FUNCTIONS.
By
K. ANANDA-RAU of MADRAS, India.
I.
I n t r o d u c t i o n .
I. Let
~ = x + i y
be a complex variable, andq=ei~;
let us write, f o l l o w i n g the n o t a t i o n of TAnNErY and MoLx 1,1 9 ~ 5
~(ol~)=2q~+2q~+2q~ +...
~3(O IT)= I + 2 q + 2 q 4 + 2q9+ "'"
# 4 ( o [ z ) ~ I - - 2 q + 2 q 4 - - 2 q ° + ...
These series are c o n v e r g e n t w h e n y > o and represent f u n c t i o n s w h i c h are ana- lytic in the half-plane y > o and w h i c h c a n n o t be c o n t i n u e d across ~he x-axis. The behaviour of these functions, w h e n ~ tends to a real n u m b e r ~ by m o v i n g along the straight line x ~ ~, has m a n y interesting f e a t u r e s . W h e n ~ is rational, ~he behaviour is f a i r l y simple and can be obtained either directly", or by effeeting on ~ a suitable linear transformation
1 TANNERY and MOLK, ]~16ments de la thdorie des Fonctions Elliptiques, Vol. 1I (1896), p. 257. We shall refer to this book (Vol. II) as F.E.
2 Cf. HARDY, On the representation of a n u m b e r as the sum of any n u m b e r of squares, and in particular of five, Transactions o f tlte American Mathematical Society, Vol. X X I (192o) pp. 255 - - 2 8 4 (p. 2519). Though the direct m e t h o d gives the result in many cases w i t h o u t necessitating an appeal to the transformation theory, t h e latter has the advantage of being applicable to all elliptic modular functions, including those for which a direct method is n o t available. See HARDY and RAMANUJAN, Asymptotic formulae in combinatory analysis, Proceedings o f the London Mathe- matical Society, (Ser. 2) Vol. 17 (I918) pp. 7 5 - - I I 5 (pp. 93, 94).
1~4 K. AnandmRau.
e + d ~
(i) T - -
a + b z '
w h e r e a, b, c. el are integers such t h a t • a d - - b e = I. I f ~ P ~ , where P, Q are i n t e g e r s prime to each other, t h e l a t t e r m e t h o d consists in t a k i n g a ~ P, b ~ - - Q a n d c, d to be a n y i n t e g e r s such t h a t P d + Q c = I. I t is t h e n easily seen f r o m (I) t h a t , w h e n ~ t r a c e s the line x - = ~ in t h e half-plane y > o , T - ~ X + i Y traces t h e line X - - d (~ in t h e half-plane Y > o ; a n d t h a t on t h e s e lines as y - - * + o , Y - ~ + ~ . N o w t h e t r a n s f o r m a t i o n t h e o r y of the T h e t a f u n c t i o n s ~ enables us to express (say) ~3 (o [~) in t e r m s of one Of the f u n c t i o n s ~2 (o [ T), ~3 (o I T), ~4 (o I T);
a n d in o r d e r to e x a m i n e t h e b e h a v i o u r of ~ ( o l z ) as • t e n d s ~ to ~, it is only n e c e s s a r y to s t u d y the b e h a v i o u r of one or o t h e r of t h e f u n c t i o n s ~2(o I T),
~3 (ol T), ~ t (o I T) as I~--~ + ~ along a fixed line parallel to t h e Y-axis. This l a t t e r i n v e s t i g a t i o n is very simple a n d leads to the r e q u i r e d r e s u l t w i t h o u t any difficulty.
2. W h e n ~ is i r r a t i o n a l t h e b e h a v i o u r of t h e T h e t a f u n c t i o n s is compli- cated, and t h e r e a p p e a r to be no results so simple as those t h a t exist w h e n ~ is rational, ttAI~DY a n d LITTLEWOOD ~ have o b t a i n e d some i n t e r e s t i n g results w h e n
is irrational. T h e y h a v e p r o v e d in this case t h a t
T h e y have f u r t h e r p r o v e d t h a t , if ~ is such t h a t , w h e n expressed as a simple c o n t i n u e d fraction, t h e p a r t i a l quotients f o r m a b o u n d e d sequence, t h e n two posi- tive constants K1, K2 exist such t h a t
K,
v; vy
I n p a r t i c u l a r t h e s e inequalities hold w h e n ~ is a q u a d r a t i c surd.
3. T h e o b j e c t of this p a p e r is to i n v e s t i g a t e m o r e f u l l y t h e case w h e n is a q u a d r a t i c surd. I t is s h o w n here t h a t t h e periodicity of t h e c o n t i n u e d frac-
1 F.E.p. 262,
To avoid constant repetition we shall understand that throughout this paper the path along which v tends to ~ is the straight line x ~ ~.
3 HARDY and ]~ITTLEWOOD, Some problems of Diophantine approximation (II), Acta Mathe- malica, Vol. 37 (r914), PP. I93--238 (PP. 225--23o).
tion for ~ is reflected also in the behaviour of the Theta functions, and enables us to obtain formulae which involve actual limits, when we break up the range of variation of y in a manner to be explained presently.
We shall confine ourselves in the following to the function ~ (o1~). A re- ference to the formulae given at the end of T x ~ R r and ~¢IOLK S book ~ will show t h a t the considerations developed i n t h i s paper are applicable, with:~sui~able modifications, to the other Theta functions a n d also tO the modular functions h (~), J(~), ~0 (~)etc.
Let ~ be a quadratic surd s, and let
-.. + + -. + - . .
l r
= [bl, b~, . . . bin, a l , a~., . . . ar],3
there being r partial quotients in a period of the continued fraction.
be the n t h convergent of the continued fraction.
suppose t h a t it is restricted to values satisfying o < y < i = q~ I L e t z/~ denote the interval
Let p__n qn As y is to tend to zero, we may
I I
- - < < ~ , ( n = I, 2, .)
q2 - - Y ""
n + l qn
so t h a t the range of variation of y may be considered to be made up of the sequence of intervals
z/l, J~, . . . J r , . . .
Now we divide this sequenc.e of intervals into a finite number of sub-sequences F.E. pp. 262, 266, 267.
We m a y suppose t h a t ~ is positive; t h i s does n o t i m p l y any loss of generality, since the function 9a(o Iv) has the period 2.
We shall use t h i s n o t a t i o n for periodic c o n t i n u e d fractions. We shall also denote t h e finite c o n t i n u e d fraction
c,+ .I~+... + II
~ c~ I Cn
b y [c~, c ~ . . . cn] and t h e infinite continued fraction
II . . + [ ~ + . . . c~+ ~ + -
b y [ci, c~ . . . . c~ .. . . . ].
