s ,
t h e E u c l i d e a n n o r m o f t h e c o e c i e n t s m u s t e x c e e d
7 1 0
1 7
. C o m p u t a t i o n s o f t h i s s o r t h a v e a l s o b e e n
a p p l i e d t o s t u d y a c e r t a i n c o n j e c t u r e r e g a r d i n g t h e
R i e m a n n z e t a f u n c t i o n . I t i s w e l l k n o w n B o r w e i n
a n d B o r w e i n 1 9 8 7 ] t h a t ( 2 ) ,
( 3 ) a n d
( 4 ) e q u a l ,
r e s p e c t i v e l y ,
3
1
X
k = 1
1
k
2
2 k
k
;
5
2
1
X
k = 1
( 1 )
k 1
k
3
2 k
k
;
3 6
1 7
1
X
k = 1
1
k
4
2 k
k
:
T h i s h a s l e d s o m e t o s u g g e s t t h a t
( 5 ) =
Z
5
1
X
k = 1
( 1 )
k 1
k
5
2 k
k
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Bailey, Borwein and Girgensohn: Experimental Evaluation of Euler Sums 25
f o r Z
5
a s i m p l e r a t i o n a l o r a l g e b r a i c n u m b e r . U n -
f o r t u n a t e l y , i n t e g e r r e l a t i o n c a l c u l a t i o n s B a i l e y ]
h a v e e s t a b l i s h e d t h a t i f Z
5
s a t i s e s a p o l y n o m i a l
o f d e g r e e 2 5 o r l e s s , t h e E u c l i d e a n n o r m o f t h e c o -
e c i e n t s m u s t e x c e e d 2
1 0
3 7
.
T h e p r e s e n t a p p l i c a t i o n o f E u l e r s u m c o n s t a n t s
i s w e l l s u i t e d t o a n a l y s i s w i t h i n t e g e r r e l a t i o n a l g o -
r i t h m s . W e w i l l p r e s e n t b u t o n e e x a m p l e o f t h e s e
c o m p u t a t i o n s . C o n s i d e r
s
a
( 2 ;
3 ) = 0 :
1 5 6 1 6 6 9 3 3 3 8 1 1 7 6 9 1 5 8 8 1 0 3 5 9 0 9 6 8 7 : : :
( s e e T a b l e 1 f o r t h e d e n i t i o n ) . B a s e d o n e x p e -
r i e n c e w i t h o t h e r c o n s t a n t s , w e c o n j e c t u r e d t h a t
t h i s c o n s t a n t s a t i s e s a r e l a t i o n i n v o l v i n g h o m o g e -
n e o u s c o m b i n a t i o n s o f ( 2 ) , ( 3 ) , ( 4 ) ,
( 5 ) , l n ( 2 ) ,
L i
4
(
1
2
) a n d L i
5
(
1
2
) , w h e r e L i
n
( x ) =
P
1
k = 1
x
k
k
n
d e -
n o t e s t h e p o l y l o g a r i t h m f u n c t i o n . T h e n u m e r i c a l
v a l u e s o f t h e s e c o n s t a n t s , t o 3 0 d e c i m a l d i g i t s , a r e :
( 2 ) = 1
: 6 4 4 9 3 4 0 6 6 8 4 8 2 2 6 4 3 6 4 7 2 4 1 5 1 6 6 6 4 6 : : :
( 3 ) = 1
: 2 0 2 0 5 6 9 0 3 1 5 9 5 9 4 2 8 5 3 9 9 7 3 8 1 6 1 5 1 1 : : :
( 4 ) = 1
: 0 8 2 3 2 3 2 3 3 7 1 1 1 3 8 1 9 1 5 1 6 0 0 3 6 9 6 5 4 1 : : :
( 5 ) = 1
: 0 3 6 9 2 7 7 5 5 1 4 3 3 6 9 9 2 6 3 3 1 3 6 5 4 8 6 4 5 7 : : :
l n ( 2 ) = 0 :
6 9 3 1 4 7 1 8 0 5 5 9 9 4 5 3 0 9 4 1 7 2 3 2 1 2 1 4 5 8 : : :
L i
4
(
1
2
) = 0 :
5 1 7 4 7 9 0 6 1 6 7 3 8 9 9 3 8 6 3 3 0 7 5 8 1 6 1 8 9 8 : : :
L i
5
(
1
2
) = 0 :
5 0 8 4 0 0 5 7 9 2 4 2 2 6 8 7 0 7 4 5 9 1 0 8 8 4 9 2 5 8 : : :
T h e t e r m s i n v o l v i n g t h e s e c o n s t a n t s w i t h d e g r e e
v e ( s e e S e c t i o n 7 f o r t h e d e n i t i o n o f t h i s t e r m )
a r e t h e f o l l o w i n g : L i
5
(
1
2
) , L i
4
(
1
2
) l n ( 2 ) , l n
5
( 2 ) , ( 5 ) ,
( 4 ) l n ( 2 ) ,
( 3 ) l n
2
( 2 ) , ( 2 ) l n
3
( 2 ) , ( 2 )
( 3 ) . W h e n
s
a
( 2 ;
3 ) i s a u g m e n t e d w i t h t h i s s e t o f t e r m s , a l l
c o m p u t e d t o 1 3 5 d e c i m a l d i g i t s a c c u r a c y , a n d t h e
