MA THEMA TICS: C . A . NICOL o f contact. I every matrix X o A ,uoB has a multiple characteristic root i t follows that C has a component which h a s t o b e counted double. Detailed proofs a nd extensions t o fields o f finite characteristic will appear elsewhere. I Motzkin, T . S . , a n d Taussky, Olga, Pairs o f Matrices with Property L, Trans. Amer. Math. Soc., 7 3 , 108-114, (1952). ON RESTRICTED PARTITIONS AND A GENERALIZATION O F THE EULER p NUMBER AND THE MOEBIUSFUNCTION B Y CHARLES A . NICOL UNIVERSITY O F TEXAS Communicated By H . S . Vandiver, June 2 3 , 1953 Introduction.-In the present paper w e shall treat the function n - I z , x ) I I I (z-x8) ( 1 ) s = 1 mainly from a n arithmetic standpoint. Functions o this type have been studied extensively i n the theory o f partitions o f positive integers' where z i s replaced b y 1 o r - 1 a n d the range o f t h e product i s infinite. F o r example, a famous result d u e t o Euler f o r l x i < 1 m ay b e written a s co ( 1 X 1 2 1 3 . . . = 1 E _j nX 1/i n 3n+1 2 n= X Th e coefficients o f t h e series admit t h e following combinatorial interpre- tation. I f E(n) denotes the number o f partitions o f n into a n even number o f unequal parts a n d U(n) t h e number o f partitions o f n into a n o d d number o f unequal parts then ( 2 ) m ay b e stated i n t he following way. E(n) = U(n) except when n = 2 k(3k 1 , when E(n) U(n) = (-1)k 2 T h e case where w e have only a finite number o f terms i n ( 2 ) h a s received comparatively little attention. I f in ( 1 ) w e l e t z = 1 w e have n- 1 F,,- 1 , x ) = I I (1 Xe). (3) s = 1 T h e coefficient o f x k f o r k < n will b e E(k) U(k). Bu t t h e coefficient o f x k f o r k > n i s the number o f partitions o f k into a n even number o f unequal parts, none o f which i s larger than n , minus t h e number o f partitions o f k VOL. 3 9 , 1953 9 6 3
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7/27/2019 Pnas01594-0071 Restricted Partitions and Generalizations of the Euler Number and the Moebius Function
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MATHEMA T I C S : C . A . NICOL
o f c o n t a c t . I f e v e r y m a t r i x X o A + , u o B h a s a m u l t i p l e c h a r a c t e r i s t i c r o o t
i t f o l l o w s t h a t C h a s a c o m p o n e n t w h i c h h a s t o b e c o u n t e d d o u b l e .
D e t a i l e d p r o o f s a n d e x t e n s i o n s t o f i e l d s o f f i n i t e c h a r a c t e r i s t i c w i l l a p p e a r
e l s e w h e r e .
I M o t z k i n , T . S . , a n d T a u s s k y , O l g a , " P a i r s o f M a t r i ce s w i t h P r o p e r t y L , " T r a n s .
A m e r . M a t h . S o c . , 7 3 , 1 0 8 - 1 1 4 , ( 1 9 5 2 ) .
