R E V I S T A D E L A

U N I O N M A T E M A T I C A A R G E N T I N A

V o l u m e n 4 2 , N u m e r o 2 , 2 0 0 1 , P a g i n a s 1 0 3 - 1 1 1

A S I M P L E P R O O F O F T H E I R R A T I O N A L I T Y O F T H E T R I L O G

P A B L O P A N Z O N E

A B S T R A C T . W e u s e o r t h o g o n a l p o l y n o m i a l s t o g i v e a s i m p l e p r o o f o f t h e i r r a t i o n a l i t y

o f t h e t r i l o g . A n a p p r o x i m a t i n g f o r m u l a f o r R i e m a n n Z e t a - f u n c t i o n i n t h e c r i t i c a l

s t r i p i s d e r i v e d .

O . I n t r o d u c t i o n .

1 0 3

I n t h i s n o t e w e g i v e a s i m p l e p r o o f o f t h e i r r a t i o n a l i t y o f t h e t r i l o g f o r c e r t a i n v a l u e s .

M o r e c o n c r e t e l y w e p r o v e t h a t L i 3 ( l j d ) = 2 : : = 1 n l d

n

i s i r r a t i o n a l f o r 1 1 7 3 : : ; ; d E N .

T h i s r e s u l t w a s f i r s t p r o v e d b y M . R a t a [ 6 , 7 } .

W e w a n t t o p o i n t o u t t h a t i m p r o v e d r e s u l t s h a d r e c e n t l y b e e n o b t a i n e d , b y M i I a d i i n

h i s t h e s i s . A l s o t h e t h e t e c h n i q u e s d e v e l o p e d b y T a n g i l y R i v o a l i n h i s t h e s i s c a n b e u s e d

e f f e c t i v e l y t o g e t I T m c h b e t t e r r e s u l t s .

T h i s n o t e i s d i v i d e d i n t w o s e c t i o n s w h i c h a r e a l m o s t i n d e p e n d e n t . I n t h e f i r s t s e c t i o n

w e g i v e a n a p p r o x i m a t i n g f o r m u l a f o r R i e m a n n Z e t a - f u n c t i o n o n t h e c r i t i c a l s t r i p . I n t h e

s e c o n d s e c t i o n w e p r o v e t h e m e n t i o n e d r e s u l t o n t h e i r r a t i o n a l i t y o f t h e t r i l o g .

I n b o t h c a s e s w e u s e o r t h o g o n a l p o l y n o m i a l s a s i n B o r w e i n a n d E r d e l i ' s b o o k [ 1 ] ,

a p p e n d i x A 2 . I n d e e d t h e p o i n t t o s t r e s s h e r e i s t h a t o r t h o g o n a l p o l y n o m i a l s h a v e a n

i n t e g r a l r e p r e s e n t a t i o n ( [ 1 ] p g . 3 7 3 ) t h a t p e r m i t s t o g u e s s , a t l e a s t i n s o m e c a s e s , w h a t

k i n d o f p o l y n o m i a l s a r e n e e d e d t o p r o v e t h e i r r a t i o n a l i t y r e s u l t s .

1 . L e t '

, r

1

X

A 1

,

1 = I ( d , ) . . 1 . ) . . 2 , ) : = 1 0 ( d X l ) A 2 d x ,

( 1 . 0 )

w i t h p a r a m e t e r s ) . . 1 , ) . . 2 , d , r a n g i n g i n c e r t a i n s e t s o f v a l u e s g i v e n b e l o w ( h e r e y A w i l l

s t a n d f o r t h e p o s i t i v e ) . . - r o o t o f y i f y , ) . . > 0 ) .

W e s o m e t i m e s u s e t h e w e l l - k n o w n n o t a t i o n ( a ) o = 1 , ( a ) n = a ( a + 1 ) . . . ( a + n - 1 ) .

I n w h a t f o l l o w s A n , B

n

, F n { x ) , a n , f 3 n , ' Y n m a y d e p e n d o n d , ) . . 1 , ) . . 2 , b u t t o s i m p l i f y t h e

n o t a t i o n d e p e n d e n c e o n n i s o n l y w r i t t e n . F o r 1 ' : : ; ; n w e d e f i n e

, ' ,

n ( - 1 ) n + i ( j + A l t 1 ) n ' l

F n ( x ) : = L ( _ . ) , . , x

J

i = O n J . J .

