pk(n), ASYMPTOTIC PARTITION FORMULAE. III. PARTITIONS INTO

ASYMPTOTIC PARTITION FORMULAE. III.

k-th P O W E R S . 1 BY

E M A I T L A N D W R I G H T of OXFORD.

PARTITIONS INTO

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

I . I . I n a well-known memoir 2 Hardy and Ramanujan obtained the asymp-

totic expansion of p(n), the number of unrestricted partitions of n, by applying Cauchy's Theorem to the generating function

o o a r

H (I - - xl) - 1 - - - I "~- Z 2 ) ( T ~ ) X n . (I. I I)

1-- 1 n = l

My object here is to find the asymptotic expansion of

pk(n),

the number of partitions of n into k-th powers. 3 The result is to some extent a generalisation of theirs, and part of this paper consists of an ~pplic~tion of their method to the new generating function

f ( x ) = H ( I - - X l k ) - - i = I -~ Z p k ( . ) X n.

/=1 n = l

1 T h e p r e v i o u s p a p e r s of t h i s series are

I. P l a n e p a r t i t i o n s . Quart. J. of Math. (Oxford series), 2 (193~), I 7 7 - - 1 8 9 . II. W e i g h t e d p a r t i t i o n s , t)roc. London Math. Soc. (2), 36 (I933) , I I 7 - - I 4 I . T h e p r e s e n t p a p e r m a y be read i n d e p e n d e n t l y of t h e s e .

s London Malh. Soc. (2), I7 (I918), 7 5 - - I I 5 . T h e r e a d e r will find it i n t e r e s t i n g to c o m p a r e t h e m e t h o d s a n d r e s u l t s of t h i s p a p e r w i t h m y w o r k here.

T h e p r o b l e m w a s s u g g e s t e d to m e b y P r o f e s s o r H a r d y , to w h o m m y t h a n k s arc also d u e for m u c h v a l u a b l e advice in t h e c o u r s e of t h e i n v e s t i g a t i o n .

144 E. Maitland Wright.

B e f o r e we can use this m e t h o d , however, we h a v e to solve t w o subsidiary p r o b l e m s which are m u c h m o r e c o m p l i c a t e d f o r g e n e r a l k t h a n in t h e p a r t i c u l a r case k ~ i c o n s i d e r e d by H a r d y a n d Ramanu.]an. I t is obvious t h a t f ( x ) i s r e g u l a r f o r I x ] < I a n d t h a t every p o i n t of t h e c i r c u m f e r e n c e I x ] = I is an e s s e n t i a l s i n g u l a r i t y of t h e f u n c t i o n . W e m u s t k n o w the n a t u r e of t h e singul- arities of f ( x ) a t t h e r a t i o n a l p o i n t s

2pzi x : e q

or, w h a t is t h e s a m e thing, t h e b e h a v i o u r of f ( x ) as x a p p r o a c h e s such a Point.

T h i s i n f o r m a t i o n is c o n t a i n e d in a c e r t a i n t r a n s f o r m a t i o n f o r m u l a .

W h e n k ~ I , f ( x ) is, a p a r t f r o m a t r i v i a l factor, t h e r e c i p r o c a l of a n elliptic m o d u l a r f u n c t i o n , a n d t h e t r a n s f o r m a t i o n t h e o r y follows a t once f r o m t h e pro- p e r t i e s of such functions. F o r g e n e r a l k this is n o t t h e case, a n d we h a v e to develop t h e t r a n s f o r m a t i o n t h e o r y of f ( x ) e n t i r e l y a f r e s h . This w o r k t a k e s up t h e whole of P a r t I of this paper. W e s t a t e t h e r e s u l t i n g f o r m u l a as a t h e o r e m ; it is a p p a r e n t l y new a n d possibly of s o m e i n t e r e s t in itself, a p a r t f r o m the a p p l i c a t i o n w i t h which we are h e r e chiefly concerned.

W e find t h a t as x a p p r o a c h e s I, f ( x ) b e h a v e s to a first a p p r o x i m a t i o n like

w h e r e l"(t) a n d ~(t) are t h e o r d i n a r y G a m m a a n d R i e m a n n - z e t a f u n c t i o n s . This b r i n g s us to o u r second s u b s i d i a r y p r o b l e m , n a m e l y t h e d e t e r m i n a t i o n of an a u x i l i a r y f u n c t i o n w i t h an i s o l a t e d s i n g u l a r i t y of t h e t y p e (I. 12) at x = I . I n a r e c e n t n o t e ~, I i n t r o d u c e d a g e n e r a l i s e d Bessel f u n c t i o n to e n a b l e m e to c o n s t r u c t such an a u x i l i a r y f u n c t i o n , a n d discussed its properties.

W e write

a ~ ] , I b - - k + I I , j = j (k) : o (k even),

j .... j ( k ) _ ( 7 I)1"2;k~1)i-(~_ i)~(k + i ) ( ~

o d d ) . 2

1 Journal London Math. Soc., 8 (I933), 7I-7-79.

l R'eaders familiar with tile memoir of Hardy and R~manujan 2 If k = I, then j - 24"

referred to a b o v e will recollect t h e a p p e a r a n c e of t h e n u m b e r . . . . I 24

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 145 Then the a p p r o p r i a t e generalised Bessel f u n c t i o n h e r e is

c o

,-0, (,o_;)

and our simplest result, f o u n d by e x a m i n i n g in detail the singularity o f f ( x ) at x =: I, is c o n t a i n e d in t h e following theorem.

T h e o r e m 1. We. have

whel'e

lb.(n) -- (n

+,}')-~'-(2 7v)-i~:q~ {l'(I 4 a)~(I + a)(n +3")"} +

O(e('t-")n"),

/Jr "--(~:~- I ) { a l ' ( I ~-a)~(I ~-a)} l-b, of = . ( k ) > O.

Using the k n o w n a s y m p t o t i c expansion ~ of q~(z) we can deduce the expan- sion of pk(n) in t e r m s of e l e m e n t a r y f u n c t i o n s of ~.

T h e o r e m 2.

w h e l ' e

We have

9 { (-(, ^{+' )" flfj}

J)k(??) = n O ( " ' + j ) b - 3 e xl(n+j)b I + ~Z_.,,j I ~-

j k'~ 1

and b,,, is a function of k a n d m only, which may be calculated with st(fficient labour for a~y given values of k and m; in particular

I i k e + I l k + 2 bl --: 24k,4 "

H a r d y and R a m a n u j a n (loe. cir. p. I I I ) give the result z ed n b

w i t h o u t proof. This is e q u i v a l e n t to T h e o r e m 2 with M:--: I, b u t w i t h o u t any s t a t e m e n t as to the o r d e r of t h e e r r o r term. A p a r t f r o m this, T h e o r e m 2 is, [ believe, new.