r e s u l t i n g v e c t o r o f l e n g t h 9 i s i n p u t t o t h e P S L Q
a l g o r i t h m , t h e r e l a t i o n ( 4 8 0 ; 1 9 2 0 ; 0 ;
1 6 ;
2 5 5 ;
6 6 0 ;
8 4 0
; 1 6 0
; 3 6 0 ) i s d e t e c t e d a t i t e r a t i o n 3 9 0 . S o l v -
i n g t h i s r e l a t i o n f o r s
a
( 2 ;
3 ) , w e o b t a i n t h e f o r m u l a
s
a
( 2 ;
3 ) = 4 L i
5
(
1
2
)
1
3 0
l n
5
( 2 )
1 7
3 2
( 5 )
1 1
8
( 4 ) l n ( 2 )
+
7
4
( 3 ) l n
2
( 2 ) +
1
3
( 2 ) l n
3
( 2 )
3
4
( 2 )
( 3 )
= 4 L i
5
(
1
2
)
1
3 0
l n
5
( 2 )
1 7
3 2
( 5 ) +
1 1
7 2 0
4
l n ( 2 )
+
7
4
( 3 ) l n
2
( 2 ) +
1
1 8
2
l n
3
( 2 )
3
2 4
2
( 3 )
( r e c a l l t h a t
( 2 n
) = ( 2 )
2 n
j B
2 n
j = ( 2 ( 2 n ) ! ) f o r
n i n -
t e g e r ) .
W h e n t h e r e l a t i o n i s d e t e c t e d , t h e m i n i m u m a n d
m a x i m u m y
v e c t o r e n t r i e s a r e 1 :
6 0
1 0
1 3 4
a n d
5 : 9 8
1 0
2 9
, w h o s e r a t i o i s o f t h e o r d e r o f 1 0
1 0 5
.
T h u s t h e c o n d e n c e l e v e l o f t h i s d e t e c t i o n i s v e r y
h i g h .
A l t h o u g h w e w e r e u s i n g 1 3 5 - d i g i t i n p u t v a l u e s
a n d 1 5 0 - d i g i t w o r k i n g p r e c i s i o n w h e n w e r s t d e -
t e c t e d t h i s r e l a t i o n , t h e f a c t t h a t t h e m a x i m u m
y - v e c t o r e n t r y i s o n l y 1 0
2 9
a t d e t e c t i o n i m p l i e s
t h a t s u c h h i g h l e v e l s o f n u m e r i c p r e c i s i o n a r e n o t
r e q u i r e d i n t h i s c a s e . I n d e e d , t h e r e l a t i o n c a n b e
s u c c e s s f u l l y d e t e c t e d b y t h e P S L Q a l g o r i t h m u s -
i n g o n l y 5 0 - d i g i t i n p u t v a l u e s a n d 5 0 - d i g i t w o r k i n g
p r e c i s i o n .
6. EXPERIMENTAL RESULTS
M a n y s p e c i a l c a s e s o f t h e p r o v e n r e s u l t s l i s t e d i n
T a b l e 2 w e r e r s t o b t a i n e d u s i n g t h e e x p e r i m e n t a l
m e t h o d p r e s e n t e d i n S e c t i o n s 2 - 4 . I n a d d i t i o n , w e
h a v e o b t a i n e d a n u m b e r o f e x p e r i m e n t a l i d e n t i t i e s
f o r w h i c h f o r m a l p r o o f s h a v e n o t y e t b e e n f o u n d .
T a b l e 3 l i s t s s o m e o f t h e m .
I t s h o u l d b e e m p h a s i z e d t h a t t h e r e s u l t s i n T a -
b l e 3 a r e n o t e s t a b l i s h e d i n a n y r i g o r o u s m a t h e m a t -
i c a l s e n s e b y t h e s e c a l c u l a t i o n s . H o w e v e r , i n e a c h
c a s e t h e \ c o n d e n c e l e v e l " ( s e e S e c t i o n 3 ) o f t h e s e
d e t e c t i o n s i s b e t t e r t h a n 1 0
5 0
, a n d i n m o s t c a s e s
i s i n t h e n e i g h b o r h o o d o f 1 0
1 0 0
. T a b l e 3 , t o g e t h e r
w i t h t h e r e s u l t s i n B o r w e i n e t a l . 1 9 9 4 ] , g i v e s a l l
v a l u e s o f s
h
( m ; n ) f o r
m + n 7 a n d
m + n = 9 , a n d
a l l v a l u e s o f t h e a l t e r n a t i n g s u m s f o r m + n
5 .
S o m e o f t h e s e i d e n t i t i e s c a n b e p r o v e d b y a d h o c
m e t h o d s , b a s e d o n L e w i n 1 9 8 1 ] ; t h e y a r e i n d i c a t e d
w i t h a n a s t e r i s k .