A

. _ r

1

( F n { x ) - F n { ( = F d ) t ) A I d

n ' - 1 0 ' ( d Xl ) A 2 , x x ,

B

n

: = F n ( { = F d ) t ) , { _ d ) l / l : = d 1 / l e ! f ,

1 9 9 1 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n . l I M x x , l l A 5 5 , l l J 7 2 .

K e y w o r d , s a n d p h r a s e s . I r r a t i o n a l i t i e s , t r i l o g , R i e m a n n Z e t a - f u n c t i o n .

R e v . U n . M a t . A r g e n t i n a . V o l . 4 2 - 2

1 0 4

P A B L O A . P A N Z O N E

P r o p o s i t i o n 1 . L e t f b e d e f i n e d b y ( 1 . 0 ) w i t h A 1 , A 2 , d E R , 0 ~ A 1 , ( ) ~ A 2 ~ 1 , 1 ~ d

a n d = 1 , 2 , 3 , . . . ( i f d = 1 a n d t h e s i g n i s - t h e n w e r e s t r i c t 0 ~ A 2 < 1 ) . T h e n t h e

f o l l o ' U J i n g h o l d s

i )

w h e r e

{

( V d - v d - = - 1 ) 2 i f t h e s i g n i s -

h ( d ) = r , ,

( V d + I - V d ) 2 i f t h e s i g n i s +

i i ) W e h a v e t h , e f o l l o w i n g r e c u r r e n c e r e l a t i o n f o r 3 ~ n ,

B " , + ( = f a n d + P n ) B

n

-

1

+ " f n B n - 2 = 0 ,

" 1

A n + ( = f a n d + P n ) A . , t - 1 + " Y n A n - 2 = = f a n / F n - l ( X ) ( d xe)~-A2;rAldx,

i o ,

w h e r e

, ' ( 2 7 1 , - l+~f!J ( 2 n " , 2 + )

a n = -

, n ( n - 1 +

A 1

t ) ,

A + 1 ( 2 n - 1 + Al+1)(2n~ 2 + A l +

1

) ( n - 2 + A l +

1

) ( 7 1 , - 1 )

P n = ( 2 n - 2 + _ 1 _ _ ) _ e , R

n ( n - 1 + A I 7 1 ) ( 2 n - 3 +

A I

7 1 ) '

( 7 1 , - 1 ) ( 2 + 2 7 1 , 2 - 3 + ( ) 2 + n ( 3 - 5 ) )

" Y n = , ,

n ( n - 1 + A 1 7 1 ) ( 2 n + A 1 7 1 - 4 ) ( 2 n - 3 + A 1 7 1 ) 2 '

i f 3 ~ n . M o r e o v e r , i f A 2 = 1 a n d 3 ~ n t h e n f o 1 F

n

-

1

( x ) ( d x

R

) 1 -

A

2 X A I d x = O .

P R O O F . W e h a v e f o r 0 ~ x < 1

T h e i d e a i s t o c o n s t r u c t p o l y n o m i a l s w h i c h a r e o r t h o g o n a l t o t h e p o w e r s o f x t h a t a p p e a r

i n ( 1 . 1 ) . F o r t h i s w e d e f i n e F n ( x ) a s

~ 1 ( t + A 1 + l ) ( t + A 1 + + l ) ( t + A 1 + 2 + 1 ) . . . ( t + A 1 + ( n - 1 ) + 1 ) : r J d t

2 r n ' Y t ( t - ) . . . ( t - n )

= ~ 1 P ( t ) x t d t ,

2 7 l ' , t ' Y

( 1 . 2 )

w h e r e " Y i s a p o s i t i v e l y o r i e n t e d c l o s e d c u r v e e n c l o s i n g 0 , , 2 , . . o r ! ' / ! l y i n g i n t h e h a l f

p l a n e R e ( t : ) > - 1 / 2 . U s i n g r e s i d u e s O I l e e a s i l y s e e s t h a t F n ( : r : ) i s o f t h e f o r m s t a t e d

R e v . U n , M m . A r g e n t i n a . V o l . 4 2 - 2

A S I M P L E P R O O F O F T H E I R R A T I O N A L I T Y O F T H E T R I L O G . 1 0 5

i n P r o p o s i t i o n 1 . N o w F n ( x ) i s o r t h o g o n a l t o x

A l

, X + A l , X

2 C

+

A l

, ; . . , x ( n - 1 ) + A l o n [ 0 , 1 ]

i . e . 1 0

1

F n ( x ) x A l + i C d x = O i f j = O , . . . , n - L T h i s i s s t r a i g h t f o r w a r d u p o n u s i n g t h e

d e f i n i t i o n o f F n { x ) , i n t e g r a t i n g f i r s t i n x , t h e n i n t = R e

i

( ) a n d t a k i n g R - - - + 0 0 ( s e e [ 1 ] ) .