I I , e l n m a 33"

1 9 - - 3 4 1 9 8 . Acta mathematica. 63. I m p r i m 6 ]e 20 juin 19349

146 E. Maitland Wright.

I. 2. W e shall, however, go m u c h f u r t h e r t h a n T h e o r e m I.

C a u c h y ' s T h e o r e m

f f(,O d :~"

p k ( n ) = 2 J r i j x '~'l '

.1"

W e h a v e by

w h e r e

I"

is a circle w i t h centre at x = o a n d r a d i u s less t h a n unity. W e shall t a k e t h e r a d i u s of F very n e a r l y unity. T h e n t h e d o m i n a n t t e r m o f p k ( n ) given in T h e o r e m I arises f r o m t h e i n t e g r a l a l o n g t h e p o r t i o n of F in t h e i m m e d i a t e n e i g h b o u r h o o d of x = I . E v e r y p o i n t of t h e c i r c u m f e r e n c e of t h e u n i t circle is an essential s i n g u l a r i t y of

f ( x ) ,

as we saw, b u t t h e s i n g u l a r i t y at x = I is the

>>heaviest>>. W h e n we consider t h e n e x t >>heaviest>> singularities, a n d also find a b e t t e r a p p r o x i m a t i o n t h a n (I. I2) f o r

f(x)

n e a r x : = t, we can find f o r any e > o an e x p a n s i o n f o r pk(n) w i t h e r r o r only O(e~"~).

T h e n u m b e r s k a n d q are positive integers. T h e n u m b e r p is also a n integer,

f o r t h e q ==

qlq~:

s a t i s f y i n g t h e conditions

I < - - p < q ,

e x c e p t t h a t , w h e n q - - I , p = o only.

all such values of p f o r t h e least positive i n t e g e r

W e define dh by

also

(p, q) - - I, (I. 2I)

W e use ~ , to d e n o t e s u m m a t i o n over

P

p a r t i c u l a r value of q in question. W e w r i t e qt- such t h a t

qlq~.

T h e n

qllq

a n d we m a y write

ph ~ ' ~ d h ( m o d q ) ,

O - - < . d h < q ;

COp, q

top, q =- I (k even),

,,,<,

^,

)l

= e x p , ~ / ~ z ~ h d,, - - 4 (q - - q~) (Z,

o,ld),

t \q"/,=1 exp \ q /

q

SI, 't = ~, e,,(lh'k),

h :-- 1

l i p , q = =

q ) n = l I I l i + l l '

Cq, o - ~ I ) / l p , q,O ~ " ~ p , q .

Asymptotic Partition Formulae. ili. Partitions into k-th Powers. 147 For values of t > o, Cq, t is a q-th r o o t of unity, d e p e n d i n g on k, q a n d t, and Av, q, t is a f u n c t i o n of k, p, q and t. F o r any p a r t i c u l a r values of these para- meters, Cq, t and _/lv, q,t m a y be calculated, b u t the process is in g e n e r a l very tedious. T h e m e t h o d of calculation wilt be described l a t e r (see w 9.4). T h e n our full r e s u l t is

T h e o r e m 3. ~ b r any e > o, lhere exist an integer qo ~ qo( k, t) and integers

Tq, t ~- T(k, q, e), such that

Tq,

p~ (n) = (n + j ) - ~ ~, ~_~ Cp, q eq (-- p n) ~,, Cq,, 9o (//v, q,t(u + j)a) + 0 (e' ,,b).

q<--qo P t--O

W h e n k ~ I, t h a t is, in the case of u n r e s t r i c t e d partitions, we have

I f we write

we have

I 7g 2

J - - 24 ql q, Tq, ~ o , zip, q - - 6 q

~o 3

~ = 0 F ( / + I ) F l - - ~r~ t f l

(

^. l D O n

a n d so the expansion becomes

1 D @n

p , ( u ) - :- d _ _

2 y ~ ] / r 2 q _ q o d n ( qn J p t%,qeq(--pn) + O(e~'e).

H a r d y and R a m a n u j a n o b t a i n e d the last result. I n fact, t h e y went f u r t h e r ; t a k i n g q0 an a p p r o p r i a t e f u n c t i o n of n , t h e y made the e r r o r t e r m only O ( n ' ~ ) . I f we a t t e m p t to make a similar i m p r o v e m e n t f o r pk(n), we find t h a t we can choose q o = q0( k, n) so t h a t t h e e r r o r t e r m is O(end), where d < b. This is n o t so good as the result f o r k--~ I, and, in view of the h e a v y analysis r e q u i r e d f o r this f u r t h e r ~tep, I am c o n t e n t to prove T h e o r e m 3. :

148 E. Maitland Wright.

Part I. Transformation of the Generating Function.

2. i. O u r f i r s t step is to find t h e t l : a n s f o r m a t i o n f o r m u l a f o r f ( x ) f o r g e n e r a l ]c, which will e x h i b i t its b e h a v i o u r n e a r t h e r a t i o n a l p o i n t s on the c i r c u m f e r e n c e ] x ] = I. O u r r e s u l t includes t h a t f o r ]c ~ I , which m a y also be d e d u c e d f r o m t h e t h e o r y of m o d u l a r functions. F o r g e n e r a l ]c no such deduc- t i o n is possible.

B e f o r e s t a t i n g t h e t r a n s f o r m a t i o n f o r m u l a , we m u s t define c e r t a i n symbols.

T h e n u m b e r s p , h, s a n d 1 are integers.

tions (I. 22) a n d

I ~ h ' ~ q , I -~-8 ~ ~, I f d,~ r o, we write

Of these, p satisfies t h e condi- l>__o.

dh (s odd), ~ , ~ - - q - & (s even).

tth, s - - q q

I f d h - - o , we t a k e #h,~ = I. H e n c e a l w a y s q~th.,~ > I.

I n c o n n e c t i o n w i t h a n y p a r t i c u l a r vMues of p a n d q we write X = x e q ( - - p ) ~ e -~J, y ~_ q k y ,

a n d we t a k e y real a n d positive w h e n X is real ~nd o < X < I. W e write also

w h e r e Y~ is t h a t k-th r o o t of Y which is regl a n d p o s i t i v e when Y is real a n d positive,

g ( h , l, 8 ) ~ - exp {2~vi ( / + lzh,s)ats} eq(-- h)

[ ( 2 7 g ) l Z c a ( ], @" #h,s) e . . . 2

= e x p ~ ... qy~ q I'

a n d finally

q k oo

p , , ~ = H I [ I l l / ~ - ~(h, ~, ~)/-'.

h : l g = l l~O

W e are n o w in a position to s t a t e o u r t r a n s f o r m a t i o n t h e o r e m .

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 149 T h e o r e m 4. I f ~ ( y ) > o, then Pv, q is convergent and

[2/lP' q l .Pp, q. (2. I I ) 1

f(x) = f ( e Yeq(p))= Cv, qy:eYVexp \ ya !

I n t h e case k = I , (2. I I) r e d u c e s to t h e m o d u l a r f u n c t i o n t r a n s f o r m a t i o n u s e d by H a r d y a n d R a m a n u j a n . 1 T h e s e a u t h o r s also s t a t e d T h e o r e m 4 in t h e p a r t i c u l a r case k = 2, q = I , p - - o, w i t h o u t proof.

To p r o v e T h e o r e m 4 we n e e d t w o l e m m a s . L e m m a 1. I f C is any positive number and i f

~(y) > c l y l ~+a, (~. ~=)

then P•, q is uniformly convergent with respect to y.

L e m m a 2. I f y is real and positive, then (2. I I) is true.

F r o m these t w o l e m m a s it follows by the" principle of a n a l y t i c c o n t i n u a t i o n t h a t (2. IX) is t r u e w h e n (2. I2) is satisfied. B u t f o r a n y p a r t i c u l a r value of y such t h a t ~(y) > o, we c a n choose C so t h a t (2.12) is satisfied. H e n c e T h e o r e m 4 is an i m m e d i a t e c o n s e q u e n c e of L e m m a s 1 a n d 2.

