Comparative prime-number theory. III

COMPARATIVE PRIME-NUMBER THEORY. III

( C O N T I N U A T I O N O F T H E S T U D Y O F C O M P A R I S O N O F T H E P R O G R E S S I O N S ~ 1 m o d k A N D --- l rood k)

By

S. KNAPOWSKI (Poznan) and P. TUR,~N (Budapest), member of the Academy

1. In the previous paper of this series 1 we established a par t o f our results concern ing the compar ison of the residue classes 1 and l m o d k. As ment ioned in I I we could deduce a (very low) lower bound for

(1 .1) Wk(T, 1, l),

the number o f sign-changes of

n(x, k, 1 ) - n(x, k, l)

for 0 < x <_- T. N o w we are going to show first how b y an appropriate modification o f the proof of the Theorem 5.1, resp, 5 .2 we can obtain a much better lower bound for W(T, 1, l) ( though these theorems themselves remain untouched being essentially best-possible ones). Most probably this will be also rather rough, but as to the modul i

(1 .2) k = 3 , 4, 5, 6, 7, 8, 9, 10, 11, 12, 19, 24

as far as we know, our result is the first theorem in this direction proved without any conjectures. Keeping the conVention of I and I I we denote by ca, c2 . . . . explici- tly calculable positive numerical constants; 2 then we assert the

THEOREM 1. 1. For T > c l we have for the moduli k in (1.2) the inequality

W k ( T , 1, l) > c2 log4 T.

More generally we have

Tr~EOREM 1. 2. I f for a k the Haselgrove-eondition 3 holds then for

(1.3)

the inequality

holds.

>maxIe,, c Wk ( T, 1, l) > k -r log4 T

1 We shall quote the previous papers of this series for the sake of brevity by I and II, respectively.

2And alsoe~(x)=e x and ev(x)=e~(e~-1(x)), further log~x=logx and logvx= = log (logv- 1 x).

3 We remind the reader that by the Haselgrove-condition for a k we mean the existence o f an A = A (k) with 0 < A (k) ~ 1 such that no L (s, Z) belonging to mod k vanishes for 0 < a < 1 and ttl<-- a(k) ( s = ~ -bit).

8 Acta~ Mathematica XIII/3--4

344 S. KNAPOWSKI AND P. T U R I N

Obviously it suffices to prove Theorem 1.2. As another immediate consequence we assert the

THEOREM 1. 3. I f for a k the Haselgrove-condition holds then the interval i

2 0 < x ~ max(e4(kc0 , e z ( ~ ) )

contains at least one sign-change of

~r(x, k, 1) - n ( x , k, l) for all l ~ 1 rood k.

w e can prove the analoga o f Theorems 1.1 and 1.2 for

1 ~,l ~ (x , k , l ) . (1.4) ~(x, k, 1) c;(k) - 1 (l. *~= 1 1 r

Denot ing the number of sign-changes of this function for 0 x = T by Sk(T) we have the

THEOREM 1. 4. I f for a k the Haselgrove-condition holds, then for

2

the inequality

Sk ( T) > k -c3 log4 T holds. ~

Since the p roo f of this theorem follows closely that of Theorem 1.2 we shall omit its details.

By these theorems steps are made towards the solution of Problems 6 and 5 o f paper I.

2. The previous results of this paper and all results of paper II refer to all l's with (l, k) = 1, l ~ 1 rood k. For certain l ' s however these theorems are capable to an essential refinement. The l ' s in question are those for which the congruence

(2. 1) x 2 = 1 m od k

has exactly as many (incongruent) solutions as the congruence

(2 .2) x 2 = 1 m od k.

Such l 's occur ,,rather often", in particular when k is a prime. Then we assert the

4 The same holds for the number of sign-changes of zffx, k, 1 ) -- ~ ~(x) and mutatis mu- g 1 ['.dv

tandis for ~(x, k, 1)-- ~ 2 log v "

COMPARATIVE PRIME-NUMBER THEORY. III 345

THEOREM 2. 1. For the k's in (1.2) and l's satisfying the condition (2. 1)--(2. 2) for T>c4 the inequalities

max {n(x,k, 1 ) -n (x ; k , l ) } > t @ e l ( - 4 2 l~176 T1/3~x~T ~ l O g 2 I

(2. 3)

and

(2. 4) Tioga T )

min {n(x, k, l) - n ( x , k, l)} < - V T e l - 4 2 l~ 2 T TI/a~x~T

hold. This is an elegant special case of the

THEOREM 2. 2. For the moduli k in (1.2) and I satisfying (2. 1 ) ( 2 . 2 ) , t f Qo = =flo + i7o with flo ~ �89 is such that L(eo, Z~)=0 with ;~(1) ~ l, then we have for

T > m a x (c5, e2(10leol)) the inequalities

(_ 421~ r l_~ T ) max {n(x,k, l)-n(x,k,l)}>TPoe~ \ l o g T

TI/3<=x<= T (2. 5)

and

(2.6) min {n(x,k, 1)-n(x,k, l)}<-TBoe~(-421~ T) . TI/3~x~T

As remarked in paper II it is again Siegel's theorem 5 (see SImEL [1]) which makes Theorem 2. 1 a special case of Theorem 2.2.

