ON THE REPRESENTATION OF A NUMBER AS THE SUM OF
TWO SQUARES AND A PRIME
BY
C. H O O L E Y
Corpus Christi College, Cambridge
1. Introduction
I n their celebrated paper "Some problems of partitio n u m e r o r u m : I I I " [6] H a r d y
and Littlewood state an asymptotic formula, suggested b y a purely formal application of their circle method, for the n u m b e r of representations of a n u m b e r n as the
sum of two squares a n d a prime number. The t r u t h of this formula would imply
that
every sufficiently large n u m b e r
is the sum of two squares and a prime.
proof, even on the extended R i e m a n n
hypothesis
No
(which we hereafter refer to as
Hypothesis R), has hitherto been found. However, in another paper [7] t h e y suggest
t h a t on Hypothesis R it should be possible to prove t h a t almost all n u m b e r s can be
so represented.
This proof was effeeted b y Miss Stanley [11], as were proofs (also on
Hypothesis R) of a s y m p t o t i c formulae for the n u m b e r of representations of a n u m b e r
as sums of greater numbers of squares and primes.
The dependence of her results on
the u n p r o v e d hypothesis was gradually r e m o v e d by later writers, in particular b y
Chowla [2], Wa]fisz [13], E s t e r m a n n [4] and H a l b e r s t a m [5].
I t is the purpose of this paper to shew t h a t the original formula of H a r d y and
Littlewood is true on Hypothesis R. Our m e t h o d depends on the fact that, as is
easily seen, the n u m b e r of representations of n in the required form is equal to the sum
X r(n-p),
p<:n
where r(v) denotes the n u m b e r of representations of v as the sum of two integral
squares.
On noting t h a t r (v) m a y be expressed as a sum over the divisors of ~, we
see t h a t our problem is related in character to the problem of determining the a s y m p totic behaviour of the sum
5
O<p+a_~x
d(p§
190
c. HOOLEY
where a is a fixed non-zero integer and d(v) denotes the n u m b e r of divisors of v.
The latter problem is due to T i t c h m a r s h [12] and was solved b y him on Hypothesis R.
Let ;r (m; b, k) denote the n u m b e r
arithmetical
sum
of primes not exceeding m which belong to the
progression b (mod k). Then the two problems are similar in t h a t each
can be expressed
as a combination of terms of the t y p e zr (m; b, k), where k
belongs to a certain range t h a t depends on a parameter m. I n each case the s t r e n g t h
of Hypothesis R is sufficient to estimate ;r (m; b, k) over nearly all the required range
of k, while elementary m e t h o d s will suffice to estimate the contributions to the sums
due to the exceptional values of k. I n our problem, however, this elementary estimation presents a more f u n d a m e n t a l difficulty and requires a different method, since it
is necessary to take
into a c c o u n t the changes of sign d u e to the presence of the
quadratic character in the expression for r (v). I n fact the m a j o r p a r t of this paper
is devoted to this estimation. We shall require repeatedly, here, the ideas of B r u n ' s
modification of the E r a t o s t h e n i a n sieve. An i m p o r t a n t feature of our m e t h o d is the
application we m a k e of a s y m p t o t i c formulae, and n o t merely upper bounds, for sums
depending on an invariant sieve.
I t is natural to ask whether it would n o t now be possible to prove this result
independently
of Hypothesis R. I t seems unlikely, however, t h a t such a proof can
be achieved at present.
The main difficulty with this problem, as with the m u c h
Goldbach problem
(concerning numbers as sums of two primes), is t h a t the
n u m b e r of representations
of a large n u m b e r is too small for the circle m e t h o d in
harder
its present form to be effective, whether or not Hypothesis R be assumed. Moreover,
our result
exponential
must
p r o b a b l y depend in some essential m a n n e r on properties of either
sums or primes in arithmetical progression t h a t can only be p r o v e d at
this time on the full strength of Hypothesis R.
We m a y consider the conjugate sum
r(p§
O<p+a<x
in a similar way. The details are a little simpler, since the term a in the s u m m a n d
is independent of the limit of summation, x. The a s y m p t o t i c formula, which is stated
without
proof in the final section, shews t h a t there are infinitely m a n y primes of
the form u S + v ~ + a , where u a n d v are integers.
I am
this paper.
v e r y m u c h indebted
to Mr I n g h a m
for reading a preliminary draft of
A N U M B E R AS T H E SUM OF T W O S Q U A R E S A N D A P R I M E
191
Notation and terminology
A~ is a positive absolute constant; the equation / = O ( ] g l )
of the
type
conditions.
I/] < A ~ / g I, true
denotes an inequality
for all values of the variables consistent with stated
A~ (a) is a constant depending at most on the parameter a.
n is a positive integer exceeding e~; p is a prime n u m b e r ; d, k, l, m, q, r, s, t,
/~ and v are positive integers; u and v are integers, except in Section 4; y is a real
n u m b e r not less t h a n 1; z is a complex variable.
[1, m] and (l, m) denote respectively the least c o m m o n multiple and the highest
c o m m o n factor of l a n d m. co (v) is the n u m b e r of different prime factors of v; s (v)
is the total n u m b e r of prime factors of v (counted according to their multiplicity).