2 . 2 . P r o o f o f L e m m a 1.

L e m m a 3. I f

I I I

- ~ < ~ < ~, ~ ( ~ + ~ ) < a - < ~ ( 3 - ~ ) ,

2 ~ ~ - ~

then

2 a

--< - - cos 4 .

7g

I + a ) - a ~ ^{, c o s -}

1 W h e n k = I , m y d e f i n i t i o n of wp, q differs in f o r m f r o m t h a t a p p e a r i n g in t h e t r a n s f o r m a - t i o n u s e d b y t h e s e a u t h o r s . T h e l a t t e r d e f i n i t i o n does n o t i n v o l v e t h e s u m

q--1

hdh.

h = l

t t . R a d e m a c h e r (,,Zur T h c o r i e d e r M o d u l f u n k t i o n e n , , Journal f u r Math., 167 (I932), 3 1 2 - - 3 3 6 ) o b t a i n e d t h e m o d u l a r f u n c t i o n t r a n s f o r m a t i o n for f ( @ w h e n k ~ I w i t h t h e f o r m of wp, q t h a t I u s e here. T h e s a m e a u t h o r s h o w e d (Journal London Math. Soc., 7 (I932), I 4 - - I 9 ) b y a n ele- m e n t a r y m e t h o d t h a t t h e t w o d e f i n i t i o n s are e q u i v a l e n t ,

1 5 0

E. Maitland Wright.

We have

I I 37~,

- ~ < I ( ~ + a ) _ a ~ _ < a _ a ~ _ < ~ ( 3 _ ~ ) _ a ~ <

2 2

and so

Also

COS

Since

we have

(2. 21)

H) I

O ~ (t 7 r ~-~ 2 ~ ,

sin {a ( ~ z - [~)[)} > 2a (~ -- z - - [~0[) ;> 2asin- (i 2 z - - [~]) = - - e o s ~ 9 . 2a (2. 23) The lemma follows from (2.2I), (2.22) and (2.23).

Lemma 4. I f (2. *2) is satisfied and if

then

4 ~ C ( 2 ~ ) ~

CI = ql +a '

[g(h, l, s)[ ~ e -c,(l+l/' .

Let us write y = l y l e i ~ , where ~O has its principal value. Since ~){(y)> o, have I ~ [ < ; z and since i - - ~ s ~ k , the number a = - z a ( z s + k - - i )

I

w e '

satisfies the condition of Lemma 3. Hence, when

~(y) > (:[~ql ,+.,

we have

Asymptotic Partition Formulae.

COS und so

since

Iii. Partitions into k-th Powers.

{ I .a(28 _{_ /r i) __ a,iD} < -- - - - 2a -- COS I/1 -~- 2 a ~ ( y ) ^' <_ ~ . c l u l 9

[ aI }]

I g (h, l, s) l - e x p

!2g)l+a(l'~ fib, s) C0S{-2:ffa(23-]- k - - I ) - - a ~ ) qlyl" 7

-~ e x p { - - 2 a

C'(27~)1+a(1 +

= exp {-- Clqa(1 + #h, 8) a} ~ e-C,(t+a) ~,

q (1 + tth, ~) >-- 1 + q#h,8 >-- 1 + i.

151

To prove L e m m a I, it is sufficient to show t h a t

. lg(h,l,s)l

l~O

is u n i f o r m l y convergent with respect to y.

since

E e--Cl (/+1) a l=O

But this follows from L e m m a 4,

is convergent and the terms are i n d e p e n d e n t of y.

Proof of Lemma 2.

3" I. The remainder of P a r t I is devoted to the proof of L e m m a 2.

work is necessarily somewhat l e n g t h y and complicated.

W e take y real a n d positive and write

-- ~:h-- - , v = vh = exp (2~itth, 1) = h eq(dh) = eq(ph~'),

q

or

) E eq

(/~IT~ a hk) ,~h = F ( I + a

m=l and, if k is odd,

jh =,)'h(q, k) -- ( - i)89 + ' "~ eq(mh),

The

152

while, if k is even, We write also

E. Maitland Wright.

j h =-= O~ O'h ~ O .

~<>;,

= ~ l o g / , 71 - ~,,, c,:p

( - ) "

(1 + ~,,)~)/,

l -0 k as

S,:

^{= ~ ,}

.~

log { I - - g (h, l, 8 ) } , s = l / = 0

where t h e l o g a r i t h m s h~ve t h e i r p r i n c i p a l values.

Now we have

x(<~t t f,)~" __ e<~ (pl/:) X (qt+h)~ == vhe,--r(l+~h )~, and so

Also, by definition,

q ~o q

--logf(x) : .,~ ,~_..~ log (,

- - x (<~+h/:) = ~_~ Sh,

h 1 l- 0 It ~ 1

?" I

f ( x ) = e,~p - ~ s,,~.

#l 1 ;

h " - 1

W e take Q a small number, real, positive ~md less t h a n 2' and r a large positive integer. W e use D to d e n o t e a positive n u m b e r , n o t always the same at each occurrence, d e p e n d i n g at most on k,p, q and y. I n P a r t I, the symbol 0 ( ) refers to the passage of (J to zero and of r to infinity, and tile c o n s t a n t implied is of the type D .

3.2. Certain differences arise between the cases k odd trnd k even, but we shall t r e a t the two cases t o g e t h e r . Most of the differences are covered by t h e n o t a t i o n ; thus, when k is odd, the functions j , j h , at, and Wp, q have various m o r e or less complicated values, while, when k is even, j , jh and ah all vanish and ,os, ~ = I. W e see t h a t the tr~msformation (2. II) has a slightly simpler f o r m w h e n k is even.

Apar~ from this, we have to consider t h r e e separate cases arising from different values of h. T h e s e are:

A s y m p t o t i c P a r t i t i o n Formulae. i I I . P a r t i t i o n s into k-th Powers.

(i) I f h = q, we h a v e ~ h - ~, #h = I , ~ h = I , a n d

oo

Sq --- ~., log' (I e-Y~i").

/:=1

153

(ii) I f h k ~ o (rood q), b u t h ~ q, we h a v e o < ~h < ^{I ,} tth, s = I , i' h = I a n d

Sh = ~ log" ( I - - e - - Y i l + ' ) k ) . / = 0

(iii) I f h k ~ o (mod q), we h a v e o < ~h < I , a n d ~h r I .

I n eases (ii) a n d (iii) w e s h a l l t r e a t & a n d Sq-~ t o g e t h e r . S i n c e h k ~ o ( m o d q) i m p l i e s t h a t (q - - h) k ~- o (rood q), h a n d q - - h a p p e a r in t h e s a m e case.

Also, i n t h e s e t w o eases,

Th -~ Tq--h ~ I ;

if k is even, w h i l e if k is o d d

~th, ~ ~ ,ttq--h, s ~ ?2h --: ~q--h

~th, s - ~ - ~ t q - - h , s = I , ~h ~q--h = I ( e a s e ( i i ) ) ,

~th, s @ ~tq--h, s = I , ~h ~q--h = I ( e a s e ( i i i ) ) .