3. Theorems 2. 1 and 2.2 are unconditional. Passing, however, to general k's, we have as in paper II to use Haselgrove-condition. Then we have

THEOREM 3. 1. If for a k the Haselgrove-condition holds and I satisfies the condition (2. 1)--(2.2) then for

T>max (c7, e2(k), e2 (A-~k)3 ) )

the inequalities (2.3) and (2.4) hold. All the theorems formulated in 2 and 3 are owing to Siegel's above-mentioned

theorem consequences of the

THEOREM 3. 2. Ifforakthe Haselgrove-condition holds andlsatisfies (2. 1)--(2. 2) and if further Qo ---flo + iyo, flo >-�89 is a zero of an L(s, Zi) with Zl(1) # I, then for

(c ) (3.1) T > max 7, ez(k), e 2 , e2(101eol)

the inequalities (2. 5)--(2. 6) hold.

5 This theorem asserts that all L(s, Z) functions belonging to primitive g 's rood k have at least one zero in the domain

1 c6 a --> --, [tl <_--

2 log3 (k q- e3 (1)) "

*8

346 S. KNAPOWSICI AND P. TURAN

As easy corollary we have for the l's in (2. 1) - - (2 .2) the following improvement of our Theorems 1.1 and 1.2.

THEOREM 3. 3, For T > c 1 we have for the rnoduli k in (I. 2) and l's satisfying (2. I)--(2. 2) the inequality

Wk(T, 1, l) > Ca log2 T.

THEOREM 3.4, I f for a k the Haselgrove-eondition holds and

2

then for the l's satisfying (2. 1)--(2. 2) the inequality

Wk(T, 1, l) > e 8 log2 T

holds.

As to a comparison of the above stated theorems to those attainable by older methods we refer to our paper I.

4. The basis of the proofs in paper II was the Lemma I. As to its proof we referred to the paper of one of us (see TUR,~N [1]) since we mentioned we shall need for our later purposes a slightly more general form of this lemma. For an exposition of this, let m be an arbitrary non-negative integer a n d

(4. 1) l = l z l l ~ l z 2 1 ~ .... ~ [z . I

further with a 0 < ~ <_- _-- for j = 1, 2, n 2 " ~ ~

(4. 2)

Let the index h be such that

~<I I~. = arc zj

4n , 4 3 , zh >

and fixed. Further we define A and the index h 1 by

(4. 4) A = min Re ~ bi h~--~<hl j=l

if there is a h 1 ~ n with

(4. 5) Izhil~ Izhl 2n


(4. 7)

such that

(4. 8)

and

(4. 9)

and

(4. 6) A = rain Re ~ ' bj h ~ _ n j = l

otherwise. Then we assert the 6

THEORI~M 4. 1. l f A >-O then there are integers v 1 and v2 with

m + l = v ~ , v2<=m+n 3 +

n

Re ~ bjz~ ~ >- A j=l 2 n + l (

24 m + n

" v2< A 12 n t2 n{ ,~,.+n(a+~) Re bjzj = .Izhl. i=1 2 n + l 4 (m -t- n i 3 + ~ ) ) } \-~--']

5. Since the proof of this theorem differs very little from that of the above mentioned Lemma I of the paper II a sketch will suffice. We can take without any change from TURXN [1] the following

/_,EMMA I. The polynomial

F(z)=zd + a i z a-1 + ... +aa

with real coefficients and with all zeros in the domain

Izl<= 1, g<- larczl<-~

can be multiplied by a polynomial 99(z) with real coefficients so that writing

F(z)qD(z)= Z~evz v we have

a) ev=>0 (v=0, 1 .... ) b) the degree of F(z)q~(z) cannot exceed 1 + -~

c) ~ e v <_- 2 d d) the coefficient of the highest power o f z in F(z)qD(z) is at least 3 -a.

Having this lemma we introduce as in TURAN [1J to our z fnumbers (which can be assumed to be all differen 0 the numbers ~ by

(5. 1) ~ 2 j - l = z j , ~2j=2/ (j---- 1, 2 . . . . . n)

6 A similar modification of the second main theorem (see TURAN [2], S6s-TURA, N [1]), which increases essentially the applicability of it, had been found by one of us (see KNAPOWSKI [1]).

348 S. KNAPOWSKI AND P. TUR/~N

and let ~/1, ~2, " " , /~l with

(5.2) 1 =l~h[----> [~/21 ~.. . ~ lr/ll

be the maximal number of different ones among the ~'s. Certainly we have beside (5.2) also

(5.3) y. <= [arc ~/j] <=~z ( j = l , ,..,I)

and

(5.4) n<-l<=2n.

