ZETA MEAN MOTIONS AND VALUES OF THE RIEMANN FUNCTION.

MEAN MOTIONS AND VALUES OF THE RIEMANN FUNCTION.

BY

VIBEKE BORCHSENIUS and BORGE JESSEN

in COPENHAGEN.

ZETA

C o n t e n t s .

Page

Introduction . . . 97 Chapter I. Mean motions and zeros o f generalized analytic almost periodic function, s 100 Ordinary analytic almost periodic functions . . . 100 The Jensen function of a type of generalized analytic almost periodic

functions . . . 109 Extension of the results to the logarithm of a generalized analytic almost

periodic function . . . 193 Chapter II. The Riemann zeta function . . . 138 Application of the previous results to the zeta function and its logarithm 138 Two types of distribution functions . . . 139 Distribution functions connected with the zeta function and its logarithm 154 Main results . . . 159 Bibliography . . . 166

Introduction.

1. I n t h e t h e o r y of a l m o s t periodic f u n c t i o n s t h e s t u d y of m e a n m o t i o n s a n d of p r o b l e m s of d i s t r i b u t i o n f o r m s an i n t e r e s t i n g chapter.

H i s t o r i c a l l y , t h e s u b j e c t begins w i t h L a g r a n g e ' s t r e a t m e n t of t h e p e r t u r b a t i o n s of t h e l a r g e planets, which leads to a s t u d y of t h e v a r i a t i o n of t h e a r g u m e n t of a t r i g o n o m e t r i c p o l y n o m i a l F ( t ) - ~ aoe~.ot+ ... § a ve~'~v t. A p a r t f r o m some cases c o n s i d e r e d by L a g r a n g e , this p r o b l e m was first t r e a t e d r i g o r o u s l y by Bohl[1] a n d W e y l [i], who by m e a n s of t h e t h e o r y of e q u i d i s t r i b u t i o n p r o v e d t h e existence of a m e a n m o t i o n w h e n e v e r t h e n u m b e r s ~1--~o . . . . , ~ x - - 4 o are linearly i n d e p e n d e n t .

9~ Vibeke Borchsenius and Borge Jessen.

Closely r e l a t e d to t h e i r m e t h o d is the t r e a t m e n t by B o h r [I] of t h e distri- b u t i o u of the values of t h e R i e m a n n zeta f u n c t i o n ~ ( s ) = ~(a +

it),

or r a t h e r of t h e f u n c t i o n lo~" ~r(,), in the half-plane a > 89 I t d e p e n d s on a c e r t a i n m e a n c o n v e r g e n c e of t h e E u l e r product, first applied by B o h r and L a n d a u [I], a n d on the linear i n d e p e n d e n c e of the l o g a r i t h m s of the primes. The m e t h o d has been f u r t h e r developed in B o h r and J e s s e n [i], [2]. T w o main results have been obtained.

One c o n c e r n s t h e d i s t r i b u t i o n of t h e values of log ~(,~.) on v e r t i c a l lines a n d states (in t h e t e r m i n o l o g y now used) t h e existence of an a s y m p t o t i c d i s t r i b u t i o n func- tion of t h e f u n c t i o n log ~(a + it) f o r every fixed a > 89 possessing a c o n t i n u o u s density

l"~(x)

which is positive in the whole x-plane when a ~ I. T h e o t h e r r e s u l t c o n c e r n s t h e d i s t r i b u t i o n of the values in vertical strips and states f o r e v e r y strip (89 ~ ) ~ 1 ~ a ~ o~ a n d every :~: t h e e x i s t e n c e of a r e l a t i v e f r e q u e n c y of t h e zeros of log ~ ( s ) - - x in t h e strip, which depends c o n t i n u o u s l y on al and a2 and is positive for all x w h e n a I ~ I.

T h e f u n c t i o n ~(s) is a l m o s t periodic in [I, + c o ] and so is, too, t h e f u n c t i o n log ~(s), whereas a c e r t a i n generalized a l m o s t p e r i o d i c i t y is p r e s e n t f o r 89 < a ~ I.

This a l m o s t p e r i o d i c i t y makes t h e results less surprising, b u t was n o t used in t h e proofs.

2. A new t r e a t m e n t of t h e d i s t r i b u t i o n of the values of log ~ ( s ) o n vertical lines was g i v e n in J e s s e n a n d W i n t n e r [I] ill c o n n e c t i o n w i t h a g e n e r a l t r e a t m e n t by m e a n s of F o u r i e r t r a n s f o r m s of d i s t r i b u t i o n f u n c t i o n s of f u n c t i o n s of a real variable which are almost periodic in t h e o r d i n a r y or in a g e n e r a l i z e d sense.