I f ~ " d e n o t e s s u m m a t i o n o v e r t h o s e v a l u e s o f h f o r w h i c h h k ~ o (rood q), w e h a v e , w h e n k is odd,

: Z H ( I - - 2 / - t h , 1) ( I - - - 2Th)

v ! ,.;

~ ~ " I - - 2 ~ ~ h , l - - 2 ~ " ~ h + 4 ~ ~h~h, 1

-- " ~ " I

- - 4 2 ~hlth, 1 - - q--1

_ 4 y h a , , - ( q - q , , ) q h=l

T h e n , w h e n k is o d d or even, w e h a v e

1 1 ,,

fDp, q = e ~ p t 4 z i ~ a h ) .

2 0 - - 3 4 1 9 8 . Acta mathematica. 63. I m p r i m 6 le 21 j u i n 1934.

(3-2x)

154

4" I. Case (i): h = q.

L e m m a 5. We h a v e ~

E. Maitland Wright.

W e

singularities tions

, I s

S~l - S<l = ~ l o g Y - - ~ k l o g 2 ~ + + j Y . W e write

Zl(U ) .... ]C ~uk--1 log (_Im C2~!'a) ]~ ~ U k-1 log (x - - e - 2 ~ )

e " ~ + - I , z ~ ( u ) . . .e Yv'k - - . I .

shall define the value of t h e logarithms precisely at a later stage. The of Zl(U) a n d of Z.2(u) in the u-plane are at the roots of the equa-

e Y u k ~ - i , e 2:~iu ~ i .

The functions have simple poles at the points

u ~ l a ts ( 8 = 1 , 2 . . . . 2k; l = 1 , 2 , . . . ) ,

and for a given value of s the poles lie along a semi-infinite s t r a i g h t line from

axis. I f k is odd, two of these lines coincide w i t h the upper and lower halves of the i m a g i n a r y axis respectively. The only other singularities of Zl(U) and of Z2(u) are logarithmic singularities at the points

u = l ( 1 - . - 2 , - - i , I , 2 , .)

on the real axis, and a singularity at the origin of a slightly more complicated character.

W e shall take q subject to the additional condition

Q<

I

It~l,

2

so t h a t the only singularity of Zl(u) or of Z.2(u) for which l u l ~ 2Q is the one at the origin. W e write R = r + I_ and choose L an integer such t h a t

2

1 I f q = I , Shen /p ~ o, ~ Y = y , h = q, a n d

S q = - - l o g f ( x ) , S q f ~ - - l o g P o , i ,

so t h a t T h e o r e m 4 f o l l o w s a t once f r o m L e m m a 5. T h i s e n a b l e s u s to s h o r t e n o u r w o r k consider- a b l y if o u r o n l y o b j e c t is t h e proof of T h e o r e m 2 (see w I 2 . 2 ) .

Asymptotic Partition Formulae. 1II. Partitions into k-th Powers.

_/_i~'i" < n < (2-}~-~)~ ^{Y -- [.}

2 J [ 2:zt: 2

1 5 5

This is a l w a y s possible f o r r g r e a t e r t h a n some n u m b e r D . T h e n we write

R'=:,l.y ~ L + 2] / We see that R < R ' ~ 2R.

W e n o w define several contours. T h e c o n t o u r a~ is c o i n c i d e n t w i t h t h e real axis f r o m u = - : 0 to ~ l = R except n e a r u .... 1 , 2 , . . . r ; n e a r such a p o i n t a].

passes r o u n d a small semi-circle above t h e point. T h e c o n t o u r fl~ consists of the i m a g i n a r y axis f r o m u = - i Q to u : i l ? ' ; if l~ is odd, Z~(~l) has poles on t h e i m a g i n a r y axis a n d fl~ is d e f o r m e d so as to pass to t h e Ic.fl of these poles.

T h e c o n t o u r 7~ has t h r e e 1)arts 7',, 7',', 7',"- On 7', we h a v e . u - - l ~ e ~

, , , , ,

a n d o ~ 0 - - < ~a~v; on 7,, u = v e ~ " ' ~ a n d l t - < - t , < - - 1~'; on 7~, u - - B ' d ~ a n d 4

I I

a z ~ 0 ~ - ~ c . Finally, 61 is t h e q u a d r a n t of t h e circle I,,l: e on which

4 2

O--~ 0--~ I 7~, a n d I" 1 is t h e closed c o n t o u r f o r m e d by c o m h i n i n g c~j, fl~, 7~ a n d 6~.

2

T h e c o n t o u r s a.~, flo, 7~, &-, a n d I'., are t h e reflexions of al, ill, 71,61 ~nd 1"~ in the real axis, except t h a t , if k is odd, ft._, passes to t h e r i g h t of t h e poles on the r e a l axis, so t h a t F~ excludes these poles.

W e t a k e t h e positive directions of a , , a.,, fl~ a n d ,3. 2 o u t w a r d s f r o m t h e origin, of 7~, 7-~, 61 a n d &2 c o u n t e r c l o c k w i s e r o u n d the origin a n d of F 1 a n d 1%

counterclockwise. T h e n we h a v e

F 1 - - cq - - fll + ;q - - dl, 1% . . . . a.~ + fl.~ + 7"., - - 6~.

(4. ii) (4. 2)

L e t us consider t h e t r a n s f o r m a t i o n z - - c '-,'~i". So long as u does n o t pass outside I'~, z will n o t pass outside a c o n t o u r in t h e z-plane consisting of t h e c i r c u m f e r e n c e of t h e u n i t circle i n d e n t e d a t z = : l , this i n d e n t a t i o n c o r r e s p o n d i n g to t h e i n d e n t a t i o n s of a~ at t h e points u = - I , 2 , . . . Then, if l o g ( I - - Z ) b e t a k e n z = - - I ( c o r r e s p o n d i n g to u = -I2), tile loo'arithm will h a v e its p r i n c i p a l

/

real a t

value so l o n g as z r e m a i n s within or on t h e contour. H e n c e , if we t a k e

156 E. Maitland Wright.

I e 2 .~ i *~)

l o g ( 1 - - e ~'~i") real at u - = - on I'1, l o g ( I - - has its principal value ~nd

2

Z,(u) is one-valued at any point on or in the interior of Ft.

I f we take log (I ^{- - e}

-2'flu)

real at u . . . . I on I',,, a similar a r g u m e n t shows

2

t h a t this logarithm has its principal value at every point on or within 1~.

W e write

l(r,)--- f z,(.)du

^"

i ;

] (r,,) = j' z..(,,)d. == -

G

1(.,) -- ~(i~,) + 1 ( 7 , ) - I(,h),

i ( a 2) + I(,L) + I(7., ) -- I ( & ) .

A little calculation shows t h a t I', includes the poles of Zt(u) at the points

u-=l"ts

( I - ~ 8 - < I ( k + 2 I); I-<.-/--<L),

while 1".2 includes the poles of Z.o(u) a.t the points u " l"tk+~

Then if k is even, by Cauehy's Theorem,

~k L

s-=l l ~ l

k .L

1(1"2) -= 27vi ~, ~_~ lo,,,, {I -- exp (-- 2~ril~tk+.,)},

~ = l k + 1 /-=1

while, if k is odd,

~ ( k + l ) L

. / ( I ' , ) . - -

27ri Z Z tog

{ I - - e x p

(2~il~'t~)},

s = l l ~ l

k L

I(1"0) =-: zTri ~, ~_~ log {I -- e x p ( - - 2~.~il"t,:+~)}.

