C OMPOSITIO M ATHEMATICA
A. Z ULAUF
On the number of representations of an
integer as a sum of primes belonging to given
arithmetical progressions
Compositio Mathematica, tome 15 (1962-1964), p. 64-69
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On the Number of
a Sum of Primes
Representations of an Integer as
belonging to given Arithmetical
Progressions
by
A. Zulauf
1. Introduction: Let K1, ..., Ks be s given positive integers,
and K their least common multiple. Let further a,, ..., as be given
integers, (au, Ku) = 1 (or 1, ..., s), and denote by K(n) the
number of sets of residues xl,
x8 (mod K) which (i) are relato
and
tively prime K,
(ii) satisfy the following system of congruences.
=
...,
Finally let N(n) denote the number of representations
positive integer n in the form
(1) n Pl+P2+
+ p., Pu = au (mod K03C3) (03C3 1,
=
=
...
where the p03C3 are odd prime numbers.
1 have proved t ) that if n = s (mod
where p
2),
and s &#x3E;
of the
...,
2:),
s)
then
denotes Euler’s function,
through the odd prime numbers, and where Q = 1,
2, according as K is even, or odd.
In this paper we shall prove an explicit formula for K(n).
2. Lemma: Let p be a prime number &#x3E; 2, let s &#x3E; 1, 1’ 1: L,
and let -K,, (.Y, 1, L ; n) denote the number of sets of residues ae1, ..., ae.
(mod pL ) which satisfy simultaneously
where p
runs
or =
t)
t)
see
If s
[lJ,
=
p. 228.
2 this results holds for "almost ail"
n.
See
[3] and [4].
65
Then
PROOF. The lemma is
pL-l,
has
or
evidently true for
no, solutions xl
=
has
or
pL-l,
or
=
1, since the system
(mod pL) according
Next, assuming the lemma true for s
also true for s
The system
s
=
t,
we
as
pin
or
pin.
shall prove that it is
t+l.
no, solutions Xt+l
(mod pL ) according
as
not. Thus
pin then pfm for every m =1= n (mod p),
assumption, from (3)
If
If pfn then
for one residue
from (3)
It
...,
obtain, by
for
our lemma is true for all s &#x3E; 1.
be defined as in the introduction. Let
let q run through all prime factors of K, and put
follows, by induction, that
(Kl,
we
(p-2) residues m # n (mod p), and plm
m # n (mod p). In this case we obtain, therefore,
pfm
3. Theorem. Let
k =
and
Ks),
K(n)
66
Then
or
K(n)
=
or
0,
according
as
not.
PROOF. Since the congruences
imply
it is trivial that
Suppose
0 if n does not satisfy the congruence (4).
that n does satisfy the congruence (4). Let
K(n)
now
=
so that
If
denotes the number of sets of residues x1,
which satisfy
Kq(n)
(mod q Lq)
...,
x,
then, obviously,
finding the value of Kq( n),
assuming that
For
there is
no
less of generality in
67
Consider first the
Since
Âq
=
lq,
case
q]k.
If
q]k
we
have 9,
=
0 and
and hence
the number of different sets of residues X2, - - " X,,
(mod rio)
satis-
fying
is
evidently given by
Now the congruence
uniquely
determines
a
residue xi (mod qLe),
and if xi is
so
deter-
mined, then, by (6), (7) and (4), also
Since
(au, Ku) 1 and ÀqU &#x3E; l, the congruences (7) and (8) imply
(xu, q) = 1 (or == 1, ..., s), and hence we conclude that
if
=
and n satisfies (4).
Now consider the case qf
qlk,
k. If qf k we have 1 = s,,
the number of different sets of
which satisfy
is
evidently given by
s-l, and
residues eXSf+l’ ..., es (mod qLq )
68
It follows that there
are
different sets of residues xl,
congruences (11) and
But
Âq(1
=
0
...,
x,
(mod rfl)
which
satisfy
the
and, consequently,
Further, the congruences (11) and the conditions (aO’, KO’)
----
1
imply
Â(l(T &#x3E; 1. Hence, if qf k, then KQ(n) is given by
expression (12). By (10) and (11), the condition
since then
is equivalent to qlmq. We deduce, therefore, from
lemma that, in the case qf k,
The truth of our theorem, when n satisfies
by (5), (9) and (13).
4. Conclusion. We have
K(n)
&#x3E; 0
if
(4),
(12)
the
and the
is thus established
simultaneously
every odd prime number q which
divides all Ku except one, Ku- say.
for
(Condition (14c) may be stated as qf mq for every odd prime number q[K for which Sq
1.)
=
69
It follows that all sufficiently large integers n satisfying the
conditions (14) can be represented as a sum of primes in the form
(1), and (2) will be an asymptotic formula for the number of such
representations. 1)
To prove the above statement about
since (au, KQ)
1,
K(n)
&#x3E;
0,
we
observe
that,,
=
provided that n satisfies (14a). Hence 82 is odd if2fm2’ and (s2 -1 )
is odd if 21M2’ It follows that, if (14a) and (14b) are satisfied, then
K(n) vanishes only if there is an odd prime number q for which
1 and qlmq
8q
=
1)
The above conclusions could also be drawn from
paper
general results proved in
my
[2].
REFERENCES
A. ZULAUF
[1] Über die Darstellung natürlicher Zahlen als Summen von Primzahlen aus
gegebenen Restklassen und Quadraten mit gegebenen Koeffizienten, I: Resultate für genügend gro03B2e Zahlen, Journ. f. Math, 192 (1954), 210-229.
[2]
[3]
-
-
-
-
-
-
ditto, II: Die singuläre Reihe, Journ. f. Math.
193
(1954),
ditto, III: Resultate für fast alle Zahlen, Journ. f. Math.
193
39-53.
(1954),
54-64.
J. G.
VAN DER
CORPUT
[4] Über Summen
(1938), 1-50.
von
Primzahlen und
Primzahlquadraten,
Math. Annalen. 116
University College, Ibadan, Nigeria.
March
(Oblatum 30-5-1959).
19th,
1959.