N t t h

. t 1 f 1 F n { x ) x

A l

d . . t .

o w w e c o m p u e e m e g r a J o ( d x t ) A 2 x m . w o w a y s :

i ) D u e t o t h e o r t h o g o n a l i t y o f F n ( x ) a n d ( 1 . 1 ) w e h a v e ( h e r e { ) = ( _ l ) n i f d + x c ; { ) = 1

i f d - x C )

- : - : : - - - ' - - - ' - : ; - : - - ; : - d x =

1

1 F n ( X . ) x

A 1

o ( d X l ) A 2

1

1 1 \ . . ( \ + 1 ) n l + A l \ ( \ + ) ( n + 1 ) + A l

= { ) F ( ) _ . _ ( / \ 2 . . . / \ 2 n - . _ x _ _ / \ 2 . . . / \ 2 . n x ) d =

o n X d A 2 d n n ! = f d n + l n + I ! + . . . x

= { ) { I F n ( x ) w ( x ) d x = ~ { 1 1 P ( t ) w ( x ) x t d t d x = ~ 1 ( I P ( t ) w ( x ) x t d x d t =

J

o

2 7 r Z J o - y . . 2 7 r Z - y J o

- { ) l

p

( ) ( A 2 . . . ( A 2 + n - l ) . A 2 . . . ( A 2 + n ) ) d

- d

A

2 2 7 r i - y t d n n ! ( t + n f + A l + 1 ) = f d n + 1 ( n + l ) ! ( t + ( n + l ) f + A l + 1 ) + . . . t .

( 1 . 3 )

I f j = q f , q = n , n + 1 , . . . a n d , , ( ' , " ( " a r e p o s i t i v e l y o r i e n t e d c u r v e s , t h e f i r s t a r o u n d

- j - A 1 - 1 o f r a d i u s E < 1 a n d t h e s e c o n d a c i r c l e c e n t e r e d a t z e r o o f r a d i u s m u c h l a r g e r

t h a n n f o r f + A l + 1 t h e n

- 1 - 1 P ( t ) . 1 d t + - 1 - 1 P ( t ) . 1 d t =

2 7 r i - y . ( t + f + A 1 + 1 ) 2 7 r i - y ' ( t + j + A 1 + 1 )

_ 1 1 P ( t ) 1 . d t .

2 7 r i - y " ( t + j + A l + 1 )

( 1 . 4 )

A l s o 2~i I - y " P ( t ) ( t + i + \ , + 1 ) d t = 0 u p o n t a k i n g t h e r a d i u s o f " ( " t e n d i n g t o i n f i n i t y .

U s i n g t h i s f a c t , ( 1 . 4 ) y i e l d s .

_ 1 l

p

( t ) 1 d t - - P ( - ' - A - 1 ) - ( q - n + l ) n ( 1 5 )

2 7 r i - y ( t + j + A l + 1 ) - J 1 . - f ( q + A l I I ) n + l ' .

L e t u s r e c a l l t h e h y p e r g e o m e t r i c f u n c t i o n f o r m u l a ( v a l i d f o r R e ( c ) > R e ( b ) > )

(

. _ f ( c ) { I b - 1 c - b - 1 - a _ ~ ( a ) n ( b ) n n

2 F 1 a , b , c , z ) . - r ( c _ b ) f ( b ) J o T ( 1 - T ) ( 1 - T Z ) d T - ~ ( c ) n n ! Z .

U s i n g t h i s f o r m u l a a n d ( 1 . 5 ) w e h a v e t h a t t h e l a s t f o r m u l a o f ( 1 . 3 ) i s e q u a l t o

{ ) ( A 2 ) n ( A 2 + n h ( n + A l 1 1 h ( A 2 + n h ( n + A l 1 1 h

d

A 2

f 2 7 r i d n ( n + ) n + l ( 1 = f ( 2 n + 1 + ) d ! d + ( 2 n + 1 + h 2 ! d

2

= f ' " =

_ { ) ( A 2 ) n f ( 2 n + l + ) ( I n + A d

l

_

1

( 1 ) n ( I T ) - n -

A 2

d

- d A 2 f 2 7 r i d n ( n + A l I I ) n + l r ( n + l ) f ( n + A I r ) J o T t - T d T .