1 . . 3) l- -1

s--2(kT

I n either case, since t~+.~----t.~, we have

A s y m p t o t i c P a r t i t i o n Formulae. i I I . P a r t i t i o n s into k-th Powers. 157

t~ L

I ( r , ) + I ( F 2 ) =

z z i ~ ~

l o g { I - - e x p

(2~il"

t.)}

S ~ I l--1 k L--1

~ i F Y, log {, - ~ ( q , t, s)}.

S ~ I / = 0

(4. I3) All t h e a b o v e l o g a r i t h m s h a v e t h e i r p r i n c i p a l v a l u e s .

4. 2. L e m m a 6.

W e h a v e

I f l is any positive integer, and ~p any real number, then u = I e~(~+ ~) ei~ - - , I > D

I f

t h e n

I f

t h e n

U 2 = i --2e-n(21+l)sinWeos

{~v(2/+

I) e o s ~ ) } + e - 2 z ( 2 / + l ) s i n ~ p I - - e - - g ( 2 / + l ) s i n w ) 2.

I

2 ( 2 / + I) ~

1 U ~ >_ (, - e - 1 " ( 2 ~ + ! ) ~ ) ~ > O

I - - I ~ s i n ~p ^---

2 ( 2 l + I) 89

1

U 2 ~ (e 1~(2/+1)~ - - I ) 2 : > D .

T h e r e r e m a i n s t h e e a s e

I f cos ~p > o ,

I f cos ~p -<- o ,

I I

< sin ~ --< - -

2 ( 2 / + I) 1 - 2(21+ I) 89

I ~ COS ~J --- sin ~ tp

I -P COS ~2 < sin" ~p --< ^I 4(21 + I)

I + cos tp - - sin ~ ~p

I ^-- cos ~p < sin~ ~p _< ^I

4(21 + I)

H e n c e , in e i t h e r c~se,

158

a n d so

T h e r e f o r e

F. Maitland Wright.

e o s ~ - - - -t- ! + 4 ( 2 / + i ) ' - - I --<~--< I ,

e o s { 7 , ( 2 [ +

i)eoslD} ---eosI+

(2l-[- I) a ~Tt:}

i

^{_ _ T E p}

(1)

. . . COS ~Jr < O.

U e - - I - - 2 4 - - " ( 2 1 § {7r(2l + I)COS ~]J}-}-e--2~("'l~l)'~inV' > I . H e n c e , f o r all v a l u e s o f s i n ~ , we h a v e U 2 > D , a n d so U > D .

L e m m a 7. I f r > D , we have

I ] ( n ) l <

-Uc-Dr,

I [ G ) I < .Do.-,,,.. (4.21)

(i) L e t .u lie on ;,'~ t h a t is, l e t .u ⁼ B d ~ o < 0 < ^{- - - -- "} I a 7 ~ . .

4

a n d , f o r R > D ,

B y L e m m a 6,

H e n c e a n d T h e n

.~)~( Y d ' ) - : ] : ' I F cos kO > Y i P c o s I ~ > D R k, 4

[e Y n ~ : - I [ > e D R 1 : - I > D 6 I)Rk

- - , ^{" = " > D .} D < [ i - - e~'i" [ < 2,

I l o g (I - e ~ " ; " ) I < / ) .

~'!

T h e n we h a v e

(ii) L e t u lie on 7 , , t h a t is, let u - v e ~I"~':, R _ < v _ < R ' < 2 R I T h e n

!1~ ( Y a ~) = Yr cos - ~ > D I F . I 4

(4.22)

Asymptotic Partition Formulae.

Hence, for /1 > D ,

Also

and ^{s o}

H e n c e

III. I'artitions into k-th Powers. 159

[e r'd: -- I [ >- e *~ze~- I > De Dl:~'.

'2:'riu] . (2--'2xrsin l et.-~ < t3--l) l:

I l o a (i - e~-'~'") I < De -l'j:.

I ( Z~(u) du] < l)l(a'c-Dle-Dr'~'<

7 1

(iii) L e t u lie on 7,,"' that is, let ^I, -

~ ' e i~ and I a e r < O ~ _.I

4 2

e~,, ~ - - , = e r , " k , ,';~ ' ' " - - , = e'-'.~'( L~ ',)~.~,~" __ I ,

where ~ p = : k O - - - r g , and so by L e m m a 6 I

2

l ( 'Yuk - I l > D.

Also

and

[ e,_,.i,, ] _<_ e-'.,.-, s' ~i,, I,,,~ <

~-me',

[ l o / ( I --

e~'i")[ < De - D w

Zl(") d , < DR'a'e-i'1r < De-*'".

*it

H e n c e

;r. Then

and (4.24).

4-3-

(4. ~3)

( 4 . 2 4 )

, t a l i t

Since 71 = 7', + 7t § 7 , , the first part of (4.2I) follows h'om (4.22),

(4. 23)

The second par{ may be proved in the same way.

L e m m a 8.

We have

I kTr, i (log

1(de) =:: ~ k e d (log

2 ~ + l o ~ e - - I ) _{9 4 ~} + O ( q l o g . o ) ,

, )

2 ~ + logq + 4~r 4- O(ologq). (4.32)

(4.3~)

These results are a m a t t e r of simple calculation and we omit the proof.

160 E. Maitland Wright.

L e m m a 9. IVe have

I(r - - 1 (fl,) .... 2 ~eij Y + 0 (d: log Q) + 0 (e-~)1~). (4.33) W e replace u in I(fl~) by - - u ; t h e n fl~ becomes ill.

at once

I(,G) = I(//x), J = o,

I f k is even, we have

so t h a t (4.33) is true, t h e r i g h t h a n d side being in f a c t zero:

I f k is odd, we have

Since

we have

flu k-1 lo~ (I --

e ''~')

I % ) = ( - - ~ ) ~ k Y e_,:i~.- - " --du

fl,

--- k Y ; uk 1 l o g ( J - - e "-~i'')

I - - e - - y.k d u .

J

I I

I - - ( ' - - ] ' u k e ) ' a k - - I

- - - I ,

i R ' .

~-= k Y f u ~'-1 log (I -- e2'~i~')du,

i0

for the i n t e g r a n d has no singularities on the i m a g i n a r y axis between u - - i Q and u : - - i R ' , and so //1 m a y be d e f o r m e d into this p o r t i o n of the i m a g i n a r y axis.

I f we p u t u = i v , we have

R r

I(fl~.) -- [(//~) --- k:

ri,: f ,,,:-i

log

(,

- ~-,~,~,,)a,

o

=-: k Y i k f d :-1 log (, --- e-~

0

+ o(Q~log(,)+

o(~-,,R').

(4.34)

Also

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 161

oe

o

r162 r

. . . . v k - - l e - - 2 n v d l )

0

. . . (2 zr)k . . . (4. 3 5 )

T h e n (4-33) follows f r o m (4. 34), (4.35) a n d t h e definition of j . 4.4- L e m m a 10. We have

z ( ~ , ) - ~ ( , , , ) = 2 ~ i & + 2~iz,, y - ~

+ ~vilog(Y,o k) + O(,ologo) + O(e-J'"), ( 4 . 4 I ) where log Sq and l o g ( Y Q ~) are real.