1 9 - 2822. Acta mathematica. 52. Imprimd lo 23 ao~t 1928.
146 K. Ananda-Rau.
in a periodic m a n n e r ; it is s h o w n t h a t t h e r e exists a positive i n t e g e r H (which is a multiple of r) such t h a t , if y is confined to a n y one of t h e f o l l o w i n g ' H sub- sequences
(2)
"~1, ,~1 +H, z ~ l + 2 H , • . . , ~ l ~ - v H ~ . . .
, d 3 + 2 H ~ . . . , J 3 + ~ H ~ • • •
Z ~ H , ~ f 2 H , , ~ f 3 H , . . . z~f(vT1) H , . . .
t h e b e h a v i o u r of ~ (o ] ~) is f a i r l y simple, a n d c a n be o b t a i n e d by m a k i n g a com- b i n e d use of some p r o p e r t i e s of periodic c o n t i n u e d f r a c t i o n s a n d of t h e f o r m u l a e of t h e t r a n s f o r m a t i o n t h e o r y of the T h e t a f u n c t i o n s . To enable us to w r i t e d o w n a s y m p t o t i c f o r m u l a e , we i n t r o d u c e a c o n t i n u o u s p a r a m e t e r a which can t a k e a n y value in the i n t e r v a l o ~ 6 < I; a n d we consider the b e h a v i o u r of ~ s ( o ] ~ ) as y t a k e s the values
(~ I - - a
~ + 2 ;
q~ q,+l
h e r e a is k e p t fixed, and. n is allowed to t e n d to infinity t h r o u g h integers, w h i c h h a v e all the same residue R to modulus H , R b e i n g in t h e i n t e r v a l I ~ R--< H . I n o t h e r words, y t e n d s to zero t h r o u g h a sequence of values, one in each of t h e intervals
. J R ~ , d R + H , , ~ f R - ] - 2 H , • • • ,
each value of y dividing t h e i n t e r v a l in which it lies in t h e c o n s t a n t r a t i o I - - q : q.
U n d e r these circumstances it is p r o v e d in this p a p e r t h a t
4
4
(where ~ y d e n o t e s t h e real positive f o u r t h root) t e n d s to a finite limit. This limit depends on R a n d 6; a n d explicit f o r m u l a e showing t h e n a t u r e of this de- p e n d e n c e are also o b t a i n e d in the course of t h e paper. As t h e sub-sequences (z) cover t h e whole r a n g e of v a r i a t i o n of y, a n d as R can take a n y one of the values i, 2 , . . . , H a n d a can t a k e any value in o - - < a < I, it is clear t h a t t h e p r o o f
of t h e existence of the above-mentioned limits, a n d the m e t h o d of t h e i r evalua- tion a m o u n t to a complete description (to the first order of approximation) of the a s y m p t o t i c behaviour of "$8 (0]7) as ,--+ g along the s t r a i g h t line x ~ - ~ .
I I .
Geometrical Preliminaries.
4. L e t ~ be a quadratic surd a n d let
= [bl, b2 . . . . bin, a l , a ~ , . . , ar],
there being r partial q u o t i e n t s in a period of t h e c o n t i n u e d fraction. L e t p~ be q~
the n t h convergent of the continued fraction, and let V~ s t a n d for (-- I)~. F u r t h e r let g be equal to r or 2 r according as r is even or o d d ' ; so t h a t g is always a n even n u m b e r and is divisible by r. L e t Q denote an integer in t h e interval
I ~ g .
L e t S,~ denote the linear t r a n s f o r m a t i o n pn--1 + *]n pn T
(3) = qn-1 + Vn q~ T '
or its e q u i v a l e n t
(4) 2' = '2.,+1 p,~-i - - q,~-~
( , = 2,3,...)
W e consider in t h i s section t h e effect of m a k i n g the t r a n s f o r m a t i o n s Sn on t h e intervals J ~ a n d on the points
where
ff I - - f f qn+l
the suffix n in S~, J , , , ~,,(a) being the same for a n y particular t r a n s f o r m a t i o n f l
1 I n o t h e r words, g is t h e l e a s t c o m m o n m u l t i p l e of 2 a n d r.
2 T h e t r a n s f o r m a t i o n s Sn a n d t h e i n t e r v a l s An are s u g g e s t e d b y t h e w o r k of HARDY a n d LITTLEWOOD, |OC. cir. 226, 220. W e h a v e n o t defined t h e t r a n s f o r m a t i o n $1 w h i c h is to be a p p l i e d to p o i n t s in A I . T h e o m i s s i o n is u n i m p o r t a n t ; w e m a y , if we w i s h to p r e s e r v e f o r m a l com- p l e t e n e s s , define S 1 to be (say) t h e identical t r a n s f o r m a t i o n v = T.
148 K. A n a n d a - R a u .
5- W e s h a l l r e q u i r e , t o s t a r t w i t h , a l e m m a o n p e r i o d i c c o n t i n u e d f r a c - t i o n s ?
L e m m a 1. Let ~, P~, qn g, Q be as defined above; let ~ be fixed and let n tend to infinity by assuming all values congruent to Q (rood. g). Then q~-~ and p~ q,~ - - ~ q~
q, tend to finite limits.
L e t u s w r i t e t h e f i n i t e s e q u e n c e
a l s o i n ~he f o r m so t h a t
(5)
F o r x =< t ~ r w e d e f i n e ~
a 1, a~, . . . a t , a 1~ a ~ . . . a r
d l , d ~ , . . . d r , d ~ + l , d ~ + 2 , . . . d 2 ~ , d l - - a l , d ~ = a ~ , . . . d ~ : a r d r + l ~ a 1 , d , + 2 = a ~ , . . , d 2 r = a ~
0t = [O, d r + r , d r + r - l , . . , dr+l],
~ t = [ d r + l , d r + 2 , . . , d t + r ] .
W e s h a l l a l s o w r i t e f o r c o n v e n i e n c e t h e c o n t i n u e d f r a c t i o n f o r ~ i n t h e f o r m
[el,e2,... en,...],
so t h a t
el ~ b i , c a ~ ba, . . . e m ~- bm, Cm+~ -~- as
w h e n ~, ~ s (rood. r), s, o f c o u r s e , b e i n g o n e o f t h e n u m b e r s I, 2 , . . . r . N o w b y a k n o w n r e s u l t w e h a v e 3
(6) q,~--lq,~ - - [o, en, c~-1, . . . ca] .