Since with qj also ~j is among our ~ls's, the polynomial

l (5. 5) do ( z ) d o f / / ( z - ~j)

j=l

has real coefficients. We apply the Lemma I to dO(z) as F(z) O. el in case d-- 1) and denoting by ep*(z) the polynomial corresponding to q~(z) we have

(5. 6) do (z) q~* (z) def Z e" z v l n

with non-negative e~-coefficients, with

(5.7) ~e~ <_- 2 t and with the leading coefficient

(5. 8) =>3 -l.

6. Let (somewhat differently from TURAN [1]) be

(6.1, 6--- IZh[ m ~ n ~ + ~_) (---->~)

and (6.2) G(z)= ( 24 ~2nl+Z+Z2"k.. ._+zt-1

q~ (z) d~f G(z) do(z) q~* (z)5 ~r Y~ve( Z ~.

W(z) is obviously a polynomial of degree

(6.3, =</ (1 + ~ ) + ( / " 1)<=n(3 + ~ ) - 1

with non-negative coefficients and if the exact degree of do(z)~v*(z) is No, then owing to (5.8)we have for v = No, No + 1 ...... No + l - i the inequality

) 2n 2n q , Z > 24 3_lfi . . . . (3+~)> ( 8 ~ 6 . . . . (3+;)

(6 .4) coe f f s , z ~ in ( ) - Izhl- 6 -" \ ~ / "


Analogously as in TURAN [1] (and as in S6s--TURAN [1]) we get the existence of an R with

(6. 5) ~ <= lzl <-- lzhi

such that on the whole periphery of the circle [z] = R the inequality

,~(z), _ - > 2 ( ~ ) t

holds and the same for all partial products

(6. 6) (Z--~ht)(Z--rh~)..,(z-~i) (I~il <iz <..,-<ii<=l). Let the index co be defined uniquely by

(6.7) 1 =l•a] => I~t2l =>... =>lq~l ~-R > Iq~+ll-->~..=~" Iq~l,

(possibly also co = l ) and let 7 as in TUR/~N [1]

l l-~o ( 6 . 8 ) Ha(z)gq 1I ( z - ~ j ) def Z c~ 1)Zj.

j = ~ + l j = O

The c~L)-coefficients are obviously real and the estimation

(6.9) [c~l,, ~ (l-~to) ( j = 0,1 . . . . , ( l - t o ) ) k J /

holds ( ( 0 ) means 1).

7. Next we consider as in Tug~N [I] the auxiliary polynomial H2(z) of degree <= t o - l , defined by the requirements (No in (6.4))

1 (7. 1) H2(qj)=t~7+I+NoH16lj~__ ( j = 1, 2, ..., to).

Writing H2(z ) in the form

(7. 2) H E (z) = C~o 2) + e(~ 2) (z - ~/x) + c~2)( z - rh) (z - ~/2),,+...

, (2 ) z (z (z �9 .. + ~ - , ( - 7 1 ) -72)... - ~ - 0

the N6rlund integral-representation

d.2>=l f dw J 2hi wm+t+NoHl(w)(w-th)...(w-qj+l)

l~l =R

gives owing to (6.6) the estimation

(7. 3) 1c)2)[ < 2 ~ ( j = 0 , 1, ..., to -- 1).

7 If co----i the product means 1.

350 S. KNAPOWSKI AND P. TURAN

Writing H2(z ) in the form

to--1

(7 .4) Hz(z) dee ~ c(~a)z ~ 7 = 0

the coefficients are owing to (7. 1) real and analogously as in TURAN [1] we get

(7 .5) [c~a)]__<_ - 1 ( 4 _~t( ~o ) 2Rm+No\ lzh l_6] \ ~ + 1 ( 7 = 0 ' 1' "" ' O~-- 1)"

We need further the polynomial

/ - I .

(7; 6) H a (z)d~rH1 (Z) H 2 (z) d~r Z c~ 4) zv' v = 0

The coefficients c~ 4) are obviously real and we have

(7. =oZ 8. Finally we consider the auxiliary polynomials (qffz) f rom (6. 2))

(8.1) qJ(z) + zNon3(z)a-~ ~vc(v5) 2 v

N O def (8. 2) V(z) - z H 3 ( z ) ~ .~c(~6)z ".

Their coefficients are obviously real; but as in TURAN [1] one can see, they are non-negative. Further we get f rom the definition of the q~(z) and (I)(z) polynomials for j = 1, 2 . . . . . /,

(5) v _ _ No ,~vCv r/j--r/j H3(r//),

Z c~6)- ~. = - n7oH3 (n j), v v " t j

resp., and hence owing to the definition of the H3(z ) polynomials as in TuRAN [1]

{~ for j = l , 2 , . . ~ (8.3) S ' c~5)n~. +1+~= "'

~'~ ~ "': for j = c o + l . . . . . l,

- 1 for j = l , 2, r (8 .4) S' rt6), 'n+l+v = ....

~ ' ~ "~ 0 for j = a ~ + l . . . . . I

resp. 8 Owing to the definition of the ~//s, {j's and z / s (8 .3) and (8.4) hold with ~'s instead of the r/'s, the two categories being

(8.5) Izjl > R [zj] < R

resp.; hence defining the index /~ by

(8.6) 1 = ]ztl-~lz21 =>...--> lz~l > g >lz,+x[ >- ...

s In (8.3) and (8.4) the second category can be empty (and also later).