ON RESTRICTED PARTITIONS AND A GENERALIZATION OFTHE EULER p NUMBER AND THE MOEBIUS FUNCTION
BY CHARLES A . N I C O L
U N I V E R S I T Y OF TEXAS
Communicated By H . S . V a n d i v e r , J u n e 2 3 , 1 9 5 3
I n t r o d u c t i o n . - I n t h e p r e s e nt p a p e r we s h a l l t r e a t t h e f u n c t i o n
n-I
( z , x ) I I I ( z - x 8 ) ( 1 )s = 1
m a i n l y f r o m a n a r i t h m e t i c s t a n d p o i n t .F u n c t i o n s o f t h i s t y p e h a v e b e e n s t u d i e d e x t e n s i v e l y i n t h e t h e o r y o f
p a r t i t i o n s o f p o s i t i v e i n t e g e r s ' w h e r e z i s r e p l a c e d b y 1 o r - 1 a n d t h e r a n g e
o f t h e p r o d u c t i s i n f i n i t e . F o r e x a m p l e , a f a m o u s r e s u l t d u e t o E u l e r f o r
l x i < 1 m a y b e w r i t t e n a s
co
( 1 - X ) ( 1 - 2 ) ( 1 - 3 ) . . . = 1 + E ( _ j ) n X ( 1 / i ) n ( 3 n + 1 ) ( 2 )n= -X
T h e c o e f f i c i e n t s o f t h e s e r i e s a d m i t t h e f o l l o w i n g c o m b i n a t o r i a l i n t e r p r e -
t a t i o n . I f E ( n ) d e n o t e s t h e n u m b e r o f p a r t i t i o n s o f n i n t o a n e v e n n u m b e ro f u n e q u a l p a r t s a n d U ( n ) t h e n u m b e r o f p a r t i t i o n s o f n i n t o a n o d d n u m b e r
o f u n e q u a l p a r t s t h e n ( 2 ) m a y b e s t a t e d i n t h e f o l l o w i n g w a y .
E ( n ) = U ( n ) e x c e p t w h e n n = 2 k ( 3 k + 1 ) , w h e n E ( n ) - U ( n ) = ( - 1 ) k2
T h e c a s e w h e r e we h a v e o n l y a f i n i t e n u m b e r o f t e r m s i n ( 2 ) h a s r e c e i v e dc o m p a r a t i v e l y l i t t l e a t t e n t i o n . I f i n ( 1 ) we l e t z = 1 we h a v e
n - 1
F , , - ( 1 , x ) = I I ( 1 -X e ) . ( 3 )s = 1
The c o e f f i c i e n t o f x k f o r k < n w i l l b e E ( k ) - U ( k ) . Bu t t h e c o e f f i c i e n t o fx k f o r k > n i s t h e number o f p a r t i t i o n s o f k i n t o an e v e n n u m b e r o f u n e q u a l
p a r t s , n o n e o f w h i c h i s l a r g e r t h a n n , m i n u s t h e n u m b e r o f p a r t i t i o n s o f k
V O L . 3 9 , 1 9 5 3 9 6 3
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MATHEMA T I C S : C . A . NICOL
i n t o a n o d d n u m b e r o f u n e q u a l p a r t s , n o n e o f w h i c h i s l a r g e r t h a n n . T h i s
t h e n i s a n e x t e n s i o n o f t h e a n a l o g o u s p r o b l e m a r i s i n g i n t h e u s e o f t h e i n -f i n i t e p r o d u c t .
A l s o i f z i s r e p l a c e d b y - 1 we c o n s i d e r t h e f o l l o w i n g f u n c t i o n .
n - i
( - ) n - l F n - _ ( - I x X ) = I ( 1 + x 8 ) . ( 4 )s - i
T h e c o e f f i c i e n t o f x " r e s u l t i n g f r o m t h e e x p a n s i o n o f t h i s p r o d u c t i s t h e
n u m b e r o f p a r t i t i o n s o f k a s a su m o f d i s t i n c t p o s i t i v e i n t e g e r s n o n e o fw h i c h i s l a r g e r t h a n n . T h i s m a y a l s o b e s t a t e d a s t h e n u m b e r o f s o l u t i o n s
o f t h e e q u a t i o n x 1 + 2 x 2 + . . . + ( n -l ) x - i= k , w h e r e f o r i = 1 , 2 , . . . ,( n - 1 ) , x i i s e i t h e r z e r o o r u n i t y .The p r o d u c t s ( 3 ) a n d ( 4 ) h a v e b e e n s t u d i e d b y C a u c h y , T . V a h l e n , v o n
S t e r n e c k , a n d o t h e r s . 2 I n p a r t i c u l a r v o n S t e r n e c k s t u d i e d t h e c a s e w h e r e
t h e p o l y n o m i a l r e s u l t i n g f r o m t h e e x p a n s i o n i s r e d u c e d m o d u l o a p o s i t i v ei n t e g e r .