I n m a n y o t h e r c a s e s w e w e r e n o t a b l e t o o b t a i n
a f o r m u l a f o r t h e E u l e r s u m e x p l i c i t l y i n t e r m s o f
v a l u e s o f t h e R i e m a n n z e t a , l o g a r i t h m a n d p o l y -
l o g a r i t h m f u n c t i o n s , b u t w e w e r e a b l e t o o b t a i n
r e l a t i o n s i n v o l v i n g t w o o r m o r e E u l e r s u m s o f t h e
s a m e d e g r e e ( w h e r e b y \ d e g r e e " w e m e a n t h e s u m
o f t h e i n d i c e s m
a n d n
o f t h e c o n s t a n t ) . S o m e o f
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26 Experimental Mathematics, Vol. 3 (1994), No. 1
s
h
( 3 ;
2 ) =
1 5
2
( 5 ) +
( 2 )
( 3 )
s
h
( 3 ;
3 ) =
3 3
1 6
( 6 ) + 2
2
( 3 )
s
h
( 3 ;
4 ) =
1 1 9
1 6
( 7 )
3 3
4
( 3 )
( 4 ) + 2
( 2 )
( 5 )
s
h
( 3 ;
6 ) =
1 9 7
2 4
( 9 )
3 3
4
( 4 )
( 5 )
3 7
8
( 3 )
( 6 ) +
3
( 3 ) + 3
( 2 )
( 7 )
s
h
( 4 ;
2 ) =
8 5 9
2 4
( 6 ) + 3
2
( 3 )
s
h
( 4 ;
3 ) =
1 0 9
8
( 7 ) +
3 7
2
( 3 )
( 4 )
5 ( 2 )
( 5 )
s
h
( 4 ;
5 ) =
2 9
2
( 9 ) +
3 7
2
( 4 )
( 5 ) +
3 3
4
( 3 )
( 6 )
8
3
3
( 3 ) 7
( 2 )
( 7 )
s
h
( 5 ;
2 ) =
1 8 5 5
1 6
( 7 ) + 3 3
( 3 )
( 4 ) +
5 7
2
( 2 )
( 5 )
s
h
( 5 ;
4 ) =
8 9 0
9
( 9 ) + 6 6
( 4 )
( 5 )
4 2 9 5
2 4
( 3 )
( 6 )
5
3
( 3 ) +
2 6 5
8
( 2 )
( 7 )
s
h
( 6 ;
3 ) =
3 0 7 3
1 2
( 9 )
2 4 3
( 4 )
( 5 ) +
2 0 9 7
4
( 3 )
( 6 ) +
6 7
3
3
( 3 )
6 5 1
8
( 2 )
( 7 )
s
h
( 7 ;
2 ) =
1 3 4 7 0 1
3 6
( 9 ) +
1 5 6 9 7
8
( 4 )
( 5 ) +
2 9 5 5 5
2 4
( 3 )
( 6 ) + 5 6
3
( 3 ) +
3 2 8 7
4
( 2 )
( 7 )
s
a
( 2 ;
2 ) = 6 L i
4
(
1
2
) +
1
4
l n
4
( 2 )
2 9
8
( 4 ) +
3
2
( 2 ) l n
2
( 2 )
s
a
( 2 ;
3 ) = 4 L i
5
(
1
2
)
1
3 0
l n
5
( 2 )
1 7
3 2
( 5 )
1 1
8
( 4 ) l n ( 2 ) +
7
4
( 3 ) l n
2
( 2 ) +
1
3
( 2 ) l n
3
( 2 )
3
4
( 2 )
( 3 )
s
a
( 3 ;
2 ) =
2 4 L i
5
(
1
2
) + 6 l n ( 2 ) L i
4
(
1
2
) +
9
2 0
l n
5
( 2 ) +
6 5 9
3 2
( 5 )
2 8 5
1 6
( 4 ) l n ( 2 ) +
5
2
( 2 ) l n
3
( 2 ) +
1
2
( 2 )
( 3 )
a
h
( 2 ;
2 ) =
2 L i
4
(
1
2
)
1
1 2
l n
4
( 2 ) +
9 9
4 8
( 4 )
7
4
( 3 ) l n ( 2 ) +
1
2
( 2 ) l n
2
( 2 )
a
h
( 2 ;
3 ) =
4 L i
5
(
1
2
) 4 l n ( 2 ) L i
4
(
1
2
)
2
1 5
l n
5
( 2 ) +
1 0 7
3 2
( 5 )
7
4
( 3 ) l n
2
( 2 ) +
2
3
( 2 ) l n
3
( 2 ) +
3
8
( 2 )
( 3 )
a
h
( 3 ;
2 ) = 6 L i
5
(
1
2
) + 6 l n ( 2 ) L i
4
(
1
2
) +
1
5
l n
5
( 2 )
3 3
8
( 5 ) +
2 1
8
( 3 ) l n
2
( 2 )
( 2 ) l n
3
( 2 )
1 5
1 6
( 2 )
( 3 )
a
a
( 2 ;
2 ) =
4 L i
4
(
1
2
)
1
6
l n
4
( 2 ) +
3 7
1 6
( 4 ) +
7
4
( 3 ) l n ( 2 )
2 ( 2 ) l n
2
( 2 )
a
a
( 2 ;
3 ) = 4 l n ( 2 ) L i
4
(
1
2
) +
1
6
l n
5
( 2 )
7 9
3 2
( 5 ) +
1 1
8
( 4 ) l n ( 2 )
1 ( 2 ) l n
3
( 2 ) +
3
8
( 2 )
( 3 )
a
a
( 3 ;
2 ) = 3 0 L i
5
(
1
2
)
1
4
l n
5
( 2 )
1 8 1 3
6 4
( 5 ) +
2 8 5
1 6
( 4 ) l n ( 2 ) +
2 1
8
( 3 ) l n
2
( 2 )
7
2
( 2 ) l n
3
( 2 ) +
3
4
( 2 )
( 3 )
TABLE 3.E x p e r i m e n t a l l y d e t e c t e d i d e n t i t i e s . T h o s e m a r k e d w i t h a n a s t e r i s k h a v e b e e n f o r m a l l y p r o v e d .