COMPARATIVE PRIME-NUMBER THEORY. III 351"

we have

(8.7) :'|~ for j = l , 2 ..... , p Z ,(5)Tm+l+v

-v -J = 1 0 for j = p + l , . . ,n,

J ' - I for j = l , 2 , . . , p ( 8 . 8 ) ~'~ r'(6)2m" + l + v = 1 0 '

~"~-~ -J f o r j = # - t - 1 . . . . , n

resp. Now we may observe first that from (6. 5) Izhl ~--Iz~l, i. e.

h<_-p;

further in the case (4, 5 ) f rom (6. 5) and (6. 1)

2n Iz.I >R--~6 = lza[- / rc'~ > IZh'l'

m + n ~ 3 + ~ - )

i . e . p < h l

and p_<-n otherwise. Hence in the case (4. 5)

(8.9) h<-_#<hl and (8.10) h<=p<-_n otherwise.

9. Now we can complete the sketch Of the proof of Theorem 4. 1. Multiplying in (8.7) by b j, summing for j = 1 . . . . . n and taking real parts we get

(9.1) 3 ' c (5) Re ~ b z re+v+1 = R e ~b j>-A ~.~v v / ~ J J / I. j = l . I j = l

from (4. 4) and (4. 6). Since the degree of the polynomial in (8. 1) (and in (8.2)) is owing to (6. 3), (5 .6) and (7.6) at most

1

we get from (9. 1) owing to the non-negativity of c~S)'s n

(9.2) max Re ~ ' b:'j >= A m+l<=v~m+n(3+~_) j = l ZvC(v 5)"

v integer

But from (8. 1), (6.2), (5.7) and (7.7) we get

~ ' e " ( 24 ~2, -, ,-~(3+L, ~ve(~ 5)<- ~ v v +Zvlc( : ) l = \ ~ j 2n6 ' " :4"+

(9. 3)

:352 S. KNAPOWSKI AND P. TURAN

Since from (6. 1) we get

48 2 4 ( m + n (3 + ~ ) )

,(9.4) Izh[--,~ n

:and from (4. 3) and (6. 1)

1 (9. 5) 6~_~ lzht,

the proof of (4. 81 follows from (9.2), (9.3), ( 9 . 4 ) a n d (9.5). Similarly for (4. 9) starting from (8.8).

10. In order to prove all the theorems stated previously it will be enough to prove the Theorems 1.2 and 3.2. We shall start with the first; the proof will have much common with that of Theorem 5.2 in paper II (these parts will be only sketched) and again some ideas of Littlewood. Ingham and Skewes (see LITTLEWOOO [1], INGHAM [1], SKEWES [I]) will be used.

We shall use from paper II the following theorem (see (3 .4 )and (3.5) there). If for a modulus k Haselgrove-condition holds and for a Z' with Z'(I) ~ 1 the

function L(s, Z') vanishes at Co ~- tro+ ito, then for

T~>max(cao,, ez(10l~of), e2(k), e2 ( A ~ k ) 3 ) ) (10. 1)

the inequalities

(10.2) max Tll/3~2r T 1

and

(10.3)

{H(r k, 1)- II(x, k, l)} > T(oe, ( - 4 1 log log2T' lOg3T~ T,) .

min (H(x ,k , 1)-Fl(x,k,l)}< -T~Oel ( - 4 1 l ~ 1Og3 T1) r',3~_x~_r, \ log2 Tt

hold. For our present aims we need this theorem in a slightly modified form. Since

obviously we have for x->2 the inequality I i{~(x, k, 1)-,~(x, k, 1)} - {n(x, k, 1) -rt(x, k, l)}I<c,,x~,

we get, if

1 log3 Tt ,(10.4) flo >- ~- + 42 log2 TI '

for the T l ' s in (10. 1) the inequalities 9

max {n(x, k, 1) - n(x, k, l)} > T(~ ( - 42

/

log T~ 1og3 T1 T'J3<=x<=T1 \ log2 Tx

) 9 Replacing if necessary C~o in (10.1). by a larger constant c.


anti

( T1 lOgaT1 T I ) : min {~(x,k, 1 ) - n ( x , k , / ) } < --T~~ -421~ 2 Tll/a~X<= T1

Hence if (10. 1) and (101 4) are satisfied, the function ~r(x, k, 1)-rr(x, k, l) has at least one sign-change in the interval

(10. 5) T1/3 ~ X ~ T~.

Now let T be in the range (1.3) and we consider the Case I. There is a character Z" with X'(1)~ 1 such that L(s, ~') has a zero o~--

= flo + i7o with

> 1 + 4 3 ! o g 3 T I (10.6) flo = ~ lOg 2 T' ]7~ 40 1Og2 T.