T h e existence of the density Fo(x) is h e r e o b t a i n e d by an e s t i m a t e of the F o u r i e r t r a n s f o r m of t h e d i s t r i b u t i o n f u n c t i o n , a n d f r o m its expression as a F o m ' i e r i n t e g r a l it followed, a m o n g o t h e r t h i n g s , t h a t it possesses c o n t i n u o u s p a r t i a l d e r i v a t i v e s of a r b i t r a r i l y h i g h order. In t h e case 89 < a ~ I it was even s h o w n t h a t it is a r e g u l a r a n a l y t i c f u n c t i o n of t h e coordinates. Similar results were o b t a i n e d r e g a r d i n g t h e d i s t r i b u t i o n of t h e values of .~(s) itself on vertical lines.

T h e d i s t r i b u t i o n of t h e zeros of an a r b i t r a r y a n a l y t i c a l m o s t periodic func- tion f ( s ) in vertical strips was studied by J e s s e n

[I]

by m e a n s of t h e so-called J e n s e n f u n c t i o n defined as the mean value

qv/(a)=~l{log]f((~+it)[};

it was

t

shown t h a t this f u n c t i o n is a c o n t i n u o u s c o n v e x f u n c t i o n a n d t h a t the r e l a t i v e f r e q u e n c y of zeros of f(.~') in a s t r i p

al~a<a.z

inside the strip of a h n o s t perio-

p / , I / ,

dicity exists and is e q u a l to

(q~f~,a2,--q~/~all)/2zr

w h e n e v e r q~f(a) is differentiable at the points o" 1 and a2. An addition on the variation of t h e aro.ument of

f(s)

Mean Motions a n d Values of the Riemann Zeta Function. 99 on vertical lines has been given by H a r t m a n [I], who p r o v e d t h a t the mean m o t i o n of f ( a +

it)

exists and is equal to ~v.~-(~) w h e n e v e r ~o/(a) is d i f f e r e n t i a b l e at the p o i n t a. A s y s t e m a t i c exposition of this subject, i n c l u d i n g a c o m p l e t e t r e a t m e n t of L a g r a n g e ' s problem, has been g i v e n in Jessen a n d T o r n e h a v e [1].

T h e s e i n v e s t i g a t i o n s of t h e J e n s e n f u n c t i o n c o n c e r n f u n c t i o n s w h i c h are almost periodic in t h e o r d i n a r y sense. I n the case of t h e zeta f u n c t i o n they are t h e r e f o r e only applicable in t h e half-plane a > I. T o g e t h e r with t h e above m e n t i o n e d result on t h e existence a n d c o n t i n u i t y in a 1 and a2 of t h e f r e q u e n c y of zeros of log ~ ( s ) - - x in

al<a<a2

t h e y show t h a t the J e n s e n f u n c t i o n ~Vlog,-~(a) of log ~ ( s ) - - x is differentiable in a > I f o r all x. I n t h e closely r e l a t e d case of an a l m o s t periodic f u n c t i o n J(s) with linearly i n d e p e n d e n t e x p o n e n t s in t h e D i r i c h l e t series an even preciser result is k n o w n . I t has been s h o w n in J e s s e n [2]

by a c o m b i n a t i o n a n d e x t e n s i o n of t h e m e t h o d f r o m t h e zeta f u n c t i o n a n d t h e F o u r i e r t r a n s f o r m m e t h o d , t h a t in this case the r e l a t i v e f r e q u e n c y of zeros of

f ( s ) - - x

exists f o r a n y strip al < a < az a n d any x and is t h e i n t e g r a l over the i n t e r v a l

a l < a

< a 2 of a c e r t a i n c o n t i n u o u s f u n c t i o n . This m e a n s t h a t t h e J e n s e n f u n c t i o n q~/_~(~) of

. f ( s ) - - x

is t w i c e differentiable with a c o n t i n u o u s second derivative.

3. T h e o b j e c t of t h e p r e s e n t p a p e r is to r o u n d off t h e p r e v i o u s work by a t r e a t m e n t of m e a n m o t i o n s on vertical lines a n d of zeros in v e r t i c a l strips of t h e f u n c t i o n s log ~ ( s ) - - x a n d ~ ( s ) - - x in t h e half-plane a > 89

Since the zeta f u n c t i o n is a h n o s t periodic only in a generalized sense in the strip 89 < a < I we m u s t first e x t e n d t h e results c o n n e c t e d w i t h t h e J e n s e n f u n c t i o n to c e r t a i n cases of g e n e r a l i z e d a l m o s t periodic f u n c t i o n s g e n e r a l e n o u g h to include t h e f u n c t i o n s log ~ ( s ) - x a n d ~ ( s ) - - x . T h i s extension, which m a y be of some i n t e r e s t in itself, is given in C h a p t e r I.