t h e s e r e l a t i o n s a r e s h o w n i n T a b l e 4 . T h i s i s n o t
a c o m p l e t e l i s t ; w e h a v e o b t a i n e d n u m e r o u s o t h e r
r e l a t i o n s o f t h i s t y p e . T h e c o n d e n c e l e v e l o f e a c h
o f t h e s e r e l a t i o n s i s b e t t e r t h a n 1 0
2 5
. T h e s e r e l a -
t i o n s a r e n o n r e d u n d a n t , i n t h e s e n s e t h a t f o r e a c h
o n e n o r e l a t i o n i n v o l v i n g f e w e r c o n s t a n t s c a n b e
d e t e c t e d ( w e c h e c k t h i s b y r e p e a t i n g t h e r u n s w i t h
e a c h c o n s t a n t r e m o v e d i n t u r n ) .
I n s t i l l o t h e r c a s e s w e w e r e n o t s u c c e s s f u l i n n d -
i n g r e l a t i o n s , b u t w e w e r e a b l e t o o b t a i n b o u n d r e -
s u l t s f r o m t h e P S L Q p r o g r a m t h a t e x c l u d e a l a r g e
c l a s s o f p o t e n t i a l r e l a t i o n s a m o n g t h e l i s t o f c a n d i -
d a t e t e r m s . T h e s e r e s u l t s d o n o t c o n c l u s i v e l y p r o v e
t h a t t h e r e i s n o s u c h r e l a t i o n , o n l y t h a t i f o n e e x -
i s t s , t h e E u c l i d e a n n o r m o f i t s c o e c i e n t s m u s t b e
l a r g e r t h a n a c e r t a i n b o u n d . S o m e o f t h e s e \ n e g a -
t i v e " r e s u l t s a r e l i s t e d i n T a b l e 5 .
O n e i n t e r e s t i n g b y p r o d u c t o f t h e b o u n d r e s u l t s
i n T a b l e 5 i s t h a t t h e r e a r e n o m o d e s t - s i z e d i n t e -
g e r r e l a t i o n s a m o n g h o m o g e n e o u s p r o d u c t s o f ( k )
w i t h d e g r e e 1 2 o r l e s s ( s e e S e c t i o n 7 ) , e x c e p t o f
c o u r s e t h e w e l l - k n o w n r e l a t i o n s w h e n a l l k
a r e e v e n
i n t e g e r s .