2. Decomposition of s u m
L e t v(n) denote the n u m b e r of representations of the integer n in the form
n=p+u2+v
Then
v(n) =
~
1 = Z
n =p+IL2+V2
Using the fact t h a t
~0~rt
2.
Z
1 = ~ r(n-p)+O(1).
?A~+V~= ~ --10
(1)
:0<n
r (v) = 4 ~" Z (1),
where Z (1) is the non-principal character (mod 4), we have
Y r(n-p)=4
lJ<n
Y z(1)
lrn=n-p
p<n
l ~n 89 log 3n
n 89
/ ~ , 8 9 log3 n
We have thus reduced our problem to t h a t of the estimation of three different
sums.
E a c h sum is considered separately, Y~Aand Nc in Section 3 and NB in Section 4.
I t will appear t h a t ~
gives rise to the d o m i n a n t term of the final a s y m p t o t i c formula,
Y~B and Nc being of a lower order of magnitude.
3. Estimation of ZA and Yv
I n order to estimate
Y'A and Yv we shall assume the extended R i e m a n n hy-
pothesis, which we state explicitly as follows:
Every zero o/ every Dirichlet's /unction
L (z) =
(m)
m=l ~ m-
'
where Zq (m) is a character (mod q), has a real part which does not exceed 89 /or all q
and all •q.
192
c. HOOLEY
I t should be noted that Lemma 1 depends on this hypothesis; but that the
remainder of the paper does not., except indirectly through this lemma.
Preliminary
lemmata
Lemma 1 is due to Titchmarsh [12].
LEMMA l. I] (a, k ) = l , then on the extended Riemann hypothesis
1
1 = ~-~-~ Li y + O (y89log 2 y).
~.
P<Y
~ a (rood k)
L E M M A 2. For any m, we have
(l, m) = 1
where
d(m;y)= ~l
1
and
a - l ( m ; y ) = ~ d.
aim
d<y
aim
d>y
We have
z (1)
l<y
(l,m)=l
all
dim
dim
d<y
I=a*
l<y
1
y. ~ (d) z (d) ~ z (t)
dim
d~
d
t <~
d
t
=Y~'~(a)z(d)(
: ~ ~ a ~+0(~)}
d<y
~ # (d) Z (d)
d<y
d>y
+0 (~d(m;y))+O{a
L E M M A
3.
For
any
m,
we
~<~ ~(l)
have
. UE(m)+O
(l,m)=l
where
~>'z
p
1)
l(m;y)}.
193
A ~TUMBEtt AS T H E S U M O F T W O SQUAI~ES AI~TD A P R I M E
and
( p - 1)~
pi~l~ p~ - p + l
E(m)=
~ 1 (mod 4)
p~- 1
p~
p~ - p - ] "
p ~ 3 (rood 4)
Also, there exist positive absolute constants A 1 and A 2 such that
A1
< E (m) < A~ log log 10 m.
log log 10 m
Let l' denote a general square-free number. Then
l(l~= ~ (1
--~1~)- 1= ~ (1_1_1~1-~1)= l,~ll(p(l1 ') .
Z (1) ~
x(l~= z_<u
~ ~,,=,
So
l<~ (p(1)
(Lm)~l
1
99
Z (l')
Z (r)
(l') ~ ,~
l, q~ (l') ,~,~9 ;
9
(I,m)=l
(I'm)=1
(r,m)=l
Hence, since
a-1 (m; y),
we have, by Lemma 2,
l',p(r)
(l',m)=l
(l, m)=l
(3)
d (m)
p /w,m)=,l'~(l')
Now,
since I
I/9~ (1) =
0
1
y ,~"
§ 0(,lI~(i'§247
}
(O'- 1 (1)} and
Y ~-~ (l) = 0 (x),
(4)
I<_x
we have, by partial summation,
1
0(~)
Z iqJ(1)=
and
~ ~/~
1 = 0 (log 2 y).
,<~
l>y
(5)
]-[ (l §
Also
(6)
vim
We deduce from
(3),
(5) a n d
(6)
~ Z(1) :~ (
Z-7! )
,<._ ~(l-~= i ,~ 1 <,. ~,
-z~(l'-)
(l~
l'~(l') + o
d
)
(m) 9
(7)
(,,m)=l
Furthermore, by Euler's theorem oR the factorisation of infinite series,
1 F o r t h e p r o o f , see, f o r e x a m p l e ,
1 3 - 573804. Acta mathematica.
HARDY and
WI~IGHT,The Theory o] Numbers, C h a p t e r X V I I I .
97. I m p r i n a 6 lo 17 j u i n 1957
194
C. H O O L E Y
(i',,~)~1
(p - 1)] =
(8)
- P + Z (P)
P>2
p>2
and
p (p - 1)
1-[ (p - 1) (p - Z (P))
p>2
(9)
p>2
The first part of the lemma follows from (7), (8) and (9).
The second part of the lemma is an easy deduction from the relation
( 1)-1
= 0 (log log 10 v).