{ log r \ Then one can see at once that replacing in (10. 1) and (10. 4) TI by el \xv83 J t / ~ / these inequalities, are amply-satisfied (choosing in ( l , 3) c3 sufficiently large) and a fortiori replacing TI by the values

T3-V with

Hence by (10. 5) the function rt(x, k, 1) - n(x, k, 1) has in the case I for 0 < x -< T at least

1 log4 T

sign-changes.

11. Case IL None of the L(s, X) functions with ~(l) # 1 vanishes for

1 , l og 3 T 1 (11.1) a_->~-n-431--~zT, Itl-<__~-~log2 r .

This part of the argument will have much resemblance to that of Theorem 5.2 of paper II but this time the parameter z will be chosen differently, as

(11.2)

c12 sufficiently large and

(11.3)

Putting

(11.4)

z = kcG

q =log2z.

n ~ l m o d k n=_lmodk

354 S. KNAPOWSKI AND P. TUR.~N

we start as in paper II (formula (15. 3)) from the inequality

(11.5) 1 Z(1-~(o) 0,~, I<c,31og~, 1 < [te[_ ~log2 T

if only r is restricted by

(11.6) 0<r_<-log3 T

(see (15.2) of paper II). Denoting again

/r sin z (-~r2-~ '/2

(11.7) co(r)~o_F~ I ~Tz--~) / k 2 ,!

we shall not work with a single m-value as in paper II but with several ones; more exactly if c14 is sufficiently large then for

,18, ..... ([ 7 0 6k c14 log 2 (k e'z) d

we restrict m~ at present only by

(11.9) q(aV+ 1)gel4 ~ my ~ q(3v+ 2)kel4

(11.8) and (11.9) give evidently that for our v's

5 (11.10) ~mv <log3 T

i.e. we can multiply in (11.5) by any of our Go, v(r)-functions and integrate over

~-mr, ~-m~ respectively. This gives, owing to

. f . z ( r - - m ) ' , 2 1 [' [ s l n ~ - ~

~2"c~_~ ~ - ~ l dr:l'

the inequality (for all of our v's) 5

~-tOv

;~fg(r)Go.(r)dr+--~Z(1-2(1)). 3 ~ t a v

S -d tOv

ltOr ~.~etz) ~ -.. / e ' - G~(r)dr < c l s l o g k ZT~Ito d I

1<Ire I ~ l l ~ 3


(12. I)

and

or proceeding as in 16 of paper II for each of our v's 5 ~- COy

f g ( ] t - ~ ) e't~176 1 1 z.~ (1 _ ~(/)) aZQ(z ) 1-- ~ logk (11.11) ] ~ ( r ) G , o ~ ( r ) d r + ~ <'to/--<~ ~ <c16 3 -~- roy

where c~6 does not depend upon the choice of c12!

12. The reasoning of 17 of paper II gives again instead of the reference to (5.3) of paper II to (1.3) of the present paper with sufficiently large c 3 - - that

�9 to~

( 1 1 z ~ ( 1 - ~ ( l ) ) -~o(x) 1 - - z - / ~ >3--~ l~

q~(k) l<lt,,l~_~

ton

( ~ - ! ) e- 'T~ 1 I log z. (12.2) of(k) Z ( 1 - Z q ) ) ~o(x) 1 - - - < -- z 1 < itol<=~ it e 327t

Next we determine for v's in (11.8) the numbers 7v so that

t o ..~ 1 (12.3) ~ ~ - q

with the q in (11.3) should hold for all 0 < t o <= z of all L(s, z)-functions belonging to mud k; here again [Ix]] stands for the distance of x from the next integer. As before we get for the number of these t~'s choosing e12 in (11.2) sufficiently large the upper bound

2 9~ (k) z log n

and hence Dirichlet's theorem assures the existence of an 7v with (12. 3) and { 3~ [~ 3"~. 2q;(k)

kCl4 q~3~+y) ~7~=>q "t~+~-'~r ~ ,log, (12. 4)

Then we have from (12. 1), (11.2) and (11.3) 1~

( ~]_)e it~ (12.5) 1 ~(l_z(/)) Yo,~, 1 - >

> 3 - ~ l o g ' c - - - ~ z~' (1 -~( / ) )~< l to /~ \

to'~ e 'ST (e 2nit~ 1 2 27~ t- T ~'~-1)l>32~l~ ~o(k)~ ~0(~)~-"

l _ ] t e l ~ r

1 �9 > ~ log z - log z log kz > ~ log z ItoL

~o Also the well-known inequality ~ o ( z ) l<c~71ogk(2-t-!~]) is used for all Z's modk.

356 S. K N A P O W S K I AND P. TUR,~N

if e12 in (11.2) is sufficiently large. Analogously we get

(12.6) 1 (1 I-~) e% ( ~ - ~ ) < I < I ol ~ "- 402z log "c.