1 T h e r e s u l t s of L e m m a I w e r e g i v e n in a s o m e w h a t different f o r m in m y paper, S o m e Dio- p h a n t i n e a p p r o x i m a t i o n s connected w i t h q u a d r a t i c s u r d s , Journal o f the Indian Mathematical So- ciety, Vol. X I V (I922), pp. I 6 i - - I 6 6 .
W h e n r = I t h e r e is o n l y one 0 a n d one ~p,
0 = [o, al, al, a~ . . . . ], ~ -- [a~, a . . . ].
8 See, for e x a m p l e , C~RYSTAL, Algebra, Vol. I I (I922), p. 433.
Since n -* ~ t h r o u g h values which differ by multiples of r, it is clear t h a t , w h e n n > m , c,~ has always a c o n s t a n t value, say at. R e m e m b e r i n g t h a t a t = dr+,., i t is easily seen f r o m (6) t h a t
qn--1 ---> [0, dr+ r, t i t + r - l , . • • dt+l ] : Or, q,~
which is t h e first r e s u l t of t h e lemma.
To p r o v e t h e second result, we write
T h e n
a n d so, r e m e m b e r i n g t h a t we have
P~ ~:P~
q~ q,~
f n + l = Ion+l, c ~ + 2 , . . . ] . f , + l p,~ + pn--1 : fi~+l q,~ + qn--1 '
.Pn qn--1 - - ~)n--1 qn = ~n fi~+ip~ + p ~ - 1 __
q,~ (fi~+~ q,~ + q~-~) '
qn - - f n + l qn + q n - l '
I
(7) Pn qn - - ~ q~ - - ~ n f n + l + ~]n qn--l.qn
Since n has t h e same residue to m o d u l u s g, w h i c h is an even n u m b e r , it follows t h a t n has t h e same p a r i t y , a n d so V,~ has t h e c o n s t a n t Value ~e. Also by t h e r e m a r k m a d e above, w h e n n > m, c~ has a constant~ value at; so t h a t , c~+1 has t h e c o n s t a n t value dt+l. T h e r e f o r e f~+l h a s t h e c o n s t a n t value q~t. A n d so u s i n g t h e first r e s u l t of t h e l e m m a we see f r o m (7) t h a t
I
This limit c a n n o t be zero, since Ot > o, ~0t > I. H e n c e p,~q,.--~q,~ t e n d s to a finite limit.
T h e limits of q,~-i a n d p ~ q , ~ - - ~ q ~ d e p e n d on Q and we shall d e n o t e t h e m qn
respectively by L e a n d //e, (¢)= I, 2 / . . . g ) . W e shall also have by definition
150 K. Ananda-Rau.
. d g + l ~ - d l , .l/0 ~ - / / g ,
L~+I = L1, L0 - - Lg.
6. L e t us n o w r e t u r n to t h e r e l a t i o n (3) b e t w e e n ~ a n d T a n d c o n s i d e r t h e c u r v e t r a c e d by T = X + i Y w h e n ~ -~ x + i y describes t h e p a r t of t h e line I I - W e shall r e g a r d t h e ~-plane a n d x = ~ w h i c h lies b e t w e e n y - ~ a n d y - ~ 2
q~ q,+l
t h e T- p l a n e as coincident, so t h a t t h e x- a n d X - a x e s coincide, a n d t h e y- a n d Y-axes coincide (in p o s i t i o n us well as in direction). T h e r e will be no c o n f u s i o n in d o i n g so, as v is confined to t h e fixed line x = ~ a n d T t r a c e s c i r c u l a r arcs w h o s e positions we will n o w describe. W r i t i n g v ---- ~ + i y , T = X + i ~ in (3), a n d e q u a t i n g t h e r e a l p a r t s , we see a f t e r a n easy c a l c u l a t i o n t h a t , as • describes t h e semi-infi- nite line x ~ - - ~ , y > o, T describes t h e semi-circle, whose e q u a t i o n is
(8) - - ~ qn)+ ~,, X ( p n qn-1 +pn--1 qn - - 2 ~qn--1 qn)
-~-(pn--I qn--1 - - ~
qn-1)
2 = O,a n d w h i c h lies in t h e h a l f p l a n e Y > o.
N o w t h e r e l a t i o n (4) gives t h e f o l l o w i n g e q u a t i o n s c o n n e c t i n g X , Y, x, y;
- - q n x ) -{-qny2] = ~ n + l [(pn--1 - -
qn--lX) (~3n-- qnX) ~- qn--lqny2], (9) x [(p~ ~
(i o) Y [(p. - - q . x ) + q . y2] == y . ~
F r o m t h e second of these e q u a t i o n s we see t h a t if t h e T-points, w h i c h corres- p o n d r e s p e c t i v e l y to
~ + i , ~ " = ~ i
, _~ + ~ ,
q~ q~ + 1
are X ' + i Y ' a n d X " + i Y " , t h e n
r , [(~, _ q.[)~+~ ] I
qn
02) y" pn-q.~)~÷ [( - 2
iq,~+l] qn+l
T h e T-curve t h a t c o r r e s p o n d s to t h e p a r t of t h e s t r a i g h t line x ~ ~ be-
I I
t w e e n y--~ ~ a n d y -~ 2 is t h e p a r t of t h e semi-circle (8) w h i c h is i n t e r c e p t e d
q,~ q. + 1
b e t w e e n X = X ' a n d X ~ X " . W e shall call this circular arc C~, a n d shall p r o v e t h a t it lies in t h e h a l f plane Y > _I. 2 Since C. has its c e n t r e on t h e X- axis a n d is t h e r e f o r e concave t o w a r d s this axis, it is clear t h a t in o r d e r to p r o v e t h a t C, lies wholly in the half p l a n e Y > _I it is sufficient to show t h a t ~
2
I T , > I i t , , I
2 2
N o w b y a k n o w n p r o p e r t y of c o n t i n u e d f r a c t i o n s I p , , - q,~ gl < ~ ;
q n + 1
so t h e coefficient of Y ' in (I~) is less t h a n
I I 2
2 -~-~ < - ~ ' qn + 1 q~, q~
f r o m w h i c h it follows t h a t Y ' less t h a n
a n d so Y "
> - " I Similarly t h e coefficient of Y " in (I2) is
2
I 017 2 I
qn + 1 qn + I qn + I
I 2
L e t us n o w consider the points
I 2
2 2
q n + l q n + l
T = n,(o)= X~(a)+¢ }.~ (~) (. ->_ 2)
o b t a i n e d by effeeting t h e t r a n s f o r m a t i o n s S~ on t h e pointsw h e r e
= ~,, (~) - - g + i v,, (~),
(I I - - ( 7
y , , ( ~ ) = ~ + ~
~/n q n +1
* Cf. H A R D Y a n d LITTLEWOOD, loc. c i t . , p. 229.