T h e b o u n d r e s u l t f o r a
h
( 1 ;
5 ) i n T a b l e 5 c o n -
r m s t h e o b s e r v a t i o n i n B o r w e i n e t a l . 1 9 9 4 ] t h a t
a
h
( 1 ; n ) , w h i c h e q u a l s
h
( 1 ; n ) , d o e s n o t a p p e a r t o
p o s s e s s a n e x p l i c i t e v a l u a t i o n w h e n n
i s o d d a n d
g r e a t e r t h a n t h r e e . T h e b o u n d r e s u l t s f o r
h
( 2 ;
6 )
a n d
h
( 2 ;
8 ) c o n r m t h e o b s e r v a t i o n i n B o r w e i n e t
a l . 1 9 9 4 ] t h a t
h
( 2 ; n ) d o e s n o t a p p e a r t o p o s s e s s
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Bailey, Borwein and Girgensohn: Experimental Evaluation of Euler Sums 27
8 4 5 4 9 s
h
( 1 ;
7 ) + 2 1 1 4 6 8 s
h
( 2 ;
6 ) + 1 4 8 9 0 2 s
h
( 3 ;
5 ) 1 3 3 6 0 s
h
( 4 ;
4 ) 1 9 7 8 s
h
( 5 ;
3 )
1 2 7
( 8 ) + 3 3 6
( 3 )
( 5 )
1 2 0
( 2 )
2
( 3 )
2 4 s
h
( 2 ;
6 )
9 6 s
h
( 3 ;
5 )
2 7 1 8 5 8 7 s
h
( 1 ;
8 ) 1 6 4 5 2 5 6 6 4 s
h
( 2 ;
7 ) 1 7 8 0 4 2 9 4 4 s
h
( 3 ;
6 ) 8 8 9 4 7 8 6 2 s
h
( 4 ;
5 ) + 3 8 6 3 9 4 0 s
h
( 5 ;
4 ) + 6 7 2 1 0 0 s
h
( 6 ;
3 )
5 1 3 8 s
h
( 1 ;
8 ) 5 6 6 6 5 6 s
h
( 2 ;
7 ) 6 2 4 0 1 6 s
h
( 3 ;
6 ) 3 1 6 9 8 8 s
h
( 4 ;
5 ) + 6 4 8 0 s
h
( 5 ;
4 ) + 3 3 6 0 5
( 3 )
( 6 )
1 4 2 6 9 4 0 8 s
h
( 1 ;
9 ) + 2 5 7 8 4 7 0 s
h
( 2 ;
8 ) + 2 8 1 5 3 7 6 s
h
( 3 ;
7 ) + 5 8 1 4 5 5 0 s
h
( 4 ;
6 )
+ 6 2 3 8 8 8 4 s
h
( 5 ;
5 ) + 3 9 3 8 9 1 2 s
h
( 6 ;
4 ) + 1 1 2 2 7 8 4 s
h
( 7 ;
3 ) 1 8 6 0 s
h
( 8 ;
2 ) + 6 3 1 6 4 2 8 5
( 1 0 )
3 2 1 ( 1 0 )
4 4 0
2
( 5 )
7 2 0
( 3 )
( 7 )
8 0
2
( 3 )
( 4 ) + 5 6 0
( 2 )
( 3 )
( 5 )
4 0 s
h
( 2 ;
8 ) + 1 6 0 s
h
( 3 ;
7 )
1 6 9 1 7 5 5 5 0 3 s
h
( 1 ;
1 0 ) 3 1 7 2 5 8 9 6 8 8 s
h
( 2 ;
9 ) + 8 3 7 5 1 1 5 0 4 s
h
( 3 ;
8 ) 7 3 0 2 7 1 7 5 7 6 s
h
( 4 ;
7 )
1 3 9 5 8 6 6 0 0 1 6 s
h
( 5 ;
6 ) 1 2 9 1 0 4 6 6 0 6 4 s
h
( 6 ;
5 ) 7 0 9 9 3 3 2 9 1 2 s
h
( 7 ;
4 ) 1 7 7 3 2 1 2 6 8 8 s
h
( 8 ;
3 )
+ 6 5 8 3 6 0 s
h
( 9 ;
2 ) + 5 3 4 9 1 4 3 4 6 7 9 ( 1 1 ) 2 1 8 6 8 2 4 8 9 7 1
( 2 )
( 9 )
5 8 9
( 1 1 ) + 3 2 2
( 5 )
( 6 ) + 7 5 6
( 4 )
( 7 ) + 2 5 4
( 3 )
( 8 )
3 3 6
2
( 3 )
( 5 )
3 6 8
( 2 )
( 9 ) + 8 0
( 2 )
3
( 3 )
1 6 s
h
( 3 ;
8 )
4 8 s
h
( 4 ;
7 )
7 0 6 6 3 ( 1 2 ) 1 6 5 8 4 0 ( 5 )
( 7 )
1 2 1 6 1 6 ( 3 )
( 9 )
3 3 1 6 8
2
( 3 )
( 6 ) + 5 5 2 8
4
( 3 )
+ 4 9 7 5 2
( 2 )
2
( 5 ) + 9 9 5 0 4
( 2 )
( 3 )
( 7 ) 1 6 5 8 4 s
h
( 2 ;
1 0 ) + 2 2 1 1 2 s
h
( 3 ;
9 )
1 1 5 2 s
a
( 2 ;
4 ) + 6 4 0 s
a
( 3 ;
3 )
7 6 8 0 l n ( 2 ) L i
5
( 1 =
2 ) + 6 4 l n
6
( 2 ) 1 8 8 1
( 6 )
+ 7 4 4 0
( 5 ) l n ( 2 ) 1 6 8 0
( 4 ) l n
2
( 2 ) 1 1 2 0
( 3 ) l n
3
( 2 ) + 8 6 4
( 3 )
( 3 )
6 4 0
( 2 ) l n
4
( 2 )
4 3 2
( 2 )
( 3 ) l n ( 2 )
TABLE 4.E x p r e s s i o n s t h a t h a v e b e e n e x p e r i m e n t a l l y f o u n d t o v a n i s h .
a n e x p l i c i t e v a l u a t i o n f o r n
e v e n a n d g r e a t e r t h a n
f o u r .