1-I 1 Ply
Estimation of ~
B y (2), we have
EA=
:
Z(/):
l<n~log-an
1=
p=--n(modl)
p<n
:
l~n 89
3n
(l,n)~l
x(l):
1+0(
p-:n(modl)
p<n
:
l<_n89
(/,n)>l
(~od,)l)"
p~_n
p<n
Since the arithmetical progression n (mod l) contains at most one prime if (/, n ) > 1,
we have
~=
~
z(1)
l<n~log-3n
~
1+o
p~_n(modl)
p<n
(l,n)=l
Hence, by L e m m a 1,
(/,n)=l
(/,n)ffil
zzn~o~-,, q (1)
(/,n)ffil
and so, by L e m m a 3,
+o(+)
= ~ C E (n
+ 0
log log n 9
(10)
A NUMBER
AS THE
SUM OF TWO SQUARES AND A PRIME
195
Estimation of Ec
I n lee, the condition l>_n89logan implies m<n 89 log-3n.
2~ =
Y~
m<n 89
~
3n
Therefore
(11)
Zr
lm=n-p
mn 89
We denote the inner sum in (11) by Zm. The summand in Zm is 1 if l
is - 1
l(mod4),
if l ~ 3 (mod4), and is 0 otherwise. Therefore
zm=
Y
p ~ n + 3 m(mod 4 m)
~ < n - m n 89
3n
1-
Z:
1.
p-:n+m(mod4m)
p < _ n - m n 89log s n
Also the conditions (n + m, 4 m) - 1 and (n + 3 m, 4 m) = 1 are equivalent. Hence, if
(n+m, 4m)=l,
we have, by L e m m a l,
E m = 0 (n89 log n);
(12)
whereas, if (n + m, 4 m) > 1, then trivially
Era= 0(1).
(]3)
We thus have, b y (11), (12) and (13),
m< 89
4. Estimation o f EB
Our estimation of F~B depends basically on the sieve method. The virtue of the
sieve method from our point of view is t h a t we are able to prove b y its means results t h a t embody information about the distribution of primes belonging to sequences
of low density. The conventional result of this type, usually proved by either Brun's
method [1] or Selberg's method [10], is an upper or lower bound for the number of
primes in a sequence of a given class, and for the proof it is found convenient to
choose a sieve t h a t
depends on the particular sequence in question.
is in these respects rather
Lemma 4
different. I t is required for a part of our investiga-
tion (the estimation of EEL where it is necessary to work in terms of asymptotic
equalities t h a t relate throughout to the same sieve. We use here a weak variant of
the Brun sieve, in order to minimize the complications caused by our special requirements. L e m m a 4 is thus imperfect in t h a t it does not imply the best possible upper
196
c. HOOLEY
bounds; it is, however, quite adequate for our purposes, and a n y i m p r o v e m e n t would
have a negligible effect on the error term in our final result.
L e m m a 5 belongs to
the more conventional category of results.
Lemmata
on sieve
method
We commence b y defining a sieve and t h e n develop briefly its analytical formulation.
1
Let
x - n(l~176 n)';
P=
I'-[P;
p~_x
and d I a typical divisor of P .
Also for a n y positive integer represented b y an arbi-
t r a r y letter t, let
t (1)=
1-I p~;
t (2)=
pit; p ~ x
~
p~,
pit; p > x
where
t = 1-I p~.
We use the familiar relation
if v = l ,
if v > 1.
We define the function / ( , , ) ~ / , ( r )
(15)
b y the equation
/ (~) = g (v) + h (v),
where
g(v) =
1,
if v is a prime not exceeding x,
0,
otherwise,
if ~ is prime to P,
1,
and
h (v) =
0, otherwise.
Clearly ] (~) is a non-negative function which equals 1 when ~ is a prime number.
I n virtue of (15),
h(u)=
Z #(d)=
d]v(1)
~tt(dl).
(16)
d~]v
To use this formula, however, would involve considerable complications. I n s t e a d we
shall approximate to h(v) b y a formula similar to (16) b u t m u c h easier to a p p l y to
our problem.
F o r a n y r, let s~ (v) be given b y
sr (v) =
V ~ (d).
eo(d)_<r
A NUMBER AS THE SUM OF TWO SQUARES AND A PRIME
197
Then we can shew that
=1,
if ~ = 1 ,
_>0,
if v > l
and r even,
_<0,
if v > l
a n d r odd.
st(v)
}
(17)
This special case of Brun's basic formula admits of a particularly simple proof. The
case v = l
is trivial. For the case v > l , we are content to give the proof for r even,
as the proof for r odd is similar. Let then r be even. If eo(u)=t_>2r, then
since ( ~ ) i s an increasing function of s for s < r .
st(,)=-
Z ~(a)
If o J ( v ) = t < 2 r , then, by (15),
(where the sum is possibly empty)
oJ(d)>r
s~r+l
~
( t l ( - - 1)S-1)" 0 ,
S=r+I \ S ]
since (~) is a decreasing function of s for s > r.
We now have for any r
z 1),
ta(d)_<r
(18)
r
since, by (17), the left-hand side of the above equation lies betweeen sr (v) and Sr+1 (~)).
We deduce from (16) and (18)
h(v)=
a,I,
~
~/~(di)+O(a,,~
ea(d~)<r
1).