152 K. Ananda-Rau.
After multiplying each of the equations (9) and (~o) by q S, and after a slight re- arrangement we get
qn-t- 1) ]
(I3)
[ qn (pn_lqn_l__~q2n_,)(p,,q,,__~q])÷ qn-_ I{ qg l~l
= V,~+l.lq,,_ 1 q- (y -~ (i - - (Y) q 2 ÷ l j ]
y
q,~ + 1 ) .1 q,~ + 1
We n o w keep a fixed in the interval o --~ a ~ I and allow n to tend to infinity by taking all values congruent to a fixed Q (mod. g). Since n has the same parity, V,~+I has the constant v a l u e V,o+l. The coefficients of
X~(a), Y~(a)
and the ab- solute terms on the right hand sides of the equations (I3) and ( i 4 ) t e n d to finite limits by Lemma I. HenceX~ (a), Y~(a)
tend respectively to limits ~e (a), 0e (o) given by3~ e (a) [AS e + (a + (I -- a)LSe+,) e]
I-de-1 -//e ]
-~n,o+l[ L¢- + L e ( a + ( I - - a ) L ~ + l } ~ '
~() (if) [//~ + {q -f" (I - - q) n ~ + l } ~] : q + (I - - o) Le+l" ; or, on writing for shortness,
a ~ - (I s
(15)
s s ]
3~ e ( a ) ( A s s + J ~ + l ) = v e + l / L¢ + LeJe+l , A s 2
9q(")( +
Further, since z, (o) lies on the line x ~ ~ between the points z' and v", it follows that
Tn (a)
lies on the arc C,~; and since for all values of n, C~, lies in the half- plane Y > _I2, it is clear that for all values of n and a,Y,(a)
> ~; and, there- fore De(a)>--I for all values of Q and a. We have thus proved the following2 lemma.
L e m m a 2. I f a has a fixed value i n the interval o <--a < I, a~d n tends to infinity by taki,g all values congruent to a fixed e (rood. g), then the poi,nts T~, (a) tend to the definite limiting position
~Q (~) = ~¢~ (~) + i 9Q (~);
a+~d this limiting point h:es in the half plane Y >-- I.
2
I t is of i n t e r e s t to observe t h a t , if we let ~ t e n d to infinity in t h e m a n n e r described above, the arcs C. t e n d to a l i m i t i n g arc ~e, whose e q u a t i o n is
a n d w h i c h is i n t e r c e p t e d
~,o, ~,o are given b y
A o (X ~ + Ye) + D e X + A o-~ - - o,
b e t w e e n the lines X ~ =,o-' a n d X = =o-", where Do,
D e = ~ + LQ
t
[~'~ ~ -- i A 0
]
VQ+I[ L,o + G J
A e q - i 2
~¢+~ Lo + L e Le+I
÷ Le+I
I t is easily seen t h a t , if Q is k e p t fixed, a n d a varies b e t w e e n o a n d I, t h e points
~ ( ~ ) describe t h e arc ~e; so t h a t the p o i n t s ~e(a) (for all values of e a n d a u n d e r consideration) lie on a finite n u m b e r of circular arcs (~e, all of w h i c h lie in t h e half-plane Y - - > - " i
2
III.
Periodicity of Certain Sequences.
7. This section is d e v o t e d to t h e p r o o f of some c o n g r u e n c e p r o p e r t i e s of p,~, q~, the n u m e r a t o r s a n d d e n o m i n a t o r s of t h e c o n v e r g c n t s of the c o n t i n u e d
f r a c t i o n for ~, a n d also to the p r o o f of some p r o p e r t i e s of t h e n u m b e r s
2 0 - - 2 8 2 2 . A c t a mathematica. 52. Imprim4 le 22 ao~t 1928.
154 K. Ananda-Rau.
where the symbol [ b ] ( b ) " The symbol [ b ]
[
q n - ~ ] , ( n = 2 , 3 . . . . )~,, q n J
denotes a generalisation of the LEGEND~P,-JACOBI symbol appears in the formulae of the linear transformation of the Theta functions, and its definition and fundamental properties are given by TANNElCY and M O L K . 1
8. W e shall begin with a few definitions, which will facilitate the descrip- tion of what follows. Let us write the continued fraction for ~ in the two forms
[ b i , b e , . . . b , n , a l , a , , . . . arj , [Cl,Ce, • . . C n, • . .].
W e have to consider some infinite sequences for which it will be convenient to adopt two notations. Let
~ ' ~ , 7 , . , , • • . ) ' n , • • •
be the sequence, written in the usual way, the nth term being donoted by 7n. Now we associate this sequence with the sequence formed by the partial quotients of the continued fraction for ~, so that 7~ corresponds to e l , 7e to e.2, and in general 7,~ to c,. F o r n > m, this establishes a correspondence between the 7's and the a's.
We shall regard a period of the partial quotients of the continued fraction to begin e with al and end with at; so that, when we speak of the partial quotient
as (I --< s --< r) in the j t h period, the rank of the corresponding c in the c-sequence will be unambiguously determined. In fact the corresponding c will be e m + ( j - 1 ) , . + s .
Now returning to the correspondence between the 7's and the a's, we shall de- note by F~ 5) the member of the 7-sequence which corresponds to the partial quotient a~ (i~< s ~ r) in the j t h period; that is to say,
F ~ ) ~ 7 m + ( j - i ) r+s "
1 F. E. pp. x o 9 - - x I I .
2 O w i n g to t h e cyclic order in w h i c h t h e a ' s a p p e a r as p a r t i a l q u o t i e n t s in t h e c o n t i n u e d fraction, one m a y r e g a r d (in t h e n o t a t i o n g i v e n in L e m m a I) a period as b e g i n n i n g w i t h a n y d i ( I ~ t < r ) a n d e n d i n g w i t h d t + r _ 1. T h e c o n v e n t i o n a d o p t e d h e r e is n e c e s s a r y to m a k e o u r de- finitions, t h a t follow, u n a m b i g u o u s .
