—со
From (31) and (32) we see th a t
у JV2(8/1 log TV)
and hence from (18)
we obtain
< 17
i
1 ТКТ2/ГЛ Л 1
TtTX_S
(35)
'
00
/ l j
S( a, *)} (sin (л«)/(л*)}2
doc > 17
M
This together w ith Lemma 8 im m ediately implies the existence of primes
Pi, p 2, p 3 w ith
m axim um
p betw een 2 and
N inclusive such th a t (3) is satisfied;
for otherwise th e left
hand side of (35, would be zero.
I t remains therefore only to prove th a t in place of
S; 2 there exist primes for
which th e stronger inequality
p > c25 is satisfied.
For this purpose we define R(oc) to
he th e same sum as given in § 2 for 5(
c2S. We clearly have
| 5 (< V 0
— 7 ? M ! ^ c J5
-
(/ = 1 ,2 ,3 ) .
F urther, by dividing the range of integration for
oc into two parts
K lf K 2 according as
|
oc| 5S (log
N ) B or | a | > (log
N ) H we easily deduce th a t
/ I
S (a jO c )
|2 {sin
{n o c )l(n o c )Y doc
<
f
| 5(oy%) |
+ iV 2
f (n o c )~ 2doc <
- ®
x ,
K,
Baker
, On some diophantine inequalities involving primes
181
Then by Holder’s inequality we obtain
*)
/
1
3
3
U
S ( o i t x )
—
(
o m
I l - i
>-i
— 00
and from (35) it follows th a t
00
j j IT Ä(cr,a)j {sin
(
7
z o c ) l(
7
ioc
) } 2
doc
{sin
( л о с )
1
(л о с
) } 2
doc <
1
M
>
1 0 M .
Finally, by virtue of Lemma 8, the last inequality
implies th a t primes p ^ p 2, p z exist
such th a t (3) holds and for which the m axim um
p satisfies c25 <
p ^
N . This completes
the proof of th e theorem.
References
[1] B. J. Birch and H. Davenport, On a theorem of Davenport and Heilbronn, Acta Mathematica 100 (1958),
259— 279.
[2] H. Davenport and H. Heilbronn, On indefinite quadratic forms in five variables, J. London Math. Soc. 21
(1946), 185— 193.
[3] K. Prachar, Primzahl Verteilung, Berlin-Göttingen-Heidelberg 1957.
[4] H. Richert, Aus der additiven Primzahltheorie, J. Reine Angew. Math. 191 (1953), 179— 198.
[5] I. M . Vinogradov, The method of trigonometrical sums in the theory of numbers (translated by K. F. Roth
and A. Davenport), London-New York 1954.
[6] H . Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Annalen 77 (1916), 313—352.
Eingegangen 28. März 1966
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