R e v : U n . M a t . A r g e n t i n a . V o l . 4 2 - 2

1 0 6

P A B L O A . P A N Z O N E

T h e r e f o r e w e g e t t h a t o u r , i n t e g r a l i s d i f f e r e n t f r o m z e r o a n d t h i s y i e l d s t h e l e f t i n -

e q u a l i t y o f i ) . N o w (i~-ii . h a s i t s m a x i m u m a t T O = d - J d ( d - 1 ) b e i n g t h i " m a x i m u m

e q u a l t o d ( V d - y ' ( I - = 1 ) 2 . A l s o (i~-i}has i t s m a x i m u m a t T O = j d ( d + 1 ) ~ d b e i n g

t h i s m a x i m u m e q u a l t o d ( y ' d + 1 - V d ) 2 . T h e r e f o r e '

1

1 " ' j + 1 T . T ( l - T ) n j ' l " ' 1 +

1

T

T

n

+ e - 1 ( 1 - T ) n ( 1 _ ) - n - > ' 2 d T s : : ( m a x ) r e - 1 ( 1 - ) - > ' 2 d r

o d ' " T E [ O , l ] ( 1 ~) . 0 d '

A l s o ~n+1+ ~ = 1 . F r o m t h i s t h e r i g h t i n e q u a l i t y o f i ) f o l l o w s .

( n + e ) n + l r ( n + e )

i i ) W e c o m p u t e t h e i n t e g r a l i n a s e c o n d w a y s h o w i n g t h a t i t s~,ti"fics a r e c ; l l T e l l c e

r e l a t i o n .

r

l

F n ( x ) x > ' l d x =

i o ( d X

l

) > ' 2

r

l

( F n ( x ) - F

n

( ( = t = d ) l ) ) x > ' l 1 r

l

X > ' l

i o ( d X

l

) > ' 2 , d x + F n ( ( = t = d ) e ) i o , ( d : r

f

) > ' 2 d x : = A n + B n f .

( 1 . 6 )

w h e r e ( _ d ) l / f : = d l / R e T . F r o m t h e o r t h o g o n a l i t y r e l a t i o n s f o r F n ( x ) o n e h a s , i f 3 : ( n

( 1 . 7 )

f o r s o m e c o n s t a n t s c x

n

, f 3 n , 1 n . I n f a c t , t h e c o e f f i c i e n t o f x n R i n F r , ( x ) i s n o t z e r o a n d n n , f 3 n

c a n b e a d j u s t e d t o g i v e F n ( x ) + ( c x n x f + f 3 n ) F

n

-

l

( x ) = a o + a I x i ' + . . . + a n _ 2 x ( n - 2 ) .

T h i s c a n b e w r i t t e n a s a l i n e a r c o m b i n a t i o n o f F

n

-

2

( x ) , . . . , F I ( x ) , 1 . I f t h i s e q u a l i t y i s

m u l t i p l i e d b y X > ' l a n d o r t h o g o n a l i t y i s u s e d t h e n o n l y t h e c o e f f i c i e n t o f F

n

-

2

( : r : ) s u r v i v e s .

S o i f w e l e t x = d

l

/

f

o r x = ( _ d ) l / f i n ( 1 . 7 ) t h e n '

r e s p e c t i v e l y . A l s o

T h u s ,

R e v . U I I . M a t . A r g e n t i n a . V o l , 4 2 - 2

A S I M P L E P R O O F O F ~HEIRRATIONALITY O F T H E T R I L O G

1 0 7

a n d o n e g e t s t h e r e c u r r e n c e r e l a t i o n s o f i i ) . O b s e r v e t h a t i n t h e l a s t f o r m u l a t h e i n t e g r a l

i s z e r o d u e t o t h e o r t h o g o n a l i t y i f A 2 = 1 a n d 3 : : : ; ; n .

F i n a l l y , Q . n , f 3 n , ' Y n a r e obtai~ed f r o m ( 1 . 7 ) m a k i n g t h e c o e f f i c i e n t s o f x

n

l ! , x ( n - 1 ) J ! ,

x ( n - 2 ) l ! e q u a l t o z e r o . T h i s t e d i o u s c a l c u l a t i o n i s o m i t t e d .

N e x t w e g i v e t w o a p p l i c a t i o n s o f P r o p o s i t i o n 1 .