To i n d i c a t e t h a t t h e t e r m s of t h e y-sequence F - n o t u t i o n we shall w r i t e
} = { r J)
8" )" "are denoted by the alternative
W e n e x t define a periodic sequence. A s e q u e n c e
~ , 7 . ~ , - - - y n ~ . . .
is said to be p e r i o d i c , if t h e r e are t w o positive i n t e g e r s n o a n d E , so t h a t f o r e v e r y n > ~o
)'n+ £ = )'n "
E is called ~ period of t h e sequence. Obviously, if E is a period, a n y i n t e g r a l m u l t i p l e of E is also a period. 1
9. L e m m a 3. Suppose for n = i, z, 3 , . . . , ~,, is the integer such that
Then the sequence is a periodic sequence. ~
L e t
I < - - p n < - - 8
P1,~.2,... 0 , , . . .
{ } = { 1"? }
~ O n f - - " l 4%' ) "
(rood. 8).
1 W e c a n easily p r o v e t h e f o l l o w i n g p r o p e r t i e s of p e r i o d i c s e q u e n c e s . (i) I f E l , E~ a r e t w o p e r i o d s of a periodic s e q u e n c e (yn), a n d if d is t h e h i g h e s t c o m m o n d i v i s o r of E , , E 2 , t h e n d is also a period. (ii) If l is t h e l e a s t p e r i o d of t h e s e q u e n c e , t h e n e v e r y o t h e r period is a m u l t i p l e of l. To p r o v e (i) we o b s e r v e t h a t t h e r e e x i s t t w o i n t e g e r s n l , n ~ s u c h t h a t n i E ~ - - n 2 E2 = d. I f n is s u f f i c i e n t l y l a r g e we h a v e
~,n = ~/nq-n 1 E 1 (since E 1 is a period)
7 ~ + n l E 1 = 7n+~h E~--n~ E,,, (since E , is a period)
= y n + d ;
a n d so 7 n = y n + d , s h o w i n g t h a t d is a period. To p r o v e (ii) l e t E > l be a period, a n d l e t d b e t h e h i g h e s t c o m m o n d i v i s o r of E a n d 1. T h e n d - < l ; also b y (i) d is a period. If d < l , 1 w o u l d n o t be t h e l e a s t period. T h e r e f o r e d = 1 a n d E is a m u l t i p l e of l.
T h e s e q u e n c e f o r m e d b y t h e r e s i d u e s of p ~ , p ~ , . . , to a n y fixed n m d u l u s M (the r e s i d u e s l y i n g b e t w e e n I a n d M ) is also periodic. T h e p r o o f is t h e s a m e as t h a t for t h e case M = 8 g i v e n above.
156 K. Ananda-Rau.
W e shall u n d e r s t a n d t h a t t h e c o n g r u e n c e s which a p p e a r in t h e course of t h i s l e m m a a r e all to m o d u l u s 8. To p r o v e t h e l e m m a , l e t u s s u p p o s e f o r t h e mo- m e n t t h a t ~here are t w o i n t e g e r s
h, k, (k > h)
so t h a t( I 7 )
T h e n since *
it follows from (17) t h a t Now from
we deduce similarly t h a t a n d so on, till we p r o v e t h a t
#h) -.(h) ip~Z,,)
r)(h) #k)
v?)-= p?;
I o(h)
J. r - - 1 = P ( k ) l
F r o m t h e s e we deduce successively as b e f o r e
p~h+l) l)(k+l)
I t is c l e a r t h e a r g u m e n t can be r e p e a t e d indefinitely; so t h a t w h a t we h a v e p r o v e d is t h a t , if in t h e sequence
~I~2~''"
t h e r e a r e t w o sets of consecutive n u m b e r s (~h), ~ h ) ) , ( ~ k ) , ~ k ) ) SO t h a t
z I n o r d e r to s h o w clearly t h e c o n t e n t s of t h e p r o o f i t is s u p p o s e d h e r e t h a t r >--4. T h e f o r m a l a l t e r a t i o n s n e c e s s a r y w h e n r < 4 c a n b e e a s i l y seen. W h e n r = I i t will b e c o n v e n i e n t to r e g a r d t h e p e r i o d as c o n s i s t i n g of t w o e q u a l p a r t i a l q u o t i e n t s ; t h i s ' i s c l e a r l y p e r m i s s i b l e .
t h e n the sequence (I6) is a periodic one, and ( ] ~ - - h ) r is a period of the sequence.
To complete t h e p r o o f of the l e m m a it i s only n e c e s s a r y to show t h a t t h e r e are two such sets s a t i s f y i n g (I8). N o w this is obvious since, if we consider t h e sets
(~i'>, ~'>), (z-- ~, 2,...)
t h e r e can only be a finite n u m b e r of distinct sets, as every ~ is one or o t h e r of the n u m b e r s I, 2 , . . . 8.
T h e a r g u m e n t used above enables us to p r o v e t h e f o l l o w i n g l e m m a also.
L e m m a 4. Sup~)ose for n - - I , 2, 3 , ' " ", q" is the integer such that
Then the sequence
(i9)
is a periodic sequence. 1
I ~ fin ~ 8
q,~=--q~ (,~od. 8).
Q I ' ~ 2 ' ' ' "
W e shall p r o c e e d to some l e m m a s on t h e n u m b e r s [ q ' ) - ~ ] • I f a, b
IO. [ J
are two i n t e g e r s p r i m e to each o t h e r (they m a y be positive or n e g a t i v e ) t h e sym-
bo,/; /
°oo o,w h e n a, b are given, is described by TANNERY a n d MOLK. I n t h e applications t h a t follow a w i l l ~lways be positive. I n q u o t i n g t h e following p r o p e r t i e s of t h e symbol f o r f u t u r e r e f e r e n c e the f~ct t h a t a is positive is t a k e n into account. ~
(,o, [ ; ] [ q : ,
(22) ~ = exp I - 4
I~_l (2I) c, d are a n y two integers '~ such t h a t c z d - - b e = I.
1 T h e p e r i o d s of t h e p- a n d t h e q - s e q u e n c e s are, of course, n o t n e c e s s a r i l y t h e s a m e . I n t h e c o u r s e of t h e p r o o f of L e m m a 3 t h e period w h o s e e x i s t e n c e is p r o v e d is a m u l t i p l e of r. B u t s m a l l e r p e r i o d s m a y v e r y well e x i s t , w h i c h arc n o t d i v i s i b l e b y r. T h u s , f o r e x a m p l e , if e v e r y a is a m u l t i p l e o f 8, it is e a s i l y s e e n t h a t 2 is a period.
F.E., p. IO 9.
8 I h e d e p e n d e n c e of [a-+~-b- 1 on c, d is o n l y a p p a r e n t . See t h e r e m a r k s i n ' F . E . , p. IO8.
L v J .