T h e n u m e r i c a l v a l u e s o f t h e v a r i o u s E u l e r s u m
c o n s t a n t s , w h i c h w e r e u s e d t o o b t a i n t h e r e s u l t s
l i s t e d i n T a b l e s 3 { 5 , w e r e c o m p u t e d a s d e s c r i b e d i n
S e c t i o n s 2 a n d 3 . T h e e x p l i c i t s u m i n f o r m u l a ( 2 . 2 )
w a s c o m p u t e d u s i n g A l g o r i t h m 1 ( o r i t s e q u i v a l e n t
f o r a l t e r n a t i n g s u m s ) , o n t h e I n t e l P a r a g o n p a r a l l e l
c o m p u t e r s y s t e m , w i t h P
= 1 2 8 , J
= 2
1 3
, I = 1 0 0 .
T h e s y m b o l i c o p e r a t i o n s i n d i c a t e d i n ( 2 . 2 ) w e r e
p e r f o r m e d i n M a p l e . T h e n a l n u m e r i c a l v a l u e s
w e r e c h e c k e d b y c o m p a r i n g t h e m w i t h t h e v a l u e s
o b t a i n e d f r o m A l g o r i t h m 1 ( o r e q u i v a l e n t ) w i t h
I = 9 9 i n s t e a d o f 1 0 0 .
7. CONJECTURES
I t i s n o t k n o w n w h e t h e r c l o s e d - f o r m e v a l u a t i o n s
o f t h e t y p e l i s t e d i n T a b l e 3 e x i s t f o r a l l o f t h e
v a r i o u s c l a s s e s o f E u l e r s u m s s t u d i e d i n t h i s p a p e r .
I t i s p o s s i b l e t h a t s u c h f o r m u l a s a l w a y s e x i s t a n d
c o u l d b e u n c o v e r e d b y t h e t e c h n i q u e s d e s c r i b e d i n
t h i s p a p e r , i f o n e c o u l d o n l y d e d u c e t h e f o r m o f t h e
m i s s i n g t e r m s . W e p r e s e n t t h i s a s a n o p e n q u e s t i o n
f o r f u r t h e r r e s e a r c h .
O n e p r i n c i p l e w e h a v e o b s e r v e d i n t h i s w o r k i s
t h a t i n e v e r y c a s e w h e r e w e h a v e o b t a i n e d a r e -
l a t i o n , t h i s r e l a t i o n h a s a l w a y s i n v o l v e d h o m o g e -
n e o u s t e r m s , i n t h e s e n s e t h a t t h e d e g r e e m + n
o f e a c h t e r m i n v o l v e d i n t h e r e l a t i o n i s t h e s a m e .
( F o r t h e s e p u r p o s e s , t h e d e g r e e o f ( k
) i s t a k e n t o
b e k
, a s i s t h e d e g r e e o f L i
k
(
1
2
) , w h i l e t h a t o f l n ( 2 )
i s t a k e n a s o n e . )
A l t h o u g h w e b e l i e v e t h i s p r i n c i p l e m a y h o l d i n
g e n e r a l , w e h a v e n o i d e a h o w t o p r o v e i t . W e t h e r e -
f o r e p r e s e n t i t a s a c o n j e c t u r e . H o w e v e r , s i n c e i t
i s i m p o r t a n t t o l i m i t t h e n u m b e r o f c o n s t a n t s i n -
p u t t o t h e P S L Q a l g o r i t h m i n o r d e r t o e n h a n c e
t h e p o s s i b i l i t y o f d e t e c t i n g a r e l a t i o n , w e h a v e o f -
t e n u s e d t h i s p r i n c i p l e i n s e l e c t i n g t h e c a n d i d a t e
c o n s t a n t s .
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28 Experimental Mathematics, Vol. 3 (1994), No. 1