W e t a k e l o g ( i - e -Y"k) real on t h e positive h a l f of t h e real axis a n d write

~1 (~r - - - " 2 2 l ; ' i log (I - - e - y,,k) 2 ~ i log (I -- e - r,,~)

I - e - ' ~ '~ ~ ' ; . . . . ' ~,., ( u ) - e '~ "~ ~ " - 1

T h e n we see t h a t

H e n c e

I(.1)--~

- f

c q

---f

c t t

Similarly 21 --34198.

d { l o g ( i e - Y u k ) (I e2niu)}

d , , - - lo,.,. - = z , ( u ) +

~,(.),

d {log (I - e -l''tk) l o g (I - - e-2~iu)} = 7.2(,4) -}- ~2(?r du

f z (")du

~1 (u) d u + log 2 log ( I - - e - Y R~) __ log ( I - - e - Ye~) log ( I -- e 2 ~ i,o)

Aeta mathematlca. 63. Imprim6 Io 21 juin 1934.

162 E. Maitland Wright.

a n d S O

N o w

(~.,

+ o(e -~.~) 4- o(eloge);

I(,.,)--I(~)= f s.~(.,,)d,,-- f ~l(.)d,,

+ ~i log(Ye ~) + o(e lo,,, e)+ o(e-~~).

l - 1

r (zl

(4.4:)

by C u u e h y ' s T h e o r e m . Since the l o g a r i t h m s have t h e i r

H e n c e

Also

a l l ( 1

principal value, we have

r

I'~, ~ lo~ (, - ~-,,k) I

/=-1

l - r - i 1 I - - r : I

9 f

j ~ , ( . ) d . - ~,Odd.---o~-:S,, + o(,~-'~).

f ^{1~.~(.)- ~,(,.)} d . =} I lo~ (i --,,-~"~)d,.

R

= j " lo,,, ( I - e - )"'~')d u

0

l

= log ( i -- e - v,,.~:)d u + 0 (q log O) + 0 (e- D,.),

L

o

log ( I ^{- -} e - r,,k) d u I ^{e - m} l ' u k d l ~ ,

I) 0

(4.43)

(4. 44)

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 163 7

--'-- I ~ a ~ ( I -}-

a)ff

^e--'~kdu

0

= "-- r - - a I ~ ( I -I- a)~(I -}- a ) = - - ~ , q [ y - a . (4.45)

Then (4.4I) follows from ( 4 - 4 2 ) t o (4.45).

4.5. W e can now prove L e m m a 5. By (4. II) and (4. I2), I ( r , ) + I ( F ~ ) = I(cq) - I(,~) + I(3~) - - I(fl,)

-~ I(rl) Av J~Cy~)- !({~l)- I({~2).

I f we substitute the results of L e m m a s 7, 8, 9 and IO, we have 2 ~ i { I ( r , ) + I ( ~ ) 1 = s , + z~ y - o + J Y I

-~- 2Ilog ~/~-- 2I ]clog2r 0(~logQ) -~- O ( e - - D r ) .

Now let r - - ~ t h r o u g h positive integral values and let Q--~o. Then L - , ~ , and, by (4-I3),

s'~ = s~ + 2~ Y - . + j y + ! log r - ~ k log (2 ~).

2' 2

This is L e m m a 5.

5. I. Case (ii): h # q , h k--~o (modq). Here we have h ~ o (modq~).

L e m m a 11. I f h # q and h k ~ o (modq), we have

SPh @ S'q--h - - Sh - - S q - - h ~ (,~,h -~- ,~,q h) y - a _}_ (jh + j(,-,,) Y

I k {log (2 sin ,h~r)+ :log (2 sin ~q--h Jr)}.

2

The result is symmetrical in h and q - - h ; it is therefore sufficient to prove it when h < - - q . I Then we have

2

I 0 ~ x ~ ~ h ~ - 9

2

W e now t a k e

R ~ r - + - + r /~_~"~r + - - - ~ , 2

164 E. Maitland Wright.

and choose L so t h a t

- - R k Y < L ~- (2R")k Y I

2 ~ 2 ~ 2

which is clearly possible for r > D . W e write

Then

B " < R < R ' < 2 R " < 2 R .

The contour a~ is coincident w i t h the real axis from ~ to R , except t h a t it passes above the points

U = l + * ( / - ~ - O, I , 2, . . . r ) ,

while a~ is coincident with the real axis from --t~ to - - R " , except t h a t it passes above the points

u = - - l q - ~ ( l = I, 2 , . . . r ) . The contours % and % are the reflexions of a 1 and a 4 in the real axis.

W e t a k e /~ a n d r as before, except for the change in the value of R';

fls coincides with /~.,, and fl~ with fl~. The contours dl, &2, d~, d4 are the quadrants of the circle [u[ = Q on which, if u ~ Qe '~

I I I I

~ --2-7r176 --~r----_O~--2-~v, - ~ < _ 0 ~ < ~ , 2

respectively. The contours 71 and 7,2 are defined as before in terms of the new values of R and R'. The contour 73 consists of the three parts 7':~, 7~, 7~'; o n

, I ,,, l ~ , e i O I

78, u = R " e i ~ and - - - z ~ < O - - < - ~ + - a ~ , on 78, u ~ and - - ~ + - a ~ - - <

4 4

~ _ . 0 < I ,,

_ - - - - ~, and 7~ is the appropriate segment of the s t r a i g h t line

2

0 = - - z c + - a z . I

4 Finally, 74 is the reflexion of 73 in the real axis.

The positive directions of the a and the fl contours are outwards from the origin, and of the 7 a n d the d contours counterclockwise ~bout the origin.

Then we write

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 165

W e now write

I f k is even, while, if k is odd,

F I --- {Xl - - fll -It- 71 - - (~i'

k Y u ~-1 loft (l - - e 2~i(~*-*))

z , ( u ) - e r ~ - I

Z,~(u) k Y u I~-' l o g ( I - - e - 2 ~ i ( . . . . ) ) . Y u k - - I

z~(u)=z~(u), z d - ) = z , ( . ) ,

(5.

i,)

We take log

( I -

e 2zi(u-z)) real at u = . + i on F~, and log ( 1 - e -2~(~'-*)) real

2

at u ~ z + - on F2. I W e can t h e n show t h a t l o g ( I - - e 2~i(~-~)) has its principal

2

value on or w i t h i n F 1 and F~, and similarly f o r t h e o t h e r , l o g a r i t h m and the o t h e r contours.

T h e z-functions have poles at the points

ts ~a (I <_s <_ 2 k , 1 >_ I)

and at t h e origin, and l o g a r i t h m i c singularities at t h e points

W e write also

u = * + l ( I = . : . - - 2 , - - I , 0 , I , 2 , . . . ) .

~,(.)

= 2 ~ i i o g (, -

~- "~)

I - - e - 2 z i ( u - v )

e 2 a i ( u - ~ ) - - I

where l o g ( I - - e - Y~k) is real on the positive half of t h e real axis. I f k is even, we take ~8(u) a n d ~4(u) of t h e same f o r m as ~(u) a n d ~,(u) respectively, b u t t h e l o g a r i t h m is now to be t a k e n real on t h e n e g a t i v e h a l f of t h e real axis. I f k is odd, we t)ake

166 E, Maitland Wright.

2 ~ i lo.o. ( r eY'~') ' 2 7~ i log (I ~- e Y'~)~

where the l o g a r i t h m is real on the negative h a l f of the real axis.