158
L e m m a 5.
where
We apply the
K. Ananda-Rau.
TVe have
L q,~ J L~-~J e12 ,
2
formula
(2I)with the following sets of values of
a, b,c, d:a = q . - i + v q . , b - - q , , ,
Here v is given successively the values o, I, 2
. . . . ( e n + l - - I ) .value of v
a d - - bc = ~ (p. q n-1 - - p ~ - i q.) = i,
and so formula (2I) is applicable.
We have
[
q.,-1Obviously, for every
q. t qn.,
~ ~'~
q" ( q " - ~ ) (~P"-' +p") '[q,-i = I q,,-!-+ q'!] exp { ~ i ~ ~]n q. (qn-- I)
2 (2iOn_l@3p'n)}
/ q .
L q.
[ q . - l _ + 3 q n ]
= [ q . - l + 2 q . ] exp [Tri 2 l
~_ qn J [ I 2 ~n qn (qn-- I)(2pn--X + SPy) '
- ~ C . + l q . ] = [ q . - ~ + ( c . - ~ l - - , ) q . ] q,, q,~
exp { z i
~ V,~ q,~ (q,~-- I) (2p.-1 + 22
e n + l - - - I p ; ~ )}
"On multiplying these equations and cancelling the common factors which appear on both sides, and remembering that
qn--1 + en+l qn = qn+~, I + 3 + 5 + • ' " + (2On+l--I) = C n + l , 2
we get
{
[ o+11 1
a n d u s i n g t h e i d e n t i t y
2 Cn+l pn--i -~ en+l pn 2 ~ en+i (pn--1 + pn--1 + en+l pn)
we get t h e r e s u l t of t h e lemmu.
L e m m a 6. W e have
where
--
e,,+~ (p,~-i + p,,+~),
~.+1 qn+l] L?}. qnJ
wn=e,,+aq,,(qn---I)(p,,--a+pn+l)N 3~n(qn+l--I)(qn-- 2 I).
First, let n be even, so t h a t
L~],~+ 1 q,,+l]
Now using f o r m u l a e (20) and (22)
exp F r o m these two we g e t
[q. 1__ p.+q
--q~+,_l L q,, Ja n d so by L e m m a 5
[ ~ i ,
(23) [--q"
L--qn+lJ] = [q'---']
L qn Je~1~ I ~*"
R e m e m b e r i n g t h a t n is even, ~7,~---I, it result of t h e lemma.
5~ext, let n be odd, so t h a t
I q.-ll P"-~I
--qn+lJ t q ~ + l ]
{--~(qn÷1--I)(qn--')} -- f an 1 I~n÷lluqn+l_I L qn J
~ n + l q . + l J Lq~,+ 1]
exp 4 ( q ~ ' + ' - - i ) ( q n - - I) ;
• }
+'*(q.+l-~)(q,,- ~)
4
is seen t h a t (23) is e q u i v a l e n t to the
~ q,~J L - - q._l
160 K. Ananda-Rau.
By L e m m a 5
By (2o)
Also by (22)
exp
[qn+l]
_Vn
L q,~ J I_ qn J
q " - 1 1 [ q ' - l l : I .
q, J L - - q,~J
{
_ L/__ ( q , , + l - ~ ) ( q . - ~) =] l-q,,+1] l ]
- - L q. JOn multiplying these three equations we get [
1 _ qL~-j_ exp - - ~ ( q u + l I ) ( q n - - I ) = ~1 ; "
t - - q . J and so
(24) [ q n ] = [q~--l] exp { - - v , , - - - - ( q n + 1 - - I ) ( q n - - I ) ~ i ~ i } "
L q n + l l L - - q - J I 2 4
Since n is odd, V,~----I, and (24) is equivalent to the result of the lemma.
L e m m a 7. L e t
~,, = p , = l | S ;
t?2. q,,J then the sequence
C~, C,~, • • • C,,, • • • is a periodic sequence.
Let A, B be resPectively
q-sequencel Let F be the least common multiple o f 2, r, A, B.
3 and 4 there is an integer N' so t h a t for n ~ _N'
the smallest periods of the O-sequence and the By Lemmas
p,,+A ~p,,, ] (mod. 8).
qn+B ~ qn J
To prove the present lemma, suppose for the m o m e n t that there are two integers h, k satisfying the following conditions:
( 2 5 )
where a is a positive integer.
k > h > m h > N '
k - - h = c ~ F ,
(26)
w h e r e
By L e m m a 6 we h a v e
~l)h = Ch + l qh (q2 h -
I)(~9h-1 ~- ~011+1)~-
3 ~h (qh+ l - -I)(qh-- I),
Wk = C k + l qk (q~mI)(pk--I -~ pk+l) @ 3 ~k (qk+l -- I ) ( q k - I).
N o w since k - - h is a m u l t i p l e of ~', a n d /c > h > m,
since k - - h is even,
C h + l ~ Ck+l;
since k - - h is ~ m u l t i p l e of A, a n d h > N', h - - I --> 2V'
ph--I pk--1
)
(mod. 8) ; ph+l ~ pk+l
)
a n d l a s t l y since / c - - h is a multiple of B, a n d h > N '
qh ----~ qk / (rood. 8).
]
q/~+l ~ qk+l U s i n g these results we see t h a t
a n d so
Also by o u r assumption a n d t h e r e f o r e f r o m (26)
wl, ~ w/: (rood. 8),
r~ l" z i e ~ - Wh ~ e 4 - Wk.
S t a r t i n g f r o m this r e s u l t we can a r g u e similarly a n d p r o v e t h a t
a n d so on; so t h a t , t h e ~-sequence will be a periodic k - - h = a F .
21 - - 2822. A c t a m a t h e m a t i c a . 52. Imprim~ le 23 aout 1928.
sequence w i t h period
:162
two i n t e g e r s h, k, s a t i s f y i n g the conditions (25).
{C,,} - { z % F : G r , so t h a t G is an integer.
t h e n t > N'.
L e t u s consider the five n u m b e r s (27) Z(/~I), Z<I] +°), ZIfll +20),
K. Ananda~Rau.
W e wiil h a v e t h e r e f o r e p r o v e d .the lemma, if we establish t h e existence of To do this let I
F u r t h e r , let fl be an i n t e g e r so large t h a t , if Z(l ) = C,,
Z(fl+aq), ~(l~+ta) I J 1
Since every ~,, can have only one of t h e f o u r values + I , - - I, + i , - - i , t h e r e m u s t be at least two a m o n g t h e n u m b e r s ( 2 7 ) w h i c h a r e identical. T h e r e are there- fore two integers 7, 6, (d > 7) a m o n g t h e n u m b e r s o, I, 2, 3,4, so t h a t
Z(fll+y c,) = ~(fl+~l a)
I f t h e s e Z's are respectively ~h, ~ w h e n considered as m e m b e r s of the ~-sequence, t h e n clearly
k > h > m h > ~ t > N '
k - - h - - (6--7) G r = a F ,
a being the positive i n t e g e r d - - 7 . T h e n u m b e r s h, k t h e r e f o r e s a t i s f y t h e con- ditions (25), and t h e l e m m a is e o m p l e t e ! y proved.