F u n d a m e n t a l i n t h i s i n v e s t i g a t i o n w i l l b e t h e u s e o f t h e n u m b e r 3
c 1 ( k , n ) =v
( n ) _ A ( n / ( k , n ) ) , ( 5 )( p ( n / ( k , n ) )
w h e r e k a n d n a r e p o s i t i v e i n t e g e r s a n d ( k , n ) d e n o t e s t h e g r e a t e s t c o m m o nd i v i s o r o f k a n d n . I f n i s a p o s i t i v e i n t e g e r s o ( n ) d e n o t e s a s u s u a l t h e
n u m b e r o f p o s i t i v e i n t e g e r s l e s s t h a n n a n d p r i m e t o i t . ( s ( 1 ) = 1 . )A l s o , f o r n a p o s i t i v e i n t e g e r , p ( n ) i s z e r o i f n c o n t a i n s a r e p e a t e d p r i m e
f a c t o r . O t h e r w i s e u 2 ( n ) i s e q u a l t o ( - I ) T w h e r e Y i s t h e n u m b e r o f d i s -t i n c t p r i m e f a c t o r s o f n . ( , u ( 1 ) - 1 . ) N o t e t h a t ( 5 ) r e d u c e s t o , A ( n ) w h e n( k , n ) = 1 a n d p ( n ) w h e n ( k , n ) n . A l t h o u g h ( 5 ) a p p e a r s m o r e c o m p l i -
c a t e d t h a n i t s c o n s t i t u e n t s i t w i l l b e s h o w n t h a t m a n y o f t h e p r i n c i p a lt h e o r e m s c o n c e r n i n g i t a r e h a r d l y m o r e c o m p l e x t h a n t h o s e
i n v o l v i n gt h e
( p or I A n u m b e r a l o n e .T h e p r o p e r t i e s o f t h e c o e f f i c i e n t s i n t h e d e v e l o p m e n t o f ( 1 ) a r e e x t e n s i v e l y
u s e d t o o b t a i n p r o p e r t i e s o f ( 5 ) a n d v i c e v e r s a .We n o w s t a t e a n u m b e r o f t h e o r e m s w i t h o u t p r o o f . We h o p e t o p u b l i s h
t h e p r o o f s e l s e w h e r e .I n t h e f o l l o w i n g p a r a g r a p h s t h e s y m b o l [ x ] w i l l d e n o t e t h e l a r g e s t i n t e g e r
c o n t a i n e d i n t h e r e a l n u m b e r x . A l s o t h e s y m b o l $ ( a , b ) w i l l d e n o t e t h e
n u m b e r d e f i n e d i n ( 5 ) w h e r e a a n d b a r e p o s i t i v e i n t e g e r s .We h a v e i f k , r , a n d n a r e p o s i t i v e i n t e g e r s t h a t 4
E, e x p ( 2 r i r k / n ) = c 1 ( k , n ) , ( 6 )( r , n ) - 1
w h e r e t h e r a n g e o f r i s o v e r a l l p o s i t i v e i n t e g e r s l e s s t h a n n a n d p r i m e t oi t . ( j 2 = - 1 . )
9 6 4 P R O C . N . A . S .
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MATHEMA T I C S : C . A . NICOL
" , W e a l s o h a v e
T H I E O R E M 1 . -
d i nd O o t h e r w i s e .
S i m i l a r l y
( 0 i f ( k , n ) $ n .
> Z ( _ l ) d D ( k , n / d ) = ( - n i f ( k , n ) = n , n e v e n . ( 8 )d i n ( O i f ( k , n ) =n ,n o d d .
We m a y a l s o p r o v eTHEOREM 2 . L e t a . ( k ) d e n o t e t h e sum o f t h e d i v i s o r s o f k l e s s t h a n o r e q u a l
t o n . T h e n ,
n
, [ n / s ] P ( k , s ) = o a n ( k ) . ( 9 )s = 1
I n c a s e k = 1 t h i s b e c o m e s t h e w e l l - k n o w n r e l a t i o n "
n
E [ n / s ] A ( s ) = 1 .