F u r t h e r , we write

= f

_1"1

a n d so on.

= I ( ~ , ) - I(~'1) + I ( z , ) - I ( ~ , ) ,

5.2. L e m m a 12. For r > D, we have

I ICr,)I < D e - " , (5.2i)

and similarly for I(7~) , /(7+), I(7~).

T h e m e t h o d of p r o o f of these results is clear from t h a t of L e m m a 7 . L e m m a 13. We have

rot) ~k~,ilog(,-e-~'")+ o(e),

2

I(~.~1 = ~ k ~ i lo~. (i - e '~,:~) + o(~),

2

I(d~) - - ~ k ~ i l o g (~ - e ~ " ) + 0 (0),

2

z ( , ~ ) .... ~ z ~ i l o ~ (, - e - ~ , " , ) + o (~).

2

( 5 . 2 2 )

where the logarithms have their principal values.

Since t h e s i n g u l a r i t y of every Z(u) at t h e origin is now a simple pole, these results are i m m e d i a t e consequences of well-known t h e o r e m s in the t h e o r y of residues,

L e m m a 14. We have

and

I ( 3 ~ ) - I(~1) .... 2 ~ ( i ~ , - , , r + o ( e ~) + O ( e - " ) , .(5.23) (5 . 24)

Asymptotic Partition Formulae. ili. Partitions into k-th Powers9 I f k is even,

I ( f l ~ ) - I(fl~), /(fla)=-I(fl2), j , , = . ] , , ,, o ,

i

and t h e result is obvious. I f k is odd, we have

Since

- - f ( e ~ , ~ _ I)Zi(u)du

t

* U k - 1 ( I e 2 zci( . . . . ) ) d u

= k Y log - -

= k Y r '~-1

log (I ^{- =}

e-2~(~§

0

+ o (e ~) + o (ey'R').

f

co ^{V k - i} log (I - - e ~ ( ~ + i ~ ) ) d v

0

c~

- ~ - - ]~ Z e - - 2 n m i ' r

v >- l e - 2 ~ m ~ d v = ( - -

m = 1 ~ "

0

(k--~)

2 ~?q--h,

we have (5.23), a n d (5.24) m a y be proved in the same way.

5.3. L e m m a 15. W e h a v e

I(%) -- I(%) + 1(%) -- I(%)

= 2 ~ i { Y-~(Z~ + Z~_,,) + & + &_,,) + 0 (q loo. O) + 0 ( e - > ) .

W e have

/log (~ - e ~'r lo0' (, - e ' ~ % / = Z,(") + ~,("),

d u ~ "

i67

(5.31)

a n d so on. H e n c e

i68 E. Maitland Wright.

s(o,) -- _ f ( u l d . _ ,og (i ^-- e -2"'*) log (Yt) k) + E,,

(z 1

s(~D = -- f ~(~)Uu - log (~ - ~ - - ) log ( ~ 1 + E~,

~2

s(%)

= - j ~(uld.

- l o g (~ -- ~ " D log (~0 ~) + S~,

~3

I(a4) = - - f f4(u) d u - - log (I - - e -2'~i') log ( Y o k)

a4

w h e r e E l , E~., Ea, E 4 are n u m b e r s of t h e t y p e 0 (t~ loo. q) + 0 (e--S~rk).

T h e n we have

S(a~) - - S(a~) + S ( @ -- S ( @

ct~ ~4 61 aB

+ o (e log e) + o ( e - ' , ) . Now, as before, we find t h a t

f~du--f~,du

= 2 ~ i S h + 2 ~ i Y--a C(I + a)~(I -I- a)

+ o (Q log ~) + o (e-',,).

I n the same way, w h e t h e r k is odd or even,

(X3 if4

+ G ,

~4d?~ = - - 2:w.i S q _ h - - 27~.i ~ r - a / ' ( I 2- a)~(I --t- t/) + 0 (~ log ~) + 0 ( ~ - - , r ) .

Since h ;r o (rood q), we have

eq(mph':) ~- I, e q ( m p ( q - - h ) k) = I, z,~ = zq_~ = r (i + ~) ~ (~ + ~), a n d t h e l e m m a follows at once.

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 169 5.4. By Cauchy's Theorem,

lira I : {1(/11) + I(F2) + I(F3) + I(12~)}

o-*0, r ~ ~ 2 :rg ?~

- S~', + S '

- - q h* ( 5 . 4 1 )

But by (5. II) and Lemmas I2, I3, 14 and 15, we see t h u t this limit is equal to

Now

Sh + Sq-h + (s + Zq-h) Y - " + (j~, + jq-h) Y

- - ~ k { l o g ( I - - e 2~t*) + l o g ( I - - e - 2 Z i * ) } . ( 5 . 4 2 ) 2

I ]C log {(I e 2uiz) (I - - e -2zi*)} = t]g log (4 s i n '~ z z )

2 2

2 (5.43)

since ~ h = ~ a n d ~ q - h = I - - ~ . L e m m a I I follows from (5.4I), (5.42) and (5.43)- 5.5. Case (iii).

take now

I f k is even, we write

but if k is odd

We have still to consider the case h k ~ o (rood q). W e

~I(U) = vk Y u k-~ log (I -- e 2~(~-*)) vk Y u ~-1 log (I -- e -2hi(u--z))

e Y U k _

z 3 ( u ) =

These functions have poles ~t the points

u = t, (z + (i <_s.<_ 2k; l>_o)

and logarithmic singularities at the points

u = l + ~ ( l ~ . . . - - 2 , - - I , O , 1 , 2 , . . . ) ,

but the functions are regular at the origin.

22--34198. A c t a m a t h e m a t i c a . 63. Imprim6 le 22 juin 1934.

170 E. Maitland Wright.

W e n e e d n o t give t h e n e w f o r m s of t h e ~ functions. W e r e m a r k only t h a t in ~ , f o r e x a m p l e , we t a k e t h a t value of l o g ( I - ~'e -Yuk) which tends to zero as u - ~ + ~ a l o n g t h e r e a l axis.

T h e c~ a n d fl c o n t o u r s are t h e s a m e as before, while t h e ~ c o n t o u r s are u n n e c e s s a r y h e r e ; we m a y p u t Q ~ o a t once, as t h e f u n c t i o n s are all r e g u l a r at t h e origin. T h e 7 c o n t o u r s h a v e to be modified to avoid the s i n g u l a r i t i e s of the g-functions a n d to' e n a b l e us to prove a r e s u l t of t h e t y p e of L e m m a s 7 a n d I2.

T h e details of t h e w o r k will be sufficiently clear f r o m t h e f o r e g o i n g , a n d we c o n t e n t ourselves w i t h s t a t i n g L e m m a 16.

L e m m a 16. I f h k is not a m u l t @ l e o f q, we have

Sp t I

,~ + s~_~ - s ~ - s~_~ = y-~(z~ + z~_h) + r(.]~ + .i~-,,) + 4 ~i(o~ + ~_,,).

T h e t e r m - z i ( a h + aq-h) arises f r o m I

4

f {Z~ (u) + ff~ (u)} d u - - - - log (I -- v) log (, -- e 2~i~) + 0 (e -DR)

a n d t h e t h r e e s i m i l a r expressions. I f k is even, t h e s u m of t h e f o u r is zero.