(19)
~ (dl)=r+l
L E ~ ~ A 4. Let y <_n and k = 0 (no), where 0 is a positive absolute constant less
~han 1. Then
B n) y + O
9~aCmodk)
0
,
,
i/ (a(1),k)=l,
i I (a (1), k)> 1,
198
c. HOOLV.r
where B(n) depends only on n and satis[ies
B(n)=O\
(!lo log_n)2]"
logn
]
In the above conditions (a (1), k) may be replaced by (a, k(1)).
Firstly, we have
~_<y
v
=o(x)
(20)
v ~ a (rood k)
Secondly, suppose (a (1), k)= 1. Then, by (19),
v<y
,,~a (modk)
v<y
u~a (roodk) o)(dO<r
dll~
oJ(d0=r+l
oa(d,)<_r
,
\ o J ( a ~ ) = r +1
(21)
where, in the inner summations on the right-hand side, v satisfies the conditions
v < y , v~--a (mod k), v------0(mod dl).
Now the simultaneous congruence v ~ a (mod k), v ~ 0 (mod d,) has a solution, if
and only if (d1, k) la, in which case the solutions form a residue class mod ([d,, k]).
This condition is equivalent to (dl, k)]a (1), which in turn is equivalent to (d,, k)= 1,
since (a (1), k)= 1. Hence the right-hand side of (21) becomes
(dz,~k)=l#(dl)(~dl-~O(1))~-O((d~,~k)=l (~da + O (1))}
oJ(dl)<r
a)(d,)=r+l
=
Yk(d~,2
0 k )/~=dl
(dl)+
l
21 _ )
(~ oJ(d~D>r~1) + o (to(d,)<r+l
We now choose
=
+
]ry
0 (~ ~]2) + 0 (Za), say. (22)
r - [10 log log n] + 1.
We have
~1:
p~z (1 1 ) =
fi<x ( 1 - ~ ) ~ (1--~)-lp~k (1 _ 1 )
(p,k)=1
p >x
= ~ (k) B (n) ~
-
, say,
(23)
10:>X
where, by" Mertens' formula,
B(n)~logx=
o((loglog
~
io~n
]
(24)
199
A NUMBER AS THE SUM OF TWO SQUARES AND A PRIME
(1
Now
plk
p>x
~l~
1)=0(~)
'
p>X
iO>X
since k has at most log 2 k prime factors; and so
1-I 1 - =
= 1+o
=1+0
(25)
~ 9
p>X
We deduce from (23), (24) and (25)
k
( k
(log log n)2~
k
O
1
(26)
We have
~=2
y
s>r
1
t~(a,)=s d l
B y Stirling's formula
and so, if s > r ,
s~. = 0
1 ((
s-i = ~
=O
)s} ((
1
=o
e (log log n + A a
i51og]ogn
=0
0
Z 3= 0
1
11
dl < ~n(loglog n) )
< nlOglog ~.
11
Hence
0
1
/ -~\m(dD
In Za, we have
)s}
( ~ )
(n~ )
= 0
9
(28)
F r o m (22), (26), (27) and (28), we deduce the first part of
v-----a(rood k)
~_<y
0,
if
(a (1), ]r
> 1.
The second part is trivial, since (a (1), k) > 1 implies (v, P) > 1.
The first part of the lemma follows from (20) and (29); the second part from
(a (1), k) = (a, k(1)).
L EMMA 5. For any r < 89n, the number el solutions o/ the equation
n-p=p'
in primes p, p' is
0
r; p < n ;
(~ "log
lo~~~o~
~_~
(n/r)]
200
c. HOOLEY
This lemma is analogous to a result of Erd6s [3], and the proof almost identical.
If (n, r ) = l, an immediate application of Brun's method gives the number of
solutions as
0
p<(nlr) A~
P<( n/r)AI
=
{r,<(~/~)A,
-- 1)-~} = O {n.llo~ (n/r)]
- l ) 2 ~lI~
p < ( n / r ) A4
If ( n , r ) > 1, the lemma is trivial, since then the number of solutions is at
most 1.
Further lemrnata
We require a lemma concerning the distribution of numbers m, for which the
value of ~ (m) is restricted by certain conditions.
A method of H a r d y and Ra-
manujan [8] is applicable to problems of this nature.
We prefer, however, to use
another method which, though non-elementary, has the advantage of requiring less
computation. Also, for problems concerning ~)(m) our method appears to yield more
accurate
estimations,
although for those concerning o~ (m) the two methods are
equivalent in power.
LEMMA 6. I/ l<_a<_~, then
a a('~ = 0 (y log a-1 2 y).
m~y
We merely indicate the proof, as it follows very closely that of a theorem
due to Ramanujan and Wilson [14]. For R (z)> 1,
a
since the infinite product is absolutely convergent if a < 2. This product equals [~ (z)]a / (z),
where /(z) is given by a Dirichlet series that converges absolutely for R (z)> 1 - 0,
where 0 is a positive absolute constant. Hence
m<y
aa(m)=A 1 (a)y log a-1 y + O ( y log ~-z y)
(as y - , o o ) ,
from which the lemma follows.