S e t o f c o n s t a n t s t h a t d o n o t a p p e a r t o b e l i n k e d b y a r e l a t i o n B o u n d
s
h
( 2 ;
6 ) ; ( 8 ) ; ( 3 )
( 5 ) ; ( 2 )
2
( 3 ) 7:
0 6
1 0
4 1
s
h
( 3 ;
5 ) ; ( 8 ) ; ( 3 )
( 5 ) ; ( 2 )
2
( 3 ) 4:
3 1
1 0
4 1
s
h
( 4 ;
4 ) ; ( 8 ) ; ( 3 )
( 5 ) ; ( 2 )
2
( 3 ) 1:
6 3
1 0
4 0
s
h
( 2 ;
8 ) ; ( 1 0 )
;
2
( 5 ) ; ( 3 )
( 7 ) ;
2
( 3 )
( 4 ) ; ( 2 )
( 3 )
( 5 ) 1 :
2 8
1 0
2 6
s
h
( 3 ;
7 ) ; ( 1 0 )
;
2
( 5 ) ; ( 3 )
( 7 ) ;
2
( 3 )
( 4 ) ; ( 2 )
( 3 )
( 5 ) 3 :
0 3
1 0
2 6
s
h
( 4 ;
6 ) ; ( 1 0 )
;
2
( 5 ) ; ( 3 )
( 7 ) ;
2
( 3 )
( 4 ) ; ( 2 )
( 3 )
( 5 ) 3 :
3 3
1 0
2 4
s
h
( 3 ;
8 ) ; ( 1 1 )
; ( 5 )
( 6 ) ; ( 4 )
( 7 ) ; ( 3 )
( 8 ) ;
2
( 3 )
( 5 ) ; ( 2 )
( 9 ) ; ( 2 )
3
( 3 ) 2 :
0 1
1 0
1 7
s
h
( 4 ;
7 ) ; ( 1 1 )
; ( 5 )
( 6 ) ; ( 4 )
( 7 ) ; ( 3 )
( 8 ) ;
2
( 3 )
( 5 ) ; ( 2 )
( 9 ) ; ( 2 )
3
( 3 ) 1 :
8 9
1 0
1 7
s
h
( 2 ;
1 0 ) ; ( 1 2 ) ; ( 5 )
( 7 ) ; ( 3 )
( 9 ) ; ( 3 )
( 4 )
( 5 ) ;
2
( 3 )
( 6 ) ;
4
( 3 ) ; ( 2 )
2
( 5 ) ; ( 2 )
( 3 )
( 7 ) 1 :
4 3
1 0
1 5
s
h
( 3 ;
9 ) ; ( 1 2 )
; ( 5 )
( 7 ) ; ( 3 )
( 9 ) ; ( 3 )
( 4 )
( 5 ) ;
2
( 3 )
( 6 ) ;
4
( 3 ) ; ( 2 )
2
( 5 ) ; ( 2 )
( 3 )
( 7 ) 8 :
2 1
1 0
1 4
s
h
( 4 ;
8 ) ; ( 1 2 )
; ( 5 )
( 7 ) ; ( 3 )
( 9 ) ; ( 3 )
( 4 )
( 5 ) ;
2
( 3 )
( 6 ) ;
4
( 3 ) ; ( 2 )
2
( 5 ) ; ( 2 )
( 3 )
( 7 ) 1 :
0 6
1 0
1 5
s
h
( 1 ;
9 ) ; s
h
( 2 ;
8 ) ; s
h
( 3 ;
7 ) ; s
h
( 4 ;
6 ) ; s
h
( 5 ;
5 ) ; s
h
( 6 ;
4 ) ; s
h
( 7 ;
3 ) ; s
h
( 8 ;
2 ) 2 :
3 1
1 0
1 6
s
h
( 1 ;
1 0 ) ; s
h
( 2 ;
9 ) ; s
h
( 3 ;
8 ) ; s
h
( 4 ;
7 ) ; s
h
( 5 ;
6 ) ; s
h
( 6 ;
5 ) ; s
h
( 7 ;
4 ) ; s
h
( 8 ;
3 ) ; s
h
( 9 ;
2 ) 1 :
0 5
1 0
1 5
s
h
( 1 ;
1 0 ) ; s
h
( 2 ;
9 ) ; s
h
( 3 ;
8 ) ; s
h
( 4 ;
7 ) ; s
h
( 5 ;
6 ) ; s
h
( 6 ;
5 ) ; s
h
( 7 ;
4 ) ; s
h
( 8 ;
3 ) ; s
h
( 9 ;
2 ) ; ( 1 1 ) 6 :
5 4
1 0
1 3
s
h
( 1 ;
1 1 ) ; s
h
( 2 ;
1 0 ) ; s
h
( 3 ;
9 ) ; s
h
( 4 ;
8 ) ; s
h
( 5 ;
7 ) ; s
h
( 6 ;
6 ) ; s
h
( 7 ;
5 ) ; s
h
( 8 ;
4 ) ; s
h
( 9 ;
3 ) ; s
h
( 1 0 ;
2 ) 6 :
7 7
1 0
1 3
s
h
( 1 ;
1 1 ) ; s
h
( 2 ;
1 0 ) ; s
h
( 3 ;
9 ) ; s
h
( 4 ;
8 ) ; s
h
( 5 ;
7 ) ; s
h
( 6 ;
6 ) ; s
h
( 7 ;
5 ) ; s
h
( 8 ;
4 ) ; s
h
( 9 ;
3 ) ; s
h
( 1 0 ;
2 ) ; ( 1 2 ) 2 :
6 7
1 0
1 1
a
h
( 1 ;
5 ) p l u s A
6
( s e e c a p t i o n ) 7 :
2 9
1 0
1 0
s
a
( 2 ;
4 ) p l u s A
6
6 : 0 8
1 0
1 0
s
a
( 3 ;
3 ) p l u s A
6
5 : 9 5
1 0
1 0
s
a
( 2 ;
5 ) p l u s A
7
2 : 6 3
1 0
6
s
a
( 3 ;
4 ) p l u s A
7
4 : 7 3
1 0
6
s
a
( 2 ;
5 ) ; s
a
( 3 ;
4 ) ; s
a
( 4 ;
3 ) ; s
a
( 5 ;
2 ) p l u s A
7
3 : 1 6
1 0
5
h
( 2 ;
6 ) ; ( 8 ) ; ( 3 )
( 5 ) ; ( 2 )
2
( 3 ) 6:
8 1
1 0
4 1
h
( 3 ;
5 ) ; ( 8 ) ; ( 3 )
( 5 ) ; ( 2 )
2
( 3 ) 6:
2 6
1 0
4 1
h
( 2 ;
8 ) ; ( 1 0 )
;
2
( 5 ) ; ( 3 )
( 7 ) ;
2
( 3 )
( 4 ) ; ( 2 )
( 3 )
( 5 ) 3 :
9 2
1 0
2 6
h
( 3 ;
7 ) ; ( 1 0 )
;
2
( 5 ) ; ( 3 )
( 7 ) ;
2
( 3 )
( 4 ) ; ( 2 )
( 3 )
( 5 ) 2 :
7 8
1 0
2 4
TABLE 5.R e l a t i o n e x c l u s i o n b o u n d s . T h e r i g h t - h a n d c o l u m n g i v e s t h e m i n i m u m E u c l i d e a n n o r m o f a n y