Since 7, d are a m o n g the n u m b e r s o, I, 2 , 3 , 4 , it follows t h a t t h e ~-sequenee has a period a F w h e r e a is one of the n u m b e r s 1 , 2 , 3 , 4 .
IV.
Behaviour of as (o I
~).I I. W e shall apply t h e results established in the last two sections to o b t a i n t h e b e h a v i o u r of ~ (oI~) as ~--~ ~ a l o n g the line x = ~.
i The absence of the first term ~t is of no importance. As usual ~n is supposed to correspond to c n .
I f
c + d T
T - -
a + b T '
where a, b, c, d are i n t e g e r s such t h a t a d - - b e - - - , , t h e n it is k n o w n t h a t t (29) "~3 ( 0 ] T) ~t~' ] / a -]- b r ' t,tO'/# +1 ( 0 I ~ ) '
where ~" is an e i g h t h r o o t of u n i t y , tt is one of the n u m b e r s Ii 2, 3, a c c o r d i n g to the t y p e of the t r a n s f o r m a t i o n (zS), a n d t h e s q u a r e r o o t ] ~ a + b T has t h a t d e t e r m i n a t i o n w h i c h has its real p a r t positivefl T h e value of e" is given by the f o r m u l a e 3
8PP /
(30) ~ - - exp
- - ( a b + e d + 2 b c + 2 a + 2 c + 2 m " )
(3') exp
T h e n u m b e r m " in (30) is e i t h e r , I or b + d a c c o r d i n g to t h e t y p e of the trans- f o r m a t i o n (28).
T h e r e are six t y p e s of t r a n s f o r m a t i o n s a c c o r d i n g to the p a r i t y of the num- bers a, b, c, d. T h e d e p e n d e n c e of tt a n d
m"
on a, b, c, d is shown ill the f o l l o w i n g table. 4 I f a is odd, t h e n u m b e r I is e n t e r e d in the a-column; if a is even, t h e n u m b e r o is entered. S i m i l a r n o t a t i o n applies to b, c, d.T y p e a b c d
I ° , , , . . • . . . . .
2 ° . . . , . . . . . . .
o . . . . . . . . . . .
4 ° . . . . . . . .
6 ° . . . . . . . . . . .
I
I I I 0 0
0 0 I I I I
I I
I
O O I
m t t
2 - - I
3 b + d
I - - I
3 b + d
2 - - I
I - - I
1 F . E . p . 262, f o r m u l a (3) w i t h v ~ V ~ o. I h a v e s l i g h t l y a l t e r e d t h e n o t a t i o n a n d inter- c h a n g e d v a n d T.
F . E . p . 9 I .
3 F . E . p . 262 f o r m u l a e (5) a n d (7).
4 F . E . p . 241 T a b l e (6), a n d , p. 262 T a b l e (8).
164 K. Ananda-Rau.
I2. L e t us n o w consider the t r a n s f o r m a t i o n s S,~ m e n t i o n e d in Section I I ,
P,,-~ + v,P,~ T (n = 2, 3 • .).
(3 2) .t: = q,-1 + V,, qn T ' '
T h e y are all t r a n s f o r m a t i o n s of t h e m o d u l a r group, since
T h e n u m b e r s re, s", m " c o r r e s p o n d i n g to S,~ will n a t u r a l l y d e p e n d on n, a n d to indicate this d e p e n d e n c e we shall write
vv = O)ls
7~ vv = m n w.
As in L e m m a 7, let F be t h e least c o m m o n multiple of 2, r, A, B, and let H = a F be a period of t h e ~-sequence. T h e n by L e m m a s 7, 3, 4 t h e r e is an i n t e g e r N " such t h a t f o r n > - N "
~ n + H ==- ~n~
p,,+H ~-p,~+.4 ~ p , ~ t (rood. 8).
J
q n + H ==- q~+B ~ qn W e h a v e t h e f o l l o w i n g L e m m a .
L e m m a 8. L e t R be a f i x e d integer i n I <-- R <-- H . L e t n > N " a n d n ~ R (rood. H ) . T h e n f o r all values o f n w h i c h s a t i s f y these two co~ditio~s ~o,~ has a constant value 1, a n d the t r a n s f o r m a t i o n s S,t are all o f the same type.
Since n > N " a n d H is a multiple of A, all t h e n u m b e r s p,~ are c o n g r u e n t to each o t h e r to m o d u l u s 8, a n d so t h e i r p a r i t y is t h e same; similarly since n - I >-- N " , the p a r i t y of the n u m b e r s pn,1 is t h e same. Since H is a multiple of B, t h e same a r g u m e n t shows t h a t t h e p a r i t y of t h e n u m b e r s qn a n d of t h e n u m b e r s q,~-i r e m a i n u n a l t e r e d , sO t h e t r a n s f o r m a t i o n s S~ are all of the same t y p e ; M= t h e r e f o r e r e t a i n s a c o n s t a n t value. I f t h e c o m m o n t y p e of the trans- f o r m a t i o n s Sn is e i t h e r o , 3 o, 5o or 6 °, m , / ' r e t a i n s t h e c o n s t a n t value - - I . I f the c o m m o n t y p e is e i t h e r 2 ° or 4 ° , m,/' ---- v , (p, + q,) ; since H is even, n has
1 This implies that the w-sequence is periodic with period H.
t h e s a m e p a r i t y a n d ~],, h a s t h e c o n s t a n t value ~R; t h e r e f o r e f r o m t h e c o n g r u e n c e p r o p e r t i e s of pn, q,, to m o d u l u s 8, we see t&at m,," t a k e s values w h i c h differ by m u l t i p l e s of 8.
N o w by (30) a n d (3I)
w h e r e
[ q n _ l l S " i iv
= e ; " = e ]+'
W,, -~ (a b + e d + 2 b e + 2 a + 2 e + 2 m , " ) + ( a b + a e + b d - - a c b ~ ' - 3b), a = q~-~, b ~ ~ q , , c - - pn-1, d - - ~]npn.