L~.M~tA 7. I/ ~ a < l <fl<_~ and y > e e, then
(a)
~
1
- - = 0 (log~-1 y log log y),
Y~log_6y<ra<Y~log~ y m
~(m)_<~log log y
A NUMBER
AS T H E
(b)
201
SUM OF TWO SQUARES AND A PRIME
1
~
....
m~y
O (logV~y),
m
~(m)>flloglogy-1
y~ = c - c log c.
where
I f 89< a <
1, t h e n b y L e m m a 6 a n d p a r t i a l s u m m a t i o n ,
agt(m)
--y~logSy<m<yllog3y
= 0 (log ~ 1 y log log y).
m
Hence
1
--
~.~ (m)
- = 0 (log 7 ~ , " 1 y log l o g y),
y~iogSy<m<y~log3ym
~ a-~loglog
Y
y~logSy<m<y 89
(30)
~(m)_<ct log log y
where
yc,~
= a
-
c log a.
Similarly, if 1 _< a _< 3
1
m<_y
~(m)~flloglogy
a ~ (m)
m
m<y
0 (log~r ,~ y).
m
:Now for c fixed, Yc, a a t t a i n s its m i n i m u m c - c l o g
c when a=c.
t r u e , since we m a y l e g i t i m a t e l y set a - - : r a n d a = f l
LEMMA
8.
I/
(31)
1
T h e l e m m a is t h u s
in (30) a n d (31) r e s p e c t i v e l y .
(r s, m ) - 1 a n d r, s, m <_ n, t h e n
z (z)
l>j
q~ ( r s l )
rs
(l, m s)= 1
where
Rm (r; s; y) = log 2 y d (~,s,d (m; y).
y
rs
T h e p r o o f is similar to t h a t of L e m m a 3.
q~(rsl)
l~(rs)
W e use t h e r e l a t i o n
1 p~r
v a l i d for (l, s ) = 1.
This gives rise t o t h e i d e n t i t y
I
I
1
1
(q, r)~ 1
w h e r e q d e n o t e s a g e n e r a l square-free n u m b e r .
Therefore
Z q)
(r s) ,~y
~0 (r s l)
_
(l, m 8)=1
X (q t)
~>_y
(q, r s m ) = l
(t, s m ) = l
q t q0 (q)
\
rsy
/
202
c. HOOLEY
Z (t)
Z (t)
Z (q)
(q) t>_y/q - i - + (t,,.,)=l t
5
q<_u qz ~(q)
q>v
q~ (q) ;
(q, r s m ) = l
(t, s m ) = l
(q, r s m ) = l
and this, by Lemma 2, is equal to
l
qd(ms;Y)}+O{q~<ul(ms;~)}+O{]~176
=0t
llog2Yd(s) d(m;y) + 0
y
~u / ~ a
l
}
1 ms;
+0
log logn
Y
Now
1
(ms;Y)
O(t=~
1
1 5
d>yll
d<y
(1
= 0 yd(ms;y)
1
)
~
1 ~176
d>Y
+0{a_x(ms;y)}
--O(~d(s)d(m;y))+O{(r_l
(ms;y)}.
(33)
We transform ff 1 (m a; V)- If /21 m s, then /~ =/~x/~2, where /~11m and #21 s. Hence
(T-I (m S; y ) ~_ ~ __1 5
~,lm~/Zl
tql s
__1 = E __1 5 __1+ 5 __1 5 1
f12
/zl[m ~ 1
I~>YIII I
~l <_Y
tqls
Ilz> y/l~l
f12
I ~ , l m / 2 1 t t , ls/Lt$
I~t> Y
5 1--o(#ld(s)) +a l(m;Y )(r-l(8)
u~l,~/21 \ Y
(34)
Y
The lemma follows from (32), (33), (34) and the relation r s/qJ (r s) = 0 (log log n).
Lv. MMA 9. I / 1 0 < u < n ; u ' > _ u ;
and m <_n, then we have
(a)
(b)
d<u
u / d < l < ( u log s n)/d
5
5
a-1 (1)
- o'-1 (m; ~ ) = 0 {(log log n)3},
d<_u u / d < l < ( u l o g ~ n ) l d
~
(c)
d<u
d~
d. 1_
d 1
= 0 {log log n}.
uld<l<(ulog6n)ld u
1 Here, in accordance with the notation introduced in Section 1, u, u ~are not necessarily integral.
A NUMBER AS THE SUM OF TWO SQUARESAND A PRIME
203
The first sum is equal to
E
-d(1)
u/cl<l<(u log' n)/d d 1
= O {~a~ulOg 2~ -u'd (m; U~j)[log (U l~ n ) log log n]}
log- -d
(u' d
log log hi}
/ ~)}
= 0 tloglogn
[ u' a<_u'~log 22u'
~ - d ( m; ~u')} + 0 {(loglogn)
u' =a<_u,~log 2u'
~ - d (m;
Now for k = l
2 l~ k d- d
d<lt'_
9
(35)
or 2, we have
m;
= 2
\
E l~ k 2~u- '= 0
< It'
l<_u" d -- 1
log k21
1
=0
u' z
9
l<_u"
Also, as we shall shew below,
log k 2 l
llm 1
O {(log Iog 10
m)k+l}"
(37)
The first part of the lemma follows from (35), (36) and (37).