I n t h e n e x t t w o e x a m p l e s w e t a k e t h e + s i g n i n ( 1 . 0 ) . ; E x a m p l e 1 i s q u i t e w e l l k n o w n .

S e e [ 1 0 ] f o r e x a m p l e . .

E x a m p l e 1 . I f o n e p u t s A 1 = O , . e = 1 , A 2 = 1 , t h e n F n ( x ) a r e t h e L e g e n d r e p o l y n o -

m i a l s . T h e y c a n b e w r i t t e n m o r e s i m p l y a s . ,

T h u s i f A n , B n a r e a s d e f i n e d i n P r o p o s i t i o n 1 t h e n A n T n , B n E N w h e r e

T n = l c m { l , 2 , . . . , n } ( r e c a l l T n = O ( e ( 1 + e ) n ) b y t h e p r i m e r i u m b e r t h e o r e m } . T h u s b y i )

o f P r o p o s i t i o n 1 i f 1 : : : ; ; d E N

0 < I A n T n + L o g d : 1 B n T n l = 0 ( ( v ' d + 1 - v ' d ) 2 n

e

( 1 + e ) n ) ,

g i v i n g t h e i r r a t i o n a l i t y o f L o g 4 f 1 f o r 1 : : : ; ; d E N .

T h e f o l l o w i n g e x a m p l e s e e m s t o b e n e w .

E x a m p l e 2 . A s s u m e i . = 1 , A 2 = 1 , d = 1 i n ( 1 . 0 ) a n d A 1 = A . T h e n

f ( A ) = 1 1 ( 1 : x ) d x = I ! A - 2 ! A + 3 ~ A - . . .

O u r i n t e r e s t i n t h i s f u n c t i o n c o m e s f r o m t h e f a c t t h a t i f 0 < i : T < 1 , s = ( j + i t . t h e n

( s ) = s i n ( ' l r s ) { ' X ) f ( A ) A - S d A

' I r ( l - 2

1

-

8

) J o ,

w h e r e ( s ) i s t h e z e t a f u n c t i o n o f R i e m a n n ( [ 3 ] f o r m u l a ( 1 . 3 ) ) .

L e t g ( s ) :~ 1 0

0 0

f ( A ) A - S d A = s):~(~~;-.). I n t h i s c a s e i n P r o p o s i t i o n 1 o n e g e t s

A : = A ( A ) = f 1 F

n

( x ) - F

n

( - 1 ) x

A

d x a n d ( - l ) n B ' = ( - l ) n B ( A ) = " ' T ? ' (j+A:~)n

n n J o . x + 1 n . n L . J J = O n - J ! J t .

. T h e o r e m 1 . . T h e a n a l y t i c f u n c t i o n g n ( s ) : = 1 0

0 0

~:~~~A-8dA,0 < ( j < 1 , c o n v e r g e s

u n i f o r m l y t o - g ( s ) ' i n s i d e ' t h e c r i t i c a l s t r i p , i~e. , I g n ( s ) + g ( s ) 1 : : : ; ; h(~))2n ; e i f E : : : ; ;

( j : : : ; ; 1 - E a n d 0 < E < 1 / 2 .

P r o o f : N o t e t h a t A n ( A ) i s c o n t i n u o u s i n [ 0 , + 0 0 ) a n d 0 ( A n - 1 ) . A l s o ( - l W B n ( A ) i s

a p o s i t i v e , i n c r e a s i n g f u n c t i o n o n [ 0 , + 0 0 ) a n d b e h a v e s a s y m p t o t i c a l l y a s C A n a s A - t

+ 0 0 , ( C = I 0 ) . F r o m t h i s i t i s e a s i l y s e e n t h a t g n ( s ) i s a n a l y t i c i n t h e s t r i p 0 < ( j < 1 .

F r o m i ) o f P r o p o s i t i o n 1 w e g e t I A n ( A ) + B n ( A ) f ( A ) 1 : : : ; ; (~~~);;. T h u s , .

R e v . U n . M a t . A r g e n t i n a , V o l . 4 2 - 2

1 0 8

P A B L O A . P A N Z O N E '

n ( y ' 2 - 1 ) 2 n r o o ) . . - a

: : : ; ; ( - 1 ) 2 n B n ( O ) i o > . + 1 d > . .