p o s s i b l e i n t e g e r r e l a t i o n i n v o l v i n g t h e l i s t e d c o n s t a n t s . T h e a b b r e v i a t i o n s u s e d a r e : A
6
= f L i
6
(
1
2
) ; l n ( 2 ) L i
5
(
1
2
) ;
l n
2
( 2 ) L i
4
(
1
2
) ; l n
6
( 2 ) ; ( 6 ) ; ( 5 ) l n ( 2 ) ; ( 4 ) l n
2
( 2 ) ; ( 3 ) l n
3
( 2 ) ;
2
( 3 ) ; ( 2 ) l n
4
( 2 ) ; ( 2 )
( 3 ) l n ( 2 ) g ; A
7
= f L i
7
(
1
2
) ;
l n ( 2 ) L i
6
(
1
2
) ; l n
2
( 2 ) L i
5
(
1
2
) ; l n
3
( 2 ) L i
4
(
1
2
) ; l n
7
( 2 ) ; ( 7 ) ; ( 6 ) l n ( 2 ) ; ( 5 ) l n
2
( 2 ) ; ( 4 ) l n
3
( 2 ) ; ( 3 ) L i
4
(
1
2
) ; ( 3 ) l n
4
( 2 ) ;
( 3 )
( 4 ) ;
2
( 3 ) l n ( 2 ) ; ( 2 ) L i
5
(
1
2
) ; ( 2 ) l n
5
( 2 ) ; ( 2 )
( 5 ) ; ( 2 )
( 3 ) l n
2
( 2 ) g .
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Bailey, Borwein and Girgensohn: Experimental Evaluation of Euler Sums 29
T h e m o d u l a r p r o p e r t i e s o f t h e s u m s
h
( m ; n ) a r e
b e i n g i n v e s t i g a t e d b y D . Z a g i e r ( p r i v a t e c o m m u n i -
c a t i o n ) . H i s w o r k p r o v i d e s a n a l t e r n a t e , a b s t r a c t
p r o o f t h a t
h
( m ; n ) e v a l u a t e s i n t e r m s o f z e t a f u n c -
t i o n s i f m + n
i s o d d . T h i s c o r r e s p o n d s t o o u r r e -
s u l t s a n d s t i l l l e a v e s t h e c a s e m + n
e v e n a s a n o p e n
p r o b l e m .
ACKNOWLEDGMENTS
T h e a u t h o r s w i s h t o t h a n k D a v i d B o r w e i n , P e -
t e r B o r w e i n , H e l a m a n R . P . F e r g u s o n a n d P a u l
O . F r e d e r i c k s o n f o r v a l u a b l e d i s c u s s i o n s d u r i n g t h e
c o u r s e o f t h i s w o r k .
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c o n t a i n i n g ( x )
( y
) a n d ( ( x )
( y
) )
2
f o r c e r t a i n
v a l u e s o f x
a n d y
, " J . C o m p . A p p l . M a t h . 3 7 ( 1 9 9 1 ) ,
1 2 5 { 1 4 1 .
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T 0 4 5 - 1 , M o e t t F i e l d , C A 9 4 0 3 5 - 1 0 0 0 .
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f o r n
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30 Experimental Mathematics, Vol. 3 (1994), No. 1
D a v i d H . B a i l e y , N A S A p p l i e d R e s e a r c h B r a n c h , N A S A A m e s R e s e a r c h C e n t e r , M o e t t F i e l d , C A 9 4 0 3 5 - 1 0 0 0 ,
U S A ( d b a i l e y @ n a s . n a s a . g o v )
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C a n a d a ( j b o r w e i n @ c e c m . s f u . c a )
R o l a n d G i r g e n s o h n , D e p a r t m e n t o f M a t h e m a t i c s a n d S t a t i s t i c s , S i m o n F r a s e r U n i v e r s i t y , B u r n a b y , B C V 5 A 1 S 6 ,
C a n a d a ( g i r g e n @ c e c m . s f u . c a )
R e c e i v e d O c t o b e r 2 5 , 1 9 9 3 ; a c c e p t e d i n r e v i s e d f o r m M a y 1 9 , 1 9 9 4