+L In order to use the quantities y~_ 2z as ~% s in (11.11), we have to verify

(11.9) for all v's in (11.8). So far we made no restrictions upon c14 in (11.8). Let now e~2 beyond the previous requirements be so large that for the ct6 in (11.11) the inequality

C 1 2 (12.7) c16<80 ~

holds and then we fix it; then we choose c~4 in (11.9) with c~4>1 so large that

(12. 8) 2 q~ (k) -c 1 log z < ~ - k ~

and then fix it. Since owing to caz>2 we have

___ kel4 k {log 2 (k~,O } > (log 9) ~ > 2,

this means owing to (12.8) and (12.4)

(12.9) (3v+ l,75)kc14_{_ Yv ~- ~ < q 1 < : q(3V+ 2)ke14

Since further, as before, we have

3 > _ ( 3 v + ~ ) k ~ x 4 q(3v+l)kea4

(12. 10) 7 ~ - ~ ,t 1 >

the requirement (11.9) is verified indeed and hence from (11. 11), (12. 5), (12. 6) and ( 1 2 . 7 ) w e get

5

1 1 1 1 2~ J g(r)G~+~(r)dr<2~ - 40re log "c + el6 log k < - 807r log z

3 zz 4(~-~)

and analogously

5

f. 1 1 g(r)G~_ (r)dr ~ l o g z. d

3 ~z

COMPARATIVE PRIME-NUMBER THEORY. II I 3 5 7

These last two inequalities show at once that g(r) changes sign necessarily in the intervals

3 ( ~ ) 5 ( I~) ( (q3vk~ - Yv'2zz <=r---<-4 - 7V

for all of our v's; since these ranges are distinct, taking in account also (11.8), the proof of Theorem 1.2 is finished.

13. Now we turn to the proof of Theorem 3.2. Since (7. 1 ) o f paper II is by the assumptions of our theorem simply satisfied, Lemma II of the paper II is appli- cable and hence we have a Yl such that

(13.1) . 1

log 2 T<~yl <= ~ log 2 T

and for all O's of all L(s, Z) functions mod k (writing ~ = % + ite) the inequalities

I e"~,',--><8 a(k)~ (13.2) n_->_ a r c ~ - k(1 q-ltd) 6 log 3 k(2+ltol )

and

�9 t o

l t ~ e%-rl ->c~8 A(k)3 (13.3) arc - - Q k(l + Itol) 6 log 3 k(2 + Itol):

hold; we fix this y~. The integer v is restricted at present only by the requirement

(13.4) log T logO.9T ~ v<=!og T. Yl Yl

Let the residue-classes

(13.5) ~l,, ~2,, -.., so, ~1~ ~2~ . . . , 0{Q

be the incongruent solutions of the congruences

(13.6) x 2 - l m o d k, x 2 ~ l m o d k

respectively. Then we start from the integral

(13.7)

(Z)

1

-2~-, ,=,2 \ - 7 - / ~ 2 (~(~,;)-~%)) z- (s, z.) d,.

358 S. KNAPOWSKI AND P. TUR.~N

t

Inserting the known Dirichlet-series for L (s, Z) we get

evy~ evYl log`'- 1 log~ -

n n 0(13.8) J~(T,k)= X A(n) ~ A(n)

/,1~ e, lV ( V - 1 ) ! ?l~eYl,y (V - - 1 ) ! n ~ l m o d k n ~ l modk

+

evyl eVYl Q log~- a n ~ - Q log~- 1 n 2

~- Z 2 ' A(n) S' Z A(n) (v-- 1 )~ j = l n~e�89 (V - - 1)] j='a'~l 11~e�89 v

n ~ s j rood k n=-a:.i mod k

The contribution of the primes to the first two sums is obviously

(13.9)

cVYl e`'yl log"- 1 log~-

P P logp . ~ logp.

p ~ l mod k p ~ l rood k

However the contribution of the p2's to the first two sums and that of the P'S to the latter ones in (13.8) cancels, owing to (13.5). Hence we have on the one hand

I evy I evyJ log~- 1 _ _ log~- i -_~

P P ,(13.10) Jv (T , k ) - • logp. (v - l ) ! ~ Z logp. (v -D! <

p~eYl v p~eYl v p-~l mod k p-_-i mod k

{ _ } (vyl) v-1 �89 (vyl)V-1 (e�88 <c2oTSel 21 -1~g2~- ] •r (v--l)[ (e )Yiv+k (v--l)!

using (13.1), (13.4) and (3.1). Now the two sums on the left can be written in the form

eVYl log`'- 1 e~l

(13.11) (v - 1)! log x.d{n(x, k, l) -n(x , k, 1)}-= 1

eVYl eYl eVYl

1 l eYl

F o r the absolute value of the first integral we get simply (as in 10 of paper II) the upper bound

{~. log T logs T'~. ]


As to the second one, since the factor eVr~

- log ~-i .log x

X

is increasing for e rl <_-x~ eVy~ we get respectively the upper and lower bound

max {n(x, k, l) - n ( x , k, 1)} Yl �9 ~ ( ( v - ly i ) "-1, <= e vy I

Y l min {n(x, k, l) - n ( x , k, 1 ) } . ~ ((v - 1)yl) ~-a. X ~ e V Y l

Taking in account also (13.4), (13.10) gives

(13.12) max {n(x, k, l) - n ( x , k, 1)} > {J~(T, k) - r ~ ( v - 1 ) [ ( ( v - 1 ) y x ) - ~ + t x~--T Yl

and analogously for

(13. 13) min {n(x, k, 1) - n ( x , k, 1)}. x ~ T

14. We define the broken line V symmetrically to the real axis in the vertical l < <3 exactly as we did in 10 of the paper II and we shift the path of strip ~ = a = 5

integration in (13.7) to V. For the absolute value of this last integral we get as in 11 of paper II the upper bound

2 1

c2t(5V'e $ vYl k log k + 2v.k.5v.e ~ VYlk log 3 k}.