N o t i c e t h a t ( 2 ; ) : : : ; ; (~1)nB~(O) a n d 1 0

0 0

~~~ d > . : : : ; ; 2 j e . T h i s p r o v e s t h e t h e o r e m . '

2 . O n t h e t r i l o g . W e p r o v e i n t h i s s e c t i o n t h e f o l l o w i n g :

' T ' l . b L ( j d ) - \ . . . . . . 0 0 1 - 1 f 1 L o g ( x ) 2 d . . t " l . f

T h e o r e m 2 . . L n e n u m e r i 3 1 - L . m = l d " " n 3 - 2 J o d - , " , , x ~s z r r a , w n a J o r

. d E N , 1 1 7 3 : : : ; d .

P r o o f : L e t n E N a n d d e f i n e

H e r e ' Y i s a po~itively o r i e n t e d c u r v e e n c l o s i n g 0 , 1 , . . . , 3 n i n t h e h a l f p l a n e ~! < R e ( t ) .

N o t i c e t h a t F n ( x ) i s o r t h o g o n a l t o L o g ( x ) 2 , x L o g ( x ) 2 ; . . . , : r ; n - 1 L o g ( : r ) 2 .

O b s e r v e t h a t

1

1 F n ( X ) L ( ) 2 d - 1

1

F n ( x ) - F n ( d ) L ( ) 2 d F , ( d ) 1 1 L o g ( x ) 2

d

-

- d - - o g x x - d o g x x + n , d x - . .

o - x 0 - x ' o ' - x

a n d a l s o t h a t

r

1

F n ( x ) 2 1 1 1 2 X n X " + l

i o d - x L o g ( x ) d x = d i o F n ( x ) L o g ( x ) ~d" + d , , + l + . . . ) d x =

1 r 1 1 x " x " + l '

2 7 r i d i o ' Y P ( t ) x

t

L o g ( x ) 2 ( d

n

+ d n + 1 + . . . ) d t d x .

C h a n g i n g t h e o r d e r o f i n t e g r a t i o n w e o b t a i n ,

( 2 . 1 )

W e h a v e u s e d t h e s a m e i d e a o f P r o p o s i t i o n 1 t o g e t t h e e q u a l i t y i n t h e l a s t f o r m u l a ( s e e

( 1 . 5 ) a b o v e ) . B u t t t P ( t ) = P(t)(t~l + . . . + t ! n - t 2 k - . . ' . - t _ k

3

_ 3 , . ) : = P ( t ) b ( t ) .

T h e r e f o r e : f : 2 P ( t ) = P ( t ) ( b ( t ) 2 + b ' ( t ) ) . N o t i c e t h a t b ( - n - j ) 2 + b ' ( - n - j ) = h ( n , j )

w i t h h d e f i n e d b y

3 ' 3 3 1 1 1

h ( n , j ) : = ( - : - + - . - + " ' + . _ _ _ _ _ . . . _ _ ' _ ) 2 _

J J + 1 n + J - 1 n + j n + 1 + j 4 n + . i

3 3 3 1 1 1

- ( p + ( j + 1 ) 2 + . . . + ( n + j - 1 ) 2 - ( n + j ) 2 - ( n + 1 + j ) 2 - . . . - ( 4 n + J ) 2 )

R e v . u l i . M a t . A r g e n t i n a . V o l . 4 2 2

A S I M P L E P R O O F O F T H E I R R A T I O N A L I T Y O F T H E T R I L O G

2 3 4

T h u s ( 2 . 1 ) i s e q u a l t o

S ( n , d ) .

d

n

+

1

( 4 n + 1 ) (i~) (~~) ( 2 : ) ,

w i t h S ( n , d ) a s d e f i n e d i n t h e f o l l o w i n g l~mma w h i c h w e p r o v e l a t e r .

L e m m a C . L e t

S ( n d ) : = ~ ( n + j ) 3 ( n + 1 ) . . . ( n + j ) h ( n , j + 1 )

, . ~ j . ( 4 n + 2 ) . . . ( 4 n + j + 1 ) d j

3 = 0 .

a ) T h e r e e x i s t s n o s u c h t h a t S ( n , d ) i s n o n - z e r o i J 4 : : : : ; ; d , n o : : : : ; ; n

1 0 9

( 2 . 2 )

( 2 . 3 )

b ) S ( n , d ) = O ( L o g ( n ) ( 4 n + 1 ) (i~) 1 . 3 7 1 8

n

+

1