I n C h a p t e r I I t h e f u n c t i o n s log ~(.,.) a n d ~(s) are d e a l t with. W e prove t h a t the J e n s e n f u n c t i o n ~O~og :--x(a) -~ 3 I {log ]log ~ (a + i t) -- x {} of log ~ (s) - -

x

exists

t

and is a twice d i f f e r e n t i a b l e convex f u n c t i o n in t h e i n t e r v a l 89 < a < + c~. F o r a-+ 89 we have

qglog:-,,.(a)~c,o

f o r any x. F o r every a > 8 9 t h e f u n c t i o n

l o g ~ ( a + i t ) - - x

possesses a m e a n m o t i o n which is equal to r .-.~ (a), a n d f o r any strip (89 < ) a l < a < a2 t h e r e exists a r e l a t i v e f r e q u e n c y of t h e zeros of log

~(s)--x

in t h e strip, w h i c h is equal to (~'~og ;-.~(a2)--~V~og ;-2,(ax'~)/2

z.

T h e second d e r i v a t i v e ~'log ;-x (a) is o b t a i n e d

" {O. ~

in t h e f o r m ~Vlog:--.~.~ j~- 2 ~r G~(x), w h e r e

G,~(x)

is a c o n t i n u o u s f u n c t i o n of a and x, which f o r every a > 89 r e p r e s e n t s t h e d e n s i t y of a c e r t a i n d i s t r i b u t i o n f u n c t i o n

100 Vibeke Borchsenius and Borge Jessen.

analogous to the distribution function of log ~ ( a +

it).

I t has similar properties as the density

Fo(x)

mentioned above; thus it possesses continuous partial derivatives of arbitrarily high order and is in the case 89 <= o < I even a regular analytic function of the coordinates; if 8 9 it is positive for all x. Similar results are proved for the functions ~ ( s ) - x.

For the convenience of the reader we have included proofs of most of the known results which we need. In particular we have included a treatment of the asymptotic distribution functions of the functions log ~(a +

it)

and

~(a +it)

for every a > 89

Of earlier results in our subject we have mentioned above only those which are of direct importance for the present paper. A detailed account of the de- velopment of the subject has been given in the introduction to Jessen and Tornehave [I].

C H A P T E R I.

Mean Motions and Zeros o f Generalized Analytic Almost Periodic Functions.

Ordinary Analytic Almost Periodic Functions.

4. We shall begin by stating the above mentioned results of Jessen and H a r t m a n as they appear in Jessen and Tornehave If]. First we must mention certain definitions which will be used throughout.

Let

f(s)

denote an arbitrary function of the complex variable

s ~ a + it,

which is regular in an open domain G and is not identically zero. The function a r g f ( s ) is then defined mod. 2 z , by the condition f(s)---If(s)[e

iargf~),

for all s in G, with the exception of t h e zeros of

f(s).

Let L denote an orientated straight line (or segment) belonging to G. We then define the left argument a r g - f ( s ) of

f(s)

on L as an arbitrary branch of the argument, which is continuous except at the zeros of

f(s)

on L, whereas it is discontinuous with a jump of --p~r, when s passes, in the positive direction of L, a zero of

f(s)

of the order p. Similarly we define the right argument a r g + f ( s ) of

f(s)

on L as an arbitrary branch of the argument, which is continuous except at the zeros of

f(s)

on L, whereas it is discontinuous with a jump of + p ~r, when s passes, in the positive direction of L, a zero of

f(s)

of the order p.

In a discontinuity point we use as value the mean value of the limits from

Mean Motions and Values of the Riemann Zeta Function. 101 t h e two sides; the two functions a r g - f ( s ) and a r t + f (s) are hereby defined for all s on L.

I f s~ a n d s2 are points of L, so t h a t the direction from s~ to s2 coincides with t h e positive direction of L, t h e differences a r g - f ( s 2 ) - a r g - f ( s l ) a n d a r g + . f ( s 2 ) - a r g + f ( s l ) are i n d e p e n d e n t of the choice of the branches of the ar- g u m e n t s a n d are called the variation of the a r g u m e n t of

f ( s ) f r o m

sl to s~ along the left or r i g h t side of L, or simply t h e left or r i g h t variation of t h e a r g u m e n t of

f(s)

along t h e s e g m e n t f r o m s 1 to s~.

W h e n speaking of the left a n d r i g h t a r g u m e n t of a function on a vertical or horizontal line (or segment) we suppose the line o r i e n t a t e d a f t e r increasing values of t or a respectively.