We are content to sketch the proof of (37), as it depends essentially on a
method due to Ingham [9]. Let
a (z) =
Then clearly
~ l~
- - 21 < I + A
llm
5~
1
I-i 1 -
9
pim
llm
log k- l < I + A
l
--
5 ( - 1 ) kG (k)(1).
An easy calculation shews that
--G' (1)= Plm~I( 1 and
But
GH
(l)
=
~)-lp~ m (1-
~)" 1 lOgp~)
__1-1
,~-Im(1- ~) 1 { [p~m (I ~) ~ ] 2
_
~
(1--
1
2 l~ p-<
2
l~ p+
2mlog~v
p] m P
p~ logk 10m P
logk 10 m .
<
~
logkp +
p__~logk 10m ~0
I
log~ 1O m
= 0 {(log log 10 re)k).
(a ;
log p
p) 2 ~ ) } .
204
c. HOOLEY
Therefore
1)k G(~) (1) = 0 {(log log 10 m)" ~1},
(
and so (37) is established.
The second sum is equal to
~, d o'
1
m;
">'
d<u
-
u/d<l< ('~logSn)/d
by (4) and partial summation.
......
0
l
{l~176
I'
It is therefore equal to
1 ~
1
O(loglognl~,~- z.d~ )If
l~u,
the inner sum is 0 (log 2/); if l > u, it is 0 (log 2 u ) = O (log 2 l). Hence the
double sum is
0 (log log n z ~ 1-~
= 0 {(log log n)a},
by (37).
The third part of the lemma is ahnost immediate.
Inequality for EB
~re are now in a position to commence our assessment of EB.
Let
D(m)=
~
1
and
F(m)=
~
lIm
n } Iog ~ n < l < n ~ log ~ n
Ez= ~ F(n-p)=
Then
z(l)"
lIm
n ~ log s n < l < n ~ log ~ n
p<n
~
p<n
D(nt-p)
F(n-p).
~0
Therefore, by the Cauehy-Schwarz inequality,
O{(E~) 89(EE)~}, say.
D(n
(38)
p) t 0
Estimation of ED
Let ~ satisfy the inequality 1 <~_<~.
ED~
~.
D(n-p) 4-
p<•
f2(n
~
l--2D, 1 ~-ED,2, say.
(39)
p<rt
~(n-p)>o:Ioglogn
p)_<~ log log n
Wc estimate ED.1 first.
We have
Since here, and also later, sums over complicated ranges
of the variahles will occur, we shall adopt an abbreviated notation for some of the
more lengthy conditions of summation.
When possible, a capital letter will be used
to denote a condition satisfied by the corresponding small letter.
We now define
A NUMBER
AS
THE
SUM
OF
TWO
SQUARES
AND
(L), <M), (P) to be the conditions n ~log-3n<l<n'~log:~n,
lm=n-p,
respectively.
205
A PRIME
<m<n 89
n89
If (P) holds, the conditions of summation in No.1 imply
that at least one of f2 (1) and s (m) does not exceed 89~ log log n.
~2(l)~ ~ ~ log log n
Therefore
~'] ( m ) ~ 89~ l o g l o g n
In the second sum the conditions of summation imply m<n'-' log 3 n.
ED.t ~<
~
1+
~
( L), (P)
(1)< 89a l o g l o g n
1+
~
(M), (P)
l <n89 l o g ~ n
~ ( m ) < 89 r log. l o g n
m ~ n 89l o g ~ n
Clearly
Z 1 =
0
Therefore
1 = ~ + E2 + Y~3, say.
(40)
(2.2)
and
n (log log n ) ~
Now
p<n
l~(m)~ 89 ~ log log n
D(m)~
alog log n
be Lemma 4. Hence
Y~2= O ( n (l~ l~n n)3
(M)
1 ) = O (l~g n l~189 a- l n (l~ l~ n)4) '
(42)
~(m)< 89 a log log n
by Lemma 7, since 89189
Because y i ~ > 0 , we deduce from (40), (41) and (42)
ED, I = 0 ( l o ~ n log~ a-1 n (log log n)4) 9
(43)
Our estimation of ED.~, with its dependence oil Lemma 5, is due essentially to
Erd6s [3].
ED,2 --<
~
m<rt
f~ ( m ) > 10 l o g l o g n
We have
1+
~
p<?t
~ log" l o g n < ~'~ ( n - p ) < I 0 l o g log" n
1 = E 4 + Zs, say.
(44)
1 ~ m<n
~ (Ve)n(,,)=O(nlog,~_~n) '
)24 < logS~-
by Lemma 6, since 89< ]/e < ~ ; so
(45)
Let Rn be the set of numbers m which are less than n and which are such that
1
they have no prime factor exceeding n ~ ? = ' ~ .
Then
206
C. HOOLEY
~(m)<lO
(46)
1 = E 8+ET, say.
E
met~ n
n-peR
loft l o g n
Now if ~) (m) ~< 10 log log n and
~)(n-p)>~
n
log log n
m ERn, then m<_ni.
Therefore
Z 6 = 0 (n~).