m a x t E [ O l j { t n ( l - t ) 3 n } ) i f 1 0 0 0 : : : : ; ; d ( t h e

c o n s t a n t i n v o l v e d i n t h i s O - t e r m i s a b s o l u t e ) .

W e c o n t i n u e w i t h t h e p r o o f o f T h e o r e m 2 . U p t o t h i s p o i n t w e h a v e a c h a i n o f e q u a l i t i e s

s o u s i n g L e m m a C a ) w e ' g e t t h a t ( 2 . 3 ) i s n o n - z e r o i f n i s l a r g e e n o u g h a n d 4 : : : : ; ; d . U s i n g

r e s i d u e s i n t h e d e f i n i t i o n o f F n ( x ) w e o b t a i n

F ( ) = ( _ l ) n ~ ( 3 n ) ( n + j ) 3 ( _ 1 ) j j

n X ( 3 n ) ( 2 n ) ~. . \ X ,

2 n n j = O J J

a n d i t i s e a s i l y s e e n t h a t A n r n E N , B n (~~) e : ) E N i f r n = (~~) (~) ( l c m { l , 2 , . . . , 3 n } ) 3 .

R e c a l l t h a t b y t h e p r i m e n u m b e r t h e o r e m ( l c m { l , 2 " . . , 3 n } ) 3 : : : : ; ; e ( 9 + ) n f o r l a r g e n .

T h e r e f o r e w e h a v e p r o v e d t h a t

. e ( 9 + ) n I S ( n d ) 1

0 < I A n r n + B n

r

n

2 L i

3 ( 1 / d ) 1 : : : : ; ; , ( 4 ) '

d

n

+

1

( 4 n + 1 ) 3~

i f n i s l a r g e e n o u g h a n d 4 : : : : ; ; d . U s i n g L e m m a C b ) a n d S t i r l i n g ' s f o r m u l a o n e g e t s t h a t

L ' i 3 ( 1 / d ) i s i r r a t i o n a l w h e n e v e r 1 . 3 7 1 : : 9 3 3 < l o r w h a t i s t h e s a m e w h e n 1 1 7 3 : : : : ; ; d E N .

T h e p r o o f o f L e m m a C d e p e n d s o n L e m m a s A a n d B w h i c h w e g i v e b e l o w .

L e m m a A . I f 0 : : : : ; ; w : : : : ; ; 1 / 1 0 0 0 t h e n

f ( n ~ j ) 3 w i : : : : ; ; ( 1 ; ) 3 ( n + l ) : : : : ; ; 1 . 3 7 1 8 n + 1 .

j = O J

P r o o f : R e c a l l t h a t 2 : ; : 0 ( n j j ) u

j

= ( l _ ; ) n + l ' T h e r e f o r e , i f 0 : : : : ; ; w

f (n~j)3wi::::;; ( f (n~j)wi)3 = ( 1 1 n + l ) 3 .

j = o J j = o J ( 1 - : - w

3

)

T h e l e m m a f o l l o w s i f w e t a k e 0 : : : : ; ; w : : : : ; ; 1 / 1 0 0 0 .

R e v . U n . M a t . A r g e n t i n a , ' V o l . 4 2 - 2

1 1 0 P A B L O A . P A N Z O N E

L e m m a B . a ) T h e r e e x i s t s a n a t u r a l n u m b e r n o a n d a b s o l u t e c o n s t a n t s 0 < C l , C 2 s u c h

t h a t

C l ~ h ( n , j ) ,

i f 1 ~ j ~ 3 n , n o ~ n a n d

h e n , j ) ~ C 2 ,

i f n o ~ n , 3 n ~ j .

b ) W e h a v e I h ( n , j ) 1 ~ c 2 L o g ( n ) f o r a l l 1 ~ n , j .

P r o o f : a ) T h e p r o o f i s d i v i d e d i n f o u r c a s e s . R e c a l l f o r m u l a ( 2 . 2 ) :

C a s e 1 . 1 ~ j ~ L o g ( n ) . I t i s e a s i l y s e e n t h a t i n t h i s c a s e

3 L o g ( L o n ( n ) ) + 0 ( 1 ) ~ L ; L ' L a n d L a r e 0 ( 1 ) .

9 1 2 3 4

S o t h e f u n c t i o n h e n , j ) i s g r e a t e r t h a n 2 L o g ( L o ; ( n ) ) i f n i s l a r g e e n o u g h .