A l s o i f k i s r e p l a c e d b y n ! we h a v e a n o t h e r known r e s u l t 6
n
E [ n / s ] s p ( s ) = n ( n + 1 ) / 2 .s = l
THEOREM 3 . I f 5 n , t h e n
( d / ) - 1 ( k , b / d ) 0 ( m o d . n ) . ( 1 0 )
C O R O L L A R Y . I f n o w p d e n o t e s a n o d d p r i m e a n d a i s a p o s i t i v ei n t e g e r , t h e n
pa) (pa/1) ( m o d . n ) . ( l O a )
THEOREM 4 .
, R , ( d ) c I ( k , n / d ) =0 ( m o d . n ) , ( 1 1 )d l n
w h e r e
R s ( d ) =9d ) ( _ 1 ) a
a n d t h i s sum i s o v e r a l l i n t e g r a l s o l u t i o n s a t o f t h e e q u a t i o n ( n l d ) a + , = s
w h e r e O < s< n , 0 . a < d , 0 . < n / d .
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MA THEMA T I C S : C . A . NICOL
C o n s i d e r n o w t h e f u n c t i o n F , n - ( z , x ) d e f i n e d i n ( 1 ) . I f n > 1 a n d t h e
p r o d u c t i s e x p a n d e d a s a p o l y n o m i a l i n x , we m a y w r i t e
n
F P - 1 ( z , x ) = E P , ( z ) x 8 , ( 1 2 )s=O
w h e r e n i = n ( n - 1 ) / 2 a n d P . ( z ) i s a p o l y n o m i a l i n z . Then we m a y d e f i n et h e p o l y n o m i a l B , ( z ) a s
M g
B j ( z ) = E P k n + t ( Z ) , ( 1 3 )k =O
w h e r e n , = [ ( n - 1 ) / 2 -t / n ] a n d 1 < t < n .
T h e n we m a y o b t a i n
THEOREM 5 . I f z i s a n u m b e r d i f f e r e n t f r o m u n i t y , t h e n
B , ( z ) = 1 ( z n / d - ) d - W ( t , n / d ) , ( 1 4 )n ( z - 1 ) d l n
w h e r e 1 < t < n a n d B t ( z ) i s d e f i n e d i n ( 1 3 ) .D e n o t e t h e p o l y n o m i a l d e f i n e d b y t h e f u n c t i o n F . - 1 ( 1 , x ) i n ( 3 ) b y
E A o x , w h e r e n j = n ( n - 1 ) / 2 . - ( 1 5 )s-O
A l s o d e f i n e t h e n u m b e r C , b y t h e r e l a t i o n
M c
C:=E A k * + : , w h e r e n , = [ ( n - 1 ) / 2 - t / n ] . ( 1 5 a )k =O
T h e n we o b t a i n t h e f o l l o w i n gTHEOREM 6 .
C t = n ( t , ) , ( 1 6 )w h e r e C , i s d e f i n e d i n ( 1 5 a ) .THEOREM 7 .
in
i o ( n ) =- E C 2 , ( 1 7 )n t = l
w h e r e C s i s d e f i n e d i n ( 1 5 a ) .I n v i e w o f t h e o r e m 6 we m a y w r i t e t h e o r e m 7 a s
l i n
s p ( n ) = - E j 1 2 ( t , n ) . ( 1 8 )n t = i
An o b s e r v a t i o n o f p o s s i b l e i n t e r e s t m a y b e made c o n c e r n i n g t h e o r e m
7 i f i t i s n o t e d t h a t C , ( o ( n ) . T h e n ( 1 7 ) b e c o m e s a q u a d r a t i c r e l a t i o n i n( p ( n ) . E m p l o y i n g t h e q u a d r a t i c f o r m u l a we f i n d
9 6 6 P RO C . N . A . S .
