A ^^]l 0 wl0g,>=у i°g? 2 **p)[%] H V

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)

M. Ka l e c k i (Warszawa)

A short elementary proof of the prime number theorem* *

The proof based on Selberg’s theorem has some affinity with that of Brensch (х) but differs from it in substantial points and is as a result somewhat shorter. A new simple proof of a theorem equivalent to that of Selberg is included.

Th e o r e m 1 (equivalent to Selberg’s). Let us denote гр(х)-\-\ogx by Vq(#). We have

( 1 ) У л («)*,, a + \px{x)\ogx == 2x\ogx-\-0(x).

P roo f. We have

H

y,(n) I — J log— logn = [л{п) И log a

V

n I n

L J d\n

A(d)

and

Г

x~\ X YhO) = A p(n) — log — .

\_ n J n Hence

(2⁾

A ^ ^ ] l0w l0g,>= у i°g? 2 **p )[% ]

J p<ąx x ^ ■L

= 2 * * * 2 (^(^))a( - i )a V i ( 4 r ) =

x<ąrp a=l \Jr / n ^ x

* I wish to thank prof. A. Schinzel for constructive criticism of the first ver­

sion of the proof.

d) It. B re u s c h , Another p ro o f o f the prim e number theorem, Duke Math. Journ.

51 (1954), pp. 4 9 -5 3 .

But we also have

log — lo gn n

log*

2 ? (*) [ £ ] ■

*°g V - £ ^ [n) Ш

log27

»<a; w<x

% (®)log® + ^ V ( » ) { - ^ } 10

n

= y}1(x )log x + 0 (x ) — 2x ^ fc«g -

i ^ + ^ + o ' log«

X

П where A is a certain constant. Hence by (2)

п<ж '

- ) + * ( - ) t o e % l ° f + 0 W

' n^x , X_{fc < -}

n

= 2x ^{V 1} Л(т) __ 2x\ogx-\-0(x).

a—J m

m<^x

Co r o l l a r y

Vi (x)/x—l by r

1. Let us denote by e an arbitrary positive constant and (x). We have

(3) —1.4 £ < r ( # ( l + £)) — r(x) < 1.4 £ for sufficiently large x,

(4) r(a ? (l'+ o (l))) — r(x) = o ( l ) ,

(5) I ^ < - + 0 (D .

X<7l<X(l+e) P ro o f. We have by (1)

у)Лх{1 A s)) — Wi(x)

0 < n— = r (x(l + e) ( l + e) + e - r ( ® ) < 2 e + o (l).

x ' '

How, since log 2 < ip1(x)lx + o (l) < 21og2, we obtain for sufficiently large x

—1.4 e < —£21og2 + o(l) < r(a?(l + e)) — r(x) < £(2 —log2) + o(l) < 1.4fi

and г(ж(1 + о(1))) — r(x) — o(l) because of arbitrariness of s. Further­

more

1

A(n) n

щ(х(1 + е)) — ух (a?)

x < 2 e + o (l).

C^orollary 2. We have

V i Л(п) I x \

(6) r(x)logx = - > --- - r — + 0 (1 ),

ć—i n \n

(7) limsupr(£c) = — liminfr(a7) = r.

P ro o f. We have by (1)

[Wl{x)-x)\ ogx = - Л(п) l y J - H —-H + 0(X ),

n*ąx \ ' f '

which proves (6). Moreover, since limsupr(a?) > Q and liminfr(o?) < 0 we have by (5) limsupr(a?) < —liminfr(a?)+o(l); liminfr(aj) ^ — limsupr(a;) +

+ о (1) which proves (7).

C^orollary 3. I f we fix e > 0 and choose x in such a way that r(x)

= r + o(l) there exists an integer a < xe such that r(xla) = — r + o(l).

P roo f. Let us assume contrarywise, that for each n < xe we have r(xln) > — r + /3, where 0 is a positive constant. Then by (6)

(r + o(l))loga?

< { r -P )e lo g x + r { l - e ) l o g x + o {l)lo g x . Hence r < (r—/3e) + o(l) which is false.

T^heorem 2.

(8) ip(x) = x-\-o{x).

P roo f. Since ip(x) = 'ip1(x) — logx it is sufficient to prove that r = 0.