C a s e 2 . L o g ( n ) ~ j ~ n . I n t h i s c a s e

3 L o g ( 2 ) + 0 ( 1 ) ~ L ' L ~ L o g 4 + 0 ( 1 ) , L a n d L a r e 0 ( 1 ) .

1 2 3 4

N o t i c e t h a t ( 3 L o g 2 - L o g 4 ) = . 6 9 . . . . S o t h e f u n c t i o n h e n , j ) i s g r e a t e r t h a n ( 3 L o g 2 -

L o g 4 ? + o ( 1 ) i f n i s l a r g e e n o u g h .

C a s e 3 . n ~ j ~ 2 n . T h e a r g u m e n t i s t h e s a m e a s i n c a s e 2 , i . e .

3 L o g ( 3 / 2 ) + 0 ( 1 ) ~ L , L ~ L o g ( 5 / 2 ) + 0 ( 1 ) , L a n d L a r e 0 ( 1 ) .

1 2 ~~ 4

B u t 3 L o g ( 3 / 2 ) - L o g ( 5 / 2 ) = . 3 . . . . T h i s y i e l d s t h a t t h e f u n c t i o n h ( n , j ) i s g r e a t e r t h a n

( 3 L o g ( 3 / 2 ) - L o g ( 5 / 2 ) ) 2 + 0 ( 1 ) i f r i i s l a r g e e n o u g h .

C a s e 4 . 2 n ~ j ~ 3 n . S i m i l a r l y ,

3 L o g ( 4 / 3 ) + 0 ( 1 ) ~ L ' L ~ L o g ( 2 ) + 0 ( 1 ) , L a n d L a r e 0 ( 1 ) .

1 2 3 4

N o t i c e t h a t 3 L o g ( 4 / 3 ) - L o g ( 2 ) = . 1 6 9 . . . S o t h e f u n c t i o n h ( n , j ) i s g r e a t e r t h a n

( 3 L o g ( 4 / 3 ) - L o g ( 2 ) ) 2 + 0 ( 1 ) i f n i s l a r g e e n o u g h . T h i s p r o v e s t h e f i r s t i n e q u a l i t y .

I t i s e a s y t o s e e t h a t t h e s e c o n d i n e q u a l i t y a n d b ) h o l d . T h e y a r e l e f t t o t h e r e a d e r . .

P r o o f o f L e m m a C : a ) W r i t e S e n , d ) a s L~:~l + L j : 3 n = L 5 + L 6 ' T h u s , u s i n g a )

o f L e m m a B w e g e t t h a t i f n o ~ n t h e n t h e s u m L 5 i s g r e a t e r t h a n o n e o f i t s s u m m a n d s ,

n a m e l y t h a t w i t h j = n . T h u s

(

2 n ) 3 ( n + 1 ) . . . ( 2 n ) 1 " "

C l - ~ L . t .

n ( 4 n + 2 ) . . . ( 5 n + 1 ) d

n

5

( 2 . 4 )

A l s o , u s i n g a ) o f L e m m a B o n e g e t s i f 4 ~ d , ( h e r e w e u s e (n~j) ~ (i~) ( 1 Y - : 3 n i f 3 n ~ j

w h i c h i s p r o v e d u s i n g t h a t (n~~il) = n~~il (n~j))

I L I = O ( I = ( n + j ) 3 ( n + 1 ) . . . ( n + j ) 1 ) =

. j ( 4 n + 2 ) . . . ( 4 n + j + 1 ) d . 7

6 J = 3 n

R e v . U n . M a t . A r g e n t i n a , V o l . 4 2 - 2

- j

A S I M P L E P R O O F O F T H E I R R A T I O N A L I T Y O F T H E T R I L O G

I I I

= O ( ( 4 n ) 3 f (~)3(j-3n) ( n + 1 ) . . . ( n + j ) 1 . ) =

3 n j = 3 n 3 ( 4 n + 2 ) . . . ( 4 n + j + 1 ) d

J

= O ( ( 4 n )

3

( n + 1 ) . . . ( 4 n ) (~)9n f ( ( t ) 3 ) j =

3 n ( 4 n + 2 ) . . . ( 7 n + 1 ) 4 j = 3 n d

=O((~)3n ( n + 1 ) . . . ( 4 n ) ) .

3

3

d ( 4 n + 2 ) . . . ( 7 n + 1 )

F r o m t h i s a n d ( 2 . 4 ) i t i s e a s i l y s e e n t h a t 1 2 : : 6 f < 2 : : 5 i f n i s l a r g e e n o u g h . T h i s p r o v e s

a ) .

b ) N o t i c i n g t h a t ( 4 n + 1)(~~) f 0 1 t

n

+

j

( l - t )

3 n

d t = (4~:~)~::(4