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MATHEMA T I C S : C . A . NICOL
( p ( n ) = ( n ' V / n 2 - 4 G ( n ) ) / 2 , ( 1 9 )n - 1
w h e r e G ( n ) = E C.
E x c e p t i n t h e c a s e when s o ( n ) = n / 2 o n l y o n e o f t h e r o o t s o f ( 1 9 ) c o r r e -
s p o n d s t o s p ( n ) . T h e s i g n i f i c a n c e o f t h e r e m a i n i n g r o o t h a s n o t b e e n d e t e r -m i n e d a n d w o u l d s e e m t o b e o f i n t e r e s t .
I f x i s r e p l a c e d b y e x p ( i O ) , w h e r e i 2 = - 1 , we o b t a i n t h e f o l l o w i n g
THEOREM 8 .
4 ( t , n ) =2
f
{ F n _ - ( e x p ( i 6 ) )E
e x p ( - ( k n+ t ) i O ) } d O , ( 2 0 )
n - I
w h e r e F n _ i ( e x p ( i O ) ) = I ( 1 - e x p ( s i O ) ) a n d l n f = [ ( n - 1 ) / 2 - t / n ] .$=1
S i m i l a r l y f o r t h e n u m b e r s A g d e f i n e d i n ( 1 5 ) w e h a v e
1 ( 2wA t = J r 1 F n - l ( e x p ( i O ) ) ( e x p ( - t i O ) ) ) d O . ( 2 1 )
F u r t h e r m o r e t h e n u m b e r s A , d e f i n e d i n ( 1 5 ) h a v e t h e f o l l o w i ng p r o p e r -
t i e s :
n
j AA > n , ( 2 2 )s=O
w h e r e n i = n ( n - 1 ) / 2 .
A r a t h e r u n u s u a l p r o p e r t y o f t h e s e n u m b e r s i s :
T H E O R E M 9 . I f d i ( n - 1 ) , t h e n
A =0 , ( 2 3 )( s , n-1) =d
w h e r e 0 < s < n ( n - 1 ) / 2 .A b y - p r o d u c t o f t h e s e i n v e s t i g a t i o n s i s t h e f o l l o w i n g r e s u l t : I f p i s a n
o d d p r i m e , t h e i n t e g r a l r o o t s o f t h e c o n g r u e n c e
p - 1
1 + E c b ( s , p - 1)xSO0 ( m o d . P ) ( 2 4 )s= 1
a r e t h e i n c o n g r u e n t p r i m i t i v e r o o t s m o d u l o p .
I f w e c o n s i d e r t h e f u n c t i o n d e f i n e d i n ( 4 ) we m a y o b t a i n t h e f o l l o w i n g :I f n i s a n o d d p o s i t i v e i n t e g e r a n d t i s a n i n t e g e r s u c h t h a t 1 < t < n , t h e n
B t ( - 1 ) = - E 2 d 4 ( t , n l d ) , ( 2 5 )2n d i n
w h e r e B , ( z ) i s d e f i n e d - i n ( 1 3 ) .
VOL. 3 9 , 1953 9 6 7
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9 . 6 8 M A THEMA T I C S : C . A . NICOL
S i n c e B , ( z ) i s a p o l y n o m i a l w i t h i n t e g r a l c o e f f i c i e n t s t h e n u m b e r B , ( 1 )
i s a n i n t e g e r .I n p a r t i c u l a r , i f n i s o d d a n d t i s l e s s t h a n n a n d p r i m e t o i t , t h e r n
B t ( - 1 ) =-E 2 d , ( n / d ) . ( 2 6 )2 n d l - n
T h i s n u m b e r i s a g e n e r a l i z a t i o n o f t h e F e r m a t Q u o t i e n t , ( 2 P - 1 - 1 ) / p ,w h e r e p d e n o t e s a n o d d p r i m e .
A c k n o w l e d g m e n t . - T h e a u t h o r i s i n d e b t e d t o H . S . V a n d i v e r f o r h i sm a n y h e l p f u l s u g g e s t i o n s a n d e n c o u r a g e m e n t .