5~ 6. P r o o f o f L e m m a 2. W e n o w c o m b i n e L e m m a s 5, I I a n d I6. L e t

~ ' d e n o t e s u m m a t i o n o v e r t h o s e values of h f o r which h ~ - o (mod q) a n d h ~ q, a n d ~ " s u m m a t i o n o v e r those values of h f o r w h i c h h k ~ o (mod q). W e h a v e

q q q q

rEjh+- lo T

h ~ l h = l h = l " h = l 2

[ k l o g 2 z + I - z i ~ " a h - - I k ~ ' l o g ( z s i n z h ~ ) ,

2 4 2

t h e l o g a r i t h m s on the r i g h t h a n d side b e i n g real.

N o w

2 sin 1 ~ :== q2,

h = : l \

a n d so

q2--1

2 ' l o g 2 s i n q t=l q~.

Asymptotic Partition Formulae. III. Partitions into k-th Powers.

Also

Y-~ ~, A ⁼ ...

h : 1 q u a m-~- 1 ~/l+a T h e value of t h e stun

q h=l

L/p, q . ya

171

is q or o according as m is, or is not, ~ multiple of q. H e n c e , if k is odd,

and so

I f k is even,

Also, by (3.2I),

q

" ~ e ~ ( m h )

~ j h = ( - ~){('+')y r (k + i)q,,

h=l ( 2Tg)k+l m:lZ h~lZ osk+l

=jy.

q

~Zj~ o=jy.

h--I

I ~pt log ~op, q = ^- ~ i ah.

4 H e n c e we have

T a k i n g positive.

log f ( x ) - - log Pp, q = _dp, q y - a + j y + 2 I log y + log wp, q + 2 k (log q - - log q~ I log 2 ~).

e x p o n e n t i a l s a n d n o t i n g t h a t q = qlq2, we have (2. i1) f o r y r e a l a n d This is L e m m ~ 2, a n d t h e p r o o f of T h e o r e m 4 is t h u s complete.

Part II. The Asymptotic Expansion of

pk(n).

6. W e now t u r n to the p r o o f of T h e o r e m s I, ² and 3. W e shall first prove T h e o r e m 3, f r o m which t h e others m a y be readily deduced. W e first define t h e m e a n i n g of certain f u r t h e r symbols.

172 E. Maitland Wright.

W e t a k e e an a r b i t r a r y positive n u m b e r , a n d 6 a positive n u m b e r whose choice is s u b s e q u e n t to t h a t of k a n d e. N is a p o s i t i v e n u m b e r which is u l t i m a t e l y chosen as a f u n c t i o n of k; n a n d ~. T h e n u m b e r s ~ a n d 6 are to be t h o u g h t of as small, while n a n d N are to be t h o u g h t of as large. W e write

C ~ 2 k - 1 .

A a n d B are positive n u m b e r s , whose values varies f r o m one occurrence to a n o t h e r . I n a n y p a r t i c u l a r occurrence, A w i t h o u t a suffix is a n a b s o l u t e c o n s t a n t , while B d e p e n d s at m o s t on k. I f , however, A a n d B d e p e n d on o t h e r p a r a - m e t e r s t h e s e are i n d i c a t e d explicitly by suffixes, f o r e x a m p l e ,

= = B ( k , q ,

T h e c o n s t a n t i m p l i e d in t h e 0 ( ) n o t a t i o n is h e n c e f o r t h of t h e t y p e B , , e x c e p t in L e m m a 33.

I n t h e c o m p l e x x-plane, 1" is t h e circle

1

I x l = e

of o r d e r N 1-b of this circle a n d consider t w o W e t a k e t h e F a r e y dissection

k i n d s of a r c s : - -

(i) M a j o r arcs, ~)~ or ~J~p,,l, such t h a t q--< N b, (ii) M i n o r arcs, m or rap, q, such t h a t N b < q--< N i-b.

T h e r e s t r i c t i o n y r e a l a n d positive is n o w r e m o v e d and, in c o n n e c t i o n w i t h every p a i r of values of p a n d q, we write

H(_//) = exp _/1 , U (A, X) - U (A, e-") = y~- e~' H (A),

1

w h e r e y , y ~ , y a are real a n d positive f o r X real a n d o < X < I . W h e n x lies on F , we write

Y = ~ g - - i O , I

a n d f o r x on ~ p , q or on mp, q we h a v e

-Op, q <_O<_Op,~, t

w h e r e

< ' < q~T--;, ' q N~--b < 0~,, q < - - q-N1- b - - Op, q

2 ~

q Nl-b

Asymptotic Partition Formulae. III. Partitions into k-th Powers. 173 U p p e r B o u n d f o r

I logf(x)l

on t h e M i n o r A r c s .

7. I. I f k = I , t h e n b = - = 1 - - b I a n d there are no m i n o r arcs. H e n c e

2

we need only consider t h e case k--> 2 in this section.

W e take l o g f ( x ) = o at x = o. T h e n l o g f ( x ) is one-valued f o r every value of x within t h e u n i t circle, f o r f ( x ) has no zeros within the u n i t circle.

L e m m a 17. I f 7 > o, k >-- 2 and x lies on rnp, q, then

Ilogf(x) l

< B r N a-',c+r. (7. I i ) W e use [u] to denote t h e i n t e g r a l p a r t of u and write

W e have

.2.

~ = k + 3 ' ~'~=

T ( v ) ~ - W ( v , p , q , m ) = ~_j eq(plk~,).

~o ov ~0

/ = 1 / = 1 m = l 9 n

= Z E § 2; E + Y,Z

mNNb ^{/ = 1} m<Nb 1=%n+1 m>N bl.-1

= Z~ + Z2 + Z3.

L e m m a 18. On the whole circle 1",

(7. i2)

I f ~ > o ,

I Z~ I < B, log _N, Iz~l < B2r

E e-~lk ~- e--~Ukdu = ~--a e , l ~ k d u = B ~ - a .

1=1 o o

Also, on I ' , [x[--~ e ~v, a n d so

lkm a I

< BNa(l-b) -- B N ~.

174

A g a i n

H e n c e on F

E

l~cunt + I

E. Maitland Wright.

- - lkm J " e - m u k e " Y - < - I + ~v d u

(o m-F 1

PT/

"~ I -]- \ Wl ] C-- 'tk (] lt

Y' f

<--

~ + \ m I e-1'-' .,,k

0

{ Nlae__l,_ (2"l~'~

. - I + B \ m l - ~ i < I ~ 3 .

- < B~,loo. N . I

Iz,~l <- B~ Y, ,.

m "~ N b

du

Lemma 19. On a mi~zor

arc

ll1,,,q, 7~ k--~ 2,

I Z~] < BTN ~-t'~'t

W e w r i t e

p m _ p,,

q q,,

T h e n

'::, [ ( F ]

w h e r e eqm(p~, ) is a p r i m i t i v e q,~-th r o o t of u n i t y . N o w a - c; also

N b ~ q -~. N 1-~' , q,,~ ~ q ~

mqm.

H e n c e , if v --< m a~,, we haw~

L L a n d a u , V o r l e s u n g c n iiber Z a h l e n t h e o r i e , I, (I,eipzig, x927) , Satz 257.