(47)
If n - p r R= and ~ (n - p) > ~ log log n, then n - p has at least one representation in
the form r p', where p ' > n 11(2~176176
n) and ~2 (r)> ~ log log n - 1 ;
tion r
< Tb1-1/(201~176 n). T h e r e f o r e
_
~,<
X
r<nl-l/2Ologlog n
~) ( r ) > u l o g l o g n - 1
in such a representa-
i
Z
p<n
n-p=rp"
1=
0(
~
n (log log n)3]
r<nl-1/2Ologlog n
-"
r
log 2 n
]
,
Ft ( r ) > a l o g l o g rt - 1
by Lemma 5; so
E7
0 (n (log log n) a
X
\
by Lemma 7.
1}=O(lo~l~
r<_n
~(r)>a
Since ~ > 0 ,
(log log n)a),
(48)
log log n-1
(44), (45), (46), (47) and (48) imply
ED, 2=
O(lo~logr~'-ln
If we choose cr so that ~.=~ 89
(49)
(log log n)3) 9
then
~ - ~log r16289cx- 89r162 (89~),
giving
log cr = 1 - log 2; ~r = 89e,
and the condition 1 < ~-<3 is satisfied. Also
y~ = 89e log 2.
We now deduce from (39), (43) and (49)
where
n l~
]~D = 0 (\log-n
n (log log n)4),
~=l- 89
(>0).
(50)
Estimation of Z~
We have
Xs= ~F 2(n-p)~
X -~(n-v)/(r)
=
X
n 89 l o g -s n < l ~ ' , l , ' < n ~ l o g ~ n
z q;) z (t~) / (v).
207
A N U M B E R AS T H E SUM OF TWO SQUARES A N D A P R I M E
Now, for given l~, l'~, the values of n - v
(mod[/;,/~]).
where
if (l;,l;)=d,
Furthermore,
in the above sum are equivalent to 0
t h e n 1;=dl~,
l~=dl~ and [l;, l~]=dl~ la,
(/~, l~) = 1. W e now define (L~), (H), (K), (Ka) b y (L,) =- {(n 89log - ~ n ) / d < l * <
(n ~ logan)/d}, ( H ) - { ( / 1 , l z ) = 1}, (K)={(dlal~, na))= 1}, ( g ~ ) - {(dll, n u)) = 1}. Hence
x~=
~
z*(d) z ( q ) Z ( t , ) / ( v ) =
~ +
l~ It d m = n - v
d>n~l,
(LD. (L,), (H)
=ZI+Z~,
In El, we have 11 l 2 d < (n log 6 n)/d <_n '1' log 8 n.
X 1 -- (LD,(~L.,,(H)~ 2 (d) X ( l l ) Z (/2)
d~_nll8
~ (d l~ l,)
d>n ~I~
= B (n) n
(51)
Hence, using L e m m a 4,
~
/ (?))
v~n (rood d l~ l,)
v<~
Z ~ (d) Z (11) Z (12)
= B (n) n (LD. (L,).~ (H), (K)
say.
cl<n~l*
1)
(
+ 0 log s n (z,), (L,), (H) dt I l~
~
X ~ (d) Z (lO X (l~) +
(~,).(~,). (n). (K)
q) (d l~ l~)
n ~/* <d'<ri. 89log art
+ B (~) n
(L1),( Ls),(H). (K)
n89 log 3 n < d < n ~ log~ rt
NOW
(52)
0 ( l o ~ n ) , say.
:B(n)~a+B(n)~4+
~3
z "~(d) Z (l~)
(L,). (K~)
nt/8<d<_n89log ~ n
Z
(L,)
Z (I,) .
~ (d l 1 l~)
(lv l~ n(1))=1
Substituting for the inner sum by L e m m a 8, and t h e n deleting the conditions (K1)
and d_> n 'I' from the outer summation, we obtain
I
+
d~n : log a n
+ log log
-~1-
[a_i
+ (log log n)2n89 log_3 n
+a-t
(nO) n 89
d
an)]+
L)}
If we set successively m = n (1), u = n ~ log - a n ,
(53)
u'=u
and m = n a), u = n 89log -3 n,
u ' = n 89log a n, the conditions of L e m m a fl are satisfied. Therefore, by (53),
~3 = o {(log log ~)~} + o {(log log n)*} + o {(log log ~)~} = o {(log log @ ) .
(54)
208
c. HOOLEV
E~= O(
Also
O(loglogn
,n~ log -~ n < d ~ n 8 9 log :~n
l~,/z<log s rt
~
n89log~n<a<n89
1 )
d l 112
It, lz<log 6 n
(55)
= 0 {(log log n)4}.
We deduce from (52), (54) and (55)
El=O(~n
(log log n ) ' ) 9
log n
(56)
To estimate E2 we use the fact that, for given 1a, l~,
~ l / z (t) =
s t=I:
We
define
(R),
(S),
(D),
(DT)
1,
if (ll, lz) = 1,
0,
if (ll, 12) > 1.
(R) =- {(n 89
by
(S) ~ {(n ~ log "~n ) / d t < s < (n"- log '~ n ) / d t},
(D) - {d < n'/~},
~ n ) / d t <r <(n 89
(D T) ~ {d < n '/',
t < n'/'}.