I D i c k s o n , L . E . , H i s t o r y o f t h e T h e o r y o f N u m b e r s , V o l . 2 , c h a p t e r 3 , C a r n e g i e I n s t . o fW a s h i n g t o n , P u b l i ca t i o n N o . 2 5 6 ( 1 9 2 0 ) .
B a c h m a n n , P . , " N i e d e r e Z a h l e n t h e o r i e , " Z w e i t e r T e i l , c h a p t e r s 3 - 6 , B . G . T e u b n e r ,
L e i p z i g , 1 9 1 0 .
2 V a h l e n , T . , B a c h m a n n , P . , I b i d . , 1 1 6 , 1 6 7 , 2 7 3 .Vo n S t e r n e c k , S i t z u n g s b e r . d . W i e n e r A k a d . , 1 1 1 , 1 5 6 7 ( 1 9 0 2 ) ; 1 1 3 , 3 2 6 ( 1 9 0 4 ) ;
1 1 4 , 7 1 1 ( 1 9 0 5 ) .
C a u c h y , O e u v r e s D ' A u g u s t i n C a u c h y , 5 ( s e r i e s 1 ) , 8 1 - 8 5 , 1 3 5 - 1 5 2 . P a r i s , G a u t h i e r -
V i l l a r s ( 1 8 8 5 ) .3 Vo n S t e r n e c k ( P . B a c h m a n n , I b i d . ) i n t r o d u c e d a f u n c t i o n e q u i v a l e n t t o c b ( k , n ) .
H e u s e d i t t o o b t a i n r e s u l t s c o n c e r n i n g p a r t i t i o n s m o d u l o a p o s i t i v e i n t e g e r . E m p l o y i n gt h i s f u n c t i o n h e o b t a i n e d a s p e c i a l c a s e o f t h e o r e m 1 .Th e n u m b e r ' ( k , n ) was u s e d b y R . M o l l e r i n t h e f o l l o w i n g r e s u l t ( M a t h . M o n t h l y ,
5 9 , N o . 4 , 2 2 8 ( A p r i l 1 9 5 2 ) ) . I f t h e n u m b e r s g d a r e a l l o f t h e i n c o n g r u e n t i n t e g e r sb e l o n g i n g t o d m o d u l o p , p b e i n g a n o d d p r i m e a n d d a d i v i s o r o f p - 1 , t h e n f o r
any r , E g d =(r, d ) ( m o d . p )
4 T h e su m E e x p ( 2 x r i r k / n ) i s known a s R a m a n u j a n ' s su m ( c f . , H a r d y , G . H .( r , n ) = 1
a n d W r i g h t , E . M . , I n t r o d u c t i o n t o t h e N u m b e r T h e o r y , O x f o r d , 1 9 3 8 . p p . 5 5 , 2 3 7 ) . A n o t h e r
c l o s e d f o r m f o r t h i s su m was f o u n d p r e v i o u s l y b y T . M. A p o s t a l a n d D . R . A n d e r s o n
a n d s t a t e d b y them i n a n a b s t r a c t i n B u l l . Am. M a t h . S o c . , 5 8 , N o . 5 , 5 5 9 ( 1 9 5 2 ) . Th e
f o r m t h e y f o u n d i n o u r n o t a t i o n i s c J ( b ) 3 ( a ) / o p ( c ) w h e r e a = ( n , k ) : b = n / a , a n d c =( a , b ) . I f , u ( b ) . Owe h a v e t h e r e l a t i o n p ( n ) / , o ( b ) = c , ( a ) / , i ( c ) .
N a g e l l , T . , I n t r o d u c t i o n t o N u m b e r T h e o r y , U p p s a l a , 1 9 5 1 , p . 4 3 .R e v i e w o f P e r e z - C a c h o , " T h e F u n c t i o n E ( x ) i n t h e T h e o r y o f N u m b e r s , " M a t h .
R e v i e w s , 1 8 , 9 1 3 , ( 1 9 5 2 ) .
9 6 8 P R O C . N . A . S .
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