Since n > N " , n----~ R (mod. H ) , ~, r e t a i n s a constan~ value by L e m m a 7. Also V,* h a s t h e c o n s t a n t value yR. F r o m t h e c o n g r u e n c e p r o p e r t i e s (rood. 8 ) o f p,~-l, pn, q,~-~, q~, m,,", it follows t h a t W~ t a k e s values w h i c h differ by mul- tiples of 8. H e n c e , finally, ~o~, r e t a i n s a c o n s t a n t value.
T h e c o n s t a n t values of =]I~ a n d ~o,~ will, in general, d e p e n d on R a n d we shall d e n o t e t h e m r e s p e c t i v e l y by m = m (R) a n d £2 ~ Y2 (R). F u r t h e r let Z - - Z (R) be e q u a l to + I or - - i a c c o r d i n g as R is even or odd.
13. W e are n o w in a p o s i t i o n to p r o v e t h e following, w h i c h is t h e m a i n t h e o r e m of this paper.
T h e o r e m . Suppose a is a fixed number in the in ter~;al o <--a < i, and R is a fixed integer in I ~ R <-- H. L e t g be the least common multiple of 2 and r;
and let q be the integer such that I ~ e <- g, Q ~ R (rood. g). F u r t h e r let
where
(~ I - - 0 "
= +
qn qn+l
L e t n tend to infinity by taking all values congruent to R (rood. H). Then ~
4 4
- i J,o+l a m (o /
%),
1 The n u m b e r s z/Q, JQ+I = J ~ + l (a), ~0 = %Q (a) are those defined in Section II.
166 K. Ananda-Rau.
4 4:
where ]fy; V~e+ ~ denote the real positive fourth r o o t s 1, a,Td V A Q - - i J o + ~
denotes that determination of the square root .which has its real part positive.
W e m a y clearly suppose t h a t n > N " . L e t ~ ~ ~+~ (a) and T = Tn (a) be con- nected by the relation (32 ) or its equivalent
(33) T---- ~,,+I - - p~,-I -- q,--I T
W e shall apply f o r m u l a (29), the t r a n s f o r m a t i o n (32) t a k i n g the place of (28).
Since n > _N", n ~ - R (mod. H ) we have, on using the result of L e m m a 8,
(34)
the square root having its real part positive.
I t is easily verified from (33) t h a t
q,~-~ + '2,~ q,~1' (p,~---. ~ q,,) -- i q,, y R e m e m b e r i n g t h a t ~ has the c o n s t a n t value ~R, we have (35)
Now
V (qn_, + q,,T) = q'' 2 2
(p,,,q,~--~q,)-- i q , , y n --- R ( r o o d . H),
R ~ 0 (mod. g),
and so
Therefore, by L e m m a I
H ~ o (mod. g), n--~Q (mod. g).
2
J o + l q,,y -- a + ( I - - a ) - ~ - --) q" (7 + (I - - ( 7 ) L G + I = 2
qn+l
(p,,q,, . 2
-- - - z q , , y - + / l e - - i J e + 1.
4 4
' In what follows If y , ] f y , ] l j Q ~ l l , ]f~rT+ 1 denote the real positive roots.
Using these in (35) we see t h a t
V~j (qn-i + ~n qn T) -*
a n d so
(36) ]~y. ~-q,,--i
~RV~+I
_/l o - i J o + ~ '
4 + ~n qn T _.._ ~ ~ R gJ-0 ÷ 1
g-//e -- i J e + l ' where the left h a n d side has its
I/Je+i
4 is real a n d positive andreM part: positive. On the r i g h t h a n d side
has its real part ~tt posiLive. We have therefore t o choose t h a t d e t e r m i n a t i o n of 1 / ~ which makes the "real part of the r i g h t h a n d side in (36) positive. Observ- i n g t h a t Jo+~ > o a n d ~ > o, it is easi!y seen t h a t ,~ < o, altd so we should take
(37) ]ZVR -- Z (R).
N e x t since n ~ q .(rood. g) we see f r o m L e m m a 2 t h a t
r~ (~) -~ % (~) = ~,o (~) + i ~ (o),
a n d since by the same L e m m a2
it f o l l o w s from the continuity of the T h e t a functions t h a t
(38) ~ (o I T)-, a~ (o j ~ ) .
C o m b i n i n g (34), (36), (37), (38) we g e t 4
z t~ V J(,+ l
which is the result of the theorem.
I4. Before we conclude, a few remarks m a y be made on the m a g n i t u d e of the number H. R e f e r r i n g to the proof of L e m m a 3 we see t h a t a period of the
168 K. Ananda-Rau.
b-sequence exists which is n o t greater t h a n N r , where N is t h e n u m b e r of distinct sets
(1= T, 2, 3...),
the integers ~ being a m o n g the numbers I, 2, 3 , . . . 8. Now r e m e m b e r i n g t h a t p,~:-l,p,~ are prime to each other (since p~q,~-l--p,~-~ q , ~ : ± I) we see t h a t ~[~), ~t) cannot b o t h be even; a n d an easy cMculation shows t h a t a Y = 4 8 . Therefore there exists an N~--< 48 such t h a t N x r is a period of the 10-sequence. Similarly there is an 2V~--<48 such t h a t N o r is a period of the q sequence. The least common multiple of 2, r, NI r, N~ r is easily seen to be n o t greater t h a n g ' 4 7 • 4 8 - 2256 g.
Since the highest value of a is 4, it follows t h a t there exists a suitable v a l u e of H n o t g r e a t e r t h a n 9o24g.
An examination of the proof of the main theorem, however, shows t h a t it is n o t the numbers A , B by themselves t h a t play an essential role, b u t it is the period of the co-sequence t h a t is important. Of course, to ensure t h a t the trans- f o r m a t i o n s S~ (when n is sufficiently large and n---R (mod. H)) are of the same type, we should take into consideration the periods of the sequences
(39) ~ ' , ~._,', • •.,
(40) el/, q..', . . . .
where e~eh ~,~' and qn' is either I or 2 a n d
~n' ---p,~. ]
(mod. 2).
I f A ' , B ' are respectively the least periods of the sequences (39), (40), and if Z is the least period of the o)-sequenee t h e n the least common multiple of 2, r, A', B', Z will provide a suitable vMue of H ; a n d this value will be, in m a n y instances, m u c h less t h a n the b o u n d 9o24g assigned above. I t can be shown by a r g u m e n t s similar to those used for the I~-sequence t h a t 6 r is a period of the sequences (39) a n d (40). H e n c e A', B' are divisors of 6 r (see the first foot-note on page i55); a n d there is a value of H n o t greater t h a n the least common multiple of 6 r and Z-