Taking in account (13.1), (13.4) and (3.1) this is

< Z0, 45

choosing in (3.1) c7 sufficiently large. With the contribution of the residua this and (13.12) give

(14.1) max { n ( x , k , l ) - n ( x , k , 1))~ ( v - 1 ) t ( ( v _ l ) y l ) _ ~ + l . x_~T Y l

Y1

e 1 / e T ~ ~ )

iT)- and analogously for

min {n(x, k, l) --n(x, k, 1)}; x ~ _ T

here and later the dash means that the summation has to be extended only over the O's lying to the right of V.

9 Acta Mathematica x n I / 3 - 4

360 S, KNAPOWSKI AND P. TURIN

Next we estimate roughly the contribution of the Q's right to V and with 1

tQl > log ~6 T.

Owing to the already used inequality

(14.2) ~'dx) l < c 2 z logk(1 + rl) r~=to <r + l

this contribution cannot absolutely exceed

/ eyl \~ ( log T~ c23 { ~ 1 "2"<c24el k,20 ~ , ] .

\ log lo T /

Hence from (14. 1) we get

max {~ (x, k, l) - ~ ( x , k, 1)} = (v - 1)r ((v - 1 ) y , ) - ' + l - x ~ T Yl

�9 ~ ~ (.1-2(I)) Zdz) \ - ~ - ) - Itol <_-log~ i~ T

o , , } - 2 v-1 Y x~ (2(o~)--2(c9) ) Xo(x) --3T~

j=% ~ ( k ) _ "

Jtol ~ l o g l ~

For our later aims we write it in a different form. Let ~t = a l T itl be such a zero in our domain of an L(s, x)modk with z ( l ) r 1 for which

eye0 (14. 3) ~ - = maximal.

Owing to the functional-equation we have

1 (14.4) a 1 _-> ~ - .

Then we have, since our sums are obviously real, the inequality

(14. 5)

{- (v 1)~

max {~(x, k, l ) - - z~(x, k, 1)} > ((v., 1)yl) -v+x. x~=T Y i

f e',",'V ( 1 f er,(e-~',) X ~

Itel~logi0 T

2q~(k) ,_~,.= ~ (Z(. j)-X(~j)). Z'(x)• 2 ~ IO,] [t ol<_-log TM T

COMPARATIVE PRIME-NUMBER THEORY. n i 361

since from (13.1),

1 eYl (aO-~-al)

2 lel

w e l ; l a v e

05.2)

As m we chose

and analogously for

(14. 6) rain (n(x, k, l) - lr(x, k, 1)}. x~_T

15. The assertions of Theorem 3.2 wil lbe proved by two appropriate choices of the integer v which was so far restricted only by (13.4); the possibility of these choices will be given by Theorem 4. 1. The z~-numbers will be of course the numbers

(15. 1) e r,(e- ~,) 2e y'(~-r leg IQ~I;

e q

irl what follows we shall call them zi's of the first resp. of second category. That the maximal absolute value of the zj 's of first category is 1, is trivial from (14. 3);

owing to the definition of V and 0t

]Qi[ 2e ~*~" = ~ IQi ----<2e--5-----<2e-i-6--<--2e 2-E61~ 1,

max lzjt = 1. J

(15.3) m -- ~ l~ T -logO,9 T]. L Yt

As to the arguments of our zj 's owing to (13.2) and (13.3) the restriction (4. 2) is satisfied with

.4 (k) C18 - - 1 1

k(1 + log 1~ T) 6 log 3 k(2 + log t~ T)

Owing to (3. 1) this is

C2" / 3

Iog ~ T.(log 2 T) s

choosing c 7 in (3. l) sufficiently large. Hence w e may choose

2

(15.4) n---- log - ~ T.

As to n we have owing to (14. 2) the upper bound

(15.5) c28k log i~ T.log 2 T < log 1~ T.(log 2 T) 3.

As to the bj's, evidently they have the form

1 1 (15.6) q~(k)(1-~( l ) ) and ~-~(~-~(~(~)-~(~u));

9*

362 S. KNAPOWSKI AND P. TUR/~N

they can be called correspondingly to bj of the first resp. of the second category. Finally, we have to determine the indices h and hi. Let simply be

(15.7) h = 1

then (4. 3) is evidently satisfied owing to (15. 4). As tO h~ our aim is to rule out the ,,inconvenient" b/s of second Category and therefore owing to (4. 4) let h 1 be the minimal index of the zj 's of the second category. Then we have certainly

(15.8) zh~ = 2 - - [Vll- 9z

with a certain Q2 = tr2 + it2 and we have to verify (4. 5). Owing to the maximality expressed in (14. 3) we have

y o eyl(oz - a l , / yt o i ]zh~l = 2 e ~ 01]<=2e 2 ~2e-T6<=21og 2oo T

I 702 t . e~

1 1 (15.9) Izhl-lz~lt-~1.2log 200 T > - ~ .