We thus have
Z2::
~
#(t) z 'z(t) 7~2(d) y~(r)z(s)/(.v) = ~
rst*-clrn-n-v
(R), (s), (D)
+ ~
t~-nff*
(57)
= E 5 + E 6 , say.
t > n ~1.
Now in E 5 the conditions of summation imply
n
= n ~ d t log s n < n ~ log '~ n.
r t ~ d m < - ns < n~d
- 1 t x log 3 n
(58)
So
25 --
~
/t (t) Z ~ (t) Z 2 (d) Z (r)
rtedm<n~log~n
(R),(D T)
~
Z (s) / (v)
w:n--rstedm
(S)
= o {
~
I
rt~dm<n~log~n
(R),(D T)
~
z (8)/(~) I},
(59)
v~n-rst~dm
yj<~<y,
where Y2 =max ( n - n ~ r t m l o g - : ~ n ,
1) and yx= m a x ( [ n - n ~ r t m l o g a n ] + l ,
1). If we
set 2 = r t ~dm, we have t h a t v - n - 2 s .
Also Z(8) is 1 if 8~-1 {mod4), is - 1 if
s - - 3 (mod 4), and is 0 otherwise.
Hence the inner sum in (59) is
/ (r) -
v=-n --2 (rood 4 ~)
~
/ (~),
(60)
v - n - 3 ) . (rood 4 )*)
where v is the variable of summation. The conditions (n - 2, 4 2(1)) = 1 and (n
are equivalent,
42-0(n
because each is equivalent to ( n - , 1 (~))(e), 2,1(1))= 1. Also, by (58),
~ log a n ) .
Y~_--Yx~i,__2iB ( n )
3 ,t, 4 ~tr
Therefore, by L e m m a 4, the right-hand side of (60) is equal to
Yz
Y ' B ( n ) ~ - O ( n T ( 4 , 12)- 1 ~ g K n ) = O ( f~o ~ nS n ) ,
0 ( ~ - ~ - -l ~O,g, " n ]
i~
(n---,1,
42(1))~'.
if (n - '1, 4 2(1)) = 1 ,
A :NUMBER AS T H E SUM OF T W O S Q U A R E S A ~ D A P R I M E
209
Hence, by this and (59),
Es= 0
2 t~<n
r, d, m,
rdmt~-log~n
\t>n
(61)
=0
/s It<--nl t
t>_nU8
=0
nlog an
~
~
=0
(62)
log a n ) = O
t>n q8
We deduce from (57), (61) and (62)
(63)
Therefore, finally, by (51), (56) and (63),
(64)
Estimation of EB
We deduce from (38), (50) and (64)
EB =
where
O(,o
1)
(65)
n log ~ n (log log n) -~ ,
5 = 89(1 - 89e log 2)
( > 0).
5. T h e final s u m
The first part of our final result is now immediate from (1), (2), (1O), (14)
and (65).
T ~ E O R E ~ 1. On the extended Riemann hypothesis, the number o/representations
o/ the integer n in the ]orm
n p+u~+v ~
is equal to
7~n
(
~p(/~]~)
1Yggn 2 1+
(p--l) 2
rI
pin
P
p
1 (mod 4)
p2_p+
p~-I
rI
v,n
P-
1+
p-~3 (rood 4)
+0
n
~log
_~
~a\
n(loglogn) ~).
Every su/[iciently large number n is the sum o/ a prime and two integral squares.
14-563804.
Acta mathematica.
97. I m p r l m 6 le 30 jttillct 1957
210
c. HOOLEY"
The second p a r t
Lemma
3 the
of the t h e o r e m follows a t once from t h e first part, since b y
explicit t e r m
i n the f o r m u l a for the n u m b e r of r e p r e s e n t a t i o n s is
greater t h a n A6 n / l o g n log log n.
As s t a t e d
i n the
introduction, the
conjugate t h e o r e m
may
be p r o v e d b y a
similar m e t h o d .
T HEO~]~M 2.
On the extended R i e m a n n hypothesis, we have
p--1 (rood 4)
p ~ 3 (rood 4)
where a is a /ixed non-zero integer.
There exist in/initely m a n y primes o/ the /orm uP+ v 2 + a.
F i n a l l y , it m a y be of i n t e r e s t to note t h a t the n u m e r i c a l v a l u e of the c o n s t a n t
is g i v e n a p p r o x i m a t e l y b y (~ = 0 . 0 2 8 9 . . . > ~ ,
References
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[2]. S. CHOWLA, The representation of a n u m b e r as a sum of four squares and a prime. Acta
Arithmetica, 1 (1935), 115-122.
[3]. P. ERDOS, On the normal number of prime factors of p - 1 and some related problems
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[4]. T. ESTERMAN~, Proof t h a t every largo integer is the sum of two primes and a square.
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[5]. I-~. HALBERSTA~I, On the representation of large numbers as sums of squares, higher
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[7]. - - ,
Some problems of partitio numerorum; V: a further contribution to the study of
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[10]. A. SELBERC, On an elementary method in the theory of primes. Det Kongelige Norslce
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