Let us assume contrarywise that r > 0. Let us form a sequence

= (1 + е)г log ж, where s is an arbitrarily small positive constant, 0 < i < к and Tc is determined as follows:

log

Tc = logx log log a?

logo?—2 log log ж'

(9)

log(l + e) log(l + e)

Let ns denote the intervals y i,y i+1 'which contain some integers n' fulfilling the condition r(x[n') < — s by yi, yi+1 and the number of these intervals by N. Moreover, let us denote all n Ф n' < x by n". By (6) we have for x at which r(x) — r + o(l)

(10) (r+o(l))loga? = A(n') ( x \ wi Л(п") ( x \

2 j ~ 2 j n" r w 7j

< _ y i A{n')

"" z L n'

X

n' + £loga? + 0 ( l) .

According to Corollary 3 of Theorem 1 we have r(x/a) = — r + o(l) for some integer a < xe. Let us denote by yi, yi+1 the intervals yit yi+l which contain some numbers am', where m' is an integer, fulfilling the condition r(xjam') > e, and let M be their number. Moreover, let us denote all integers m Ф m' < xja by m".

By (6) we have

M+ log/ + = y i M a ' \ a ' ' m^xja m

+0⁽1^).

y i < a m ' ^ y i +l is equivalent to — < m' < ^1+1 ; thus by (5)

a a

and (7) we obtain

(11) ( r + o ( l ) ) ( l — e )lo g a ? < (r + o (l))(2 e + o ( l ) ) J f + e ^ A ^ r -

< [ 2е г ф о ( 1) ) Ш + sloga? + 0 ( l ) . Hence

(12) J f > ^ - ( i - 7 _ a *)<1 + e(1))-

Now the intervals yi, yi+1 and yi, yi+1 do not overlap since by (3) we have г (l + e)j — r j < 1.4s and by (4) r|a;(l + o(l))j — r(x) = o(l), whereas in intervals yi: yi+1 r(xln') < — s and in intervals у*, yi+1 r(xlam')

> e. Hence NĄ-M < Jc and by (9) with e < 2

(13) N < lo gx

2s 1 + б £ + Д

Let us now group yi,y%+i into segments y j,y j+h, where h + 1. Let us denote those segments уj, yj+h for which

A (n)

t

1

> 1.5e

V i< n ^ y i + h n

and for which the least r(x /n )< — 0.75 r by yu yj+h. The number of

1 ^ A (n)

intervals y{, yi+1 in other segments y,-, yj+h1 for which— 2j ---

^ Vj<n^yi+h n

< 1 .5 s or the least r(x[n) + —0.75r, let be denoted by Q. Finally let logs by ^q(z). For a segment у j, yj+h we have us denote V

n^z П

Ъ~1ш(я{У1+ъ) — Q(Vi)) > 0.5s and denoting sup\q{z)— q{w)\ by Я

<14) h < i n r -0.5s

Taking into consideration (4) we find that r(x/yj) or r(x[yj+h) >

— s + o(l). Without loss of generality of the proof we assume r(x/yj)

> — £ + 0 (1). Since r(xfy) can decline in an interval УцУг+1 according to (3) less than by 1.4s and, by definition, the least r(x [n )< —0.75r, we have for sufficiently small s

(15) h > 0 .7 5 r - s + o(l)

U s > 9 = 0.74r ---+ 1

1.4s

and also, taking into consideration (4), for yt < n ^ y i+1 we have r(xln)

> r(xlyi)—1.4 s.

Thus we obtain for у j, yj+h, taking into account (15),

-

V j< n < V j + h¹

< 2s(1 + o(1 ))(s+ 1 . 4s + o(1)- 1.4s\

~2 / + 2er (l + o (1)) (h - [g]) < 2e r { h - 0.63 g) + 2s + 5s8flr. Consequently by (14) and (15) we have

1

~hV j < n ^ y j + h

- 0 .1 5 — Г Я

for sufficiently small s. Finally we may write by (10) (r+o(l))loga?

< 2 s r (l— 0.15г/Я) (N —Q) +1.5s(r + o(l))^ + slogx < 2erdN A-slogx, where

<3 is a constant < 1 independent of s. Now by (13) we have r < rd (1 + 6s + £

Г + e>

since s is arbitrary, this is equivalent to r = 0.