Since from (15, 4 ) f o r sufficiently large c7 in (3. 1) we have

2n 2 1

rc 3+rc log x T 2 m + n 3 + - ~

(4. 5) follows from this and (15.9). Since we have from (3.1)

1 I?o[ < l o g 1~ T

there is at least one b of the first category existing and since

1

we have owing to the definition of h 1

r 8 (15.10) A = rain Re • b i > ~ .

l _~<hl j = l

Hence applying Theorem 4. 1 we obtain the existence of integer v 1 and v 2 numbers such that

log T o (15. 11) l o g T logO,9T~v~; v2~ - - l o g o'. T + Yl Yt

+ (3 + rc log ~ T)log ~ T (logz T) 3 l o g T

Yl

COMPARATIVE pRIME-NUMBER THEORY. III 363

(choosing c7 in (3.1) sufficiently large and taking in account (15.5)) i .e . (13.4) is b y these choices fulfilled. Thus

(15. 12)

- I o , i - Z

[tel ~_log*0 T

o x ~ ' " , (2 e r l (~ - '0 ).1} 1 2 (z(cq)-~(=:)) 2.0'<,) - lei l >

29~ (k) j = 1 • e ttO[<=log 10 T

.2_1 >--8 1 (Yl121~176176 l~-g T • k3 a " > e l ( . log 5 T)

3 log ig T(log2 T) 3

with (3.1) with sufficiently large c7 and analogous inequality in the opposite direction 1

with v2 instead of vl (with - e l ( - l o g 3 - T ) on the right).

16. T o conclude the proof we need a lower bound for

eYla ~vi

I--LT~I / "

Owing to the maximum-definition of 01 in (14.3) we have

(er,~'l"~Vl (er,Po'~'_(eVm)#o \ - T ~ - / - \ led ] leol ~

and hence from (15.13) and (3.1)

(16. 1) _~ TP0el(-log ~ T.log z T).(logz T) -2ul~ TP~ logz I "

Collecting (15. 14), (16. 1) and (14. 5) we got

max {rc(x, k, l) - n ( x , k, 1)} > - - x~_T

using (15. 13). Since flo =~�89 and

max {re(x, k, l) -~(x, k, li} < T �89 x ~_ TJ[

( v l 1) ,.((vx--l)yl"~. ,- ~'+1"

Yx

364 S. KNAPOWSKI AND P. TURIN: COMPARATIVE PRIME-NUMBER THEORY. n I

we get

{~t(X, k, l ) - ~ ( e l k, 1)} >TPoelt" 43 log_Tioga T] m a x r�89 log2T /

which proves the one-half of Theorem ,3.2; the second half of the proof runs after (14. 6) and (15. 14) on the same lines.

MATHEMATICAL INSTITUTE OF THE UNIVERSITY ADAM MICZKIEWICZ~ POZNAN

MATHEMATICAL INSTITUTE, EOTVOS LORAND UNIVERSITY, BUDAPEST

(Received 5 December 1961)

References

G. H. HARDY and J. E. LITTLEWOOD [1] Contributions to the theory of Riemann zeta-function and the theory of the distribution of primes, Acta Math., 41 (1918), pp. 119--196.

A. E. INGHAM [1] A note on the distribution of primes, Acta Arith., 1 (2) (1936), pp. 201--211. S. KNAPOWSKI [1] Contributions to the theory of the distribution of prime-numbers in arithmetical

progressions I, Acta Arith., 6 (1961), pp. 415 434. S. KNAPOWSKI and P. TUR.~N [1] Comparative prime-number theory. I, Acta Math. Acad. Sci.

Hung., 13 (1962), [2] Comparative prime-number theory. ]I, ibid., 13 (1962),

K. PRACHAR [1] Primzahlverteilung (Springer Verl. 1957). C. L. SIEGEL [I] On the zeros of the Dirichlet L-functions. Annals o f Math:, (1945), pp. 409--422. S. SKEWES [1] On the difference ~ ( x ) - Li x, Proc. of the London Math. Soc., 5 (17) (1955),

pp. 48 70. VERA T. S6S and P. TUR.~N [1] On some new theorems in the theory of diophantine approximations,

Acta Math. Acad. Sci. Hung., 6 (1955), pp. 241 257. P. TURAN [1] On some further one-sided theorems of new type in the theory of diophantine appro-

ximation, Acta Math. Acad. Sci. Hung., 12 (1961), pp. 455--468. [2] Eine neue Methode in der Analysis und deren Anwendungen (Akad. Kiad6, 1953).

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