5. The results referred to are now as follows.

Let f ( s ) = f ( a + it)

be almost periodic in the strip [a, fl] and n o t identically zero. T h e n the mean value ~

= lira _ I f log

[f(a-L, i t ) l d t

/(a)=Mllogt

I f ( a + it)l} a - r . !

7

exists u n i f o r m l y in the interval [a, ~] and is a convex f u n c t i o n of a. I t is called t h e

Jensen function

of

.f(s).

Moreover, if a r g - f ( a +

it)

and a r g + . f ( a +

it)

denote t h e left and r i g h t ar- g u m e n t o f f ( s ) on the line

s - ~ a + i t , - - o o < t < +o%

t h e n the lower a n d upper, left a n d r i g h t m e a n motions of

f(s)

on this line, defined by

inf

f ( a + id) -- a r g - f ( a + i 7)

~:;(a) = lim s u p a r g - ~ - - 7 and

lira inf + - - arg§ + i;,)

z F o r a n a r b i t r a r y real f u n c t i o n P(7, J) defined w h e n -- ~ ~ 7 ~ J ~ + oo we d e n o t e b y lira i n f P(7, J) t h e l e a s t u p p e r b o u n d of t h o s e n u m b e r s r for w h i c h t h e r e e x i s t s a n u m b e r (5-1) ~ oo

T f f i T ( r ) s u c h t h a t Q(7, d ) > r for ( J - - y ) > ~; a n d , s i m i l a r l y , b y lira s u p {~(7, J ) t h e g r e a t e s t (5-1) ~ o~

l o w e r b o u n d of t h o s e n u m b e r s r for w h i c h t h e r e e x i s t s a n u m b e r T ~ T ( r ) s u c h t h a t P(7, J ) ~ r for ( J - 7) ~ T. If t h e s e l i m i t s a r e e q u a l , we d e n o t e t h e i r c o m m o n v a l u e b y lira ~(7,$). W h e n P(7, J) is c o m p l e x - v a l u e d , we w r i t e lira ~ ( 7 , $ ) ~ a if t h e r e e x i s t s to e v e r y e > o a T = T(~)

($-7) ~ oo

s u c h t h a t IP(7, J ) - - a l ~ for ( J - - 7 ) ~ T. F o r a s e t of f u n c t i o n s 0 ( 7 , J) t h e l i m i t s a r e s a i d to e x i s t u n i f o r m l y , if, for a n a r b i t r a r y ~, t h e s a m e T ~ T ( ~ m a y be u s e d for all f u n c t i o n s of t h e set.

1 0 2 V i b e k e B o r c h s e n i u s a n d B o r g e J e s s e n .

satisfy the inequalities

' (~ - - o) =< (a) ,[ :~ (a) l < ~ (a) < ~.~ (a + o).

~' ::; =<

[

~; (~

= =

Finally, if N f ( a 1, az; 7, 6) denotes the n u m b e r of zeros t of .f(s) in the rectangle (a < ) a ~ < a < a ~ ( < fl), 7 < t < 6, then the lower and u p p e r relative frequencies of zeros of f(s) in the st~rip (al, a~), defined by

H ] (al, a,)] inf

Nr

(al, a,; 7, 6) /_~,. (al, a2 ) j = lim sup 6 -- 7

(5-3,)~ or satisfy the inequalities

I ^{, ~ , .} I ^{, , .}

2 ~ (~9f(O'2 - - O J - - 99y'~6r 1 -~ O)) <: / / f ( 6 r l , O'2) ~< lq](o'l, 0"2) ~ ~ (~Pf(6 2 + O) - - ~9f',6 1 - - 0 ) ) .

As a corollary we have, t h a t if f j ( a ) is differentiable at the point a, then the left and right mean motions

e l ( a ) = lira a r g - f ( a + i 6 ) - - a r g - f ( a + i7) and

c+(a) = lira a r ~ + ' f ( a + i d ) - - a r g + f ( a + i7)

.f" 6

both exist and are d e t e r m i n e d by

q (o) = g;~ (~,) = r

@.

Similarly, if 9~s(a) is differentiable at al and a~, then the relative frequency of zeros

Hf(al, a2) = lim N](al a~; 7, d) exists and is determined by

H,(ai, a , ) = I ,~ ,, ,

2~ (~I',a~

^--

9o/,a~ ).

The latter formula is called the Jense~ formula for almost periodic functions.

I f for m > o we pnt

If(8) I,.

= max

{If(,s.)I, .~}

the aensen function ~t'(a) is also d e t e r m i n e d as the limit of the mean value M { l o g I/(,~ + it)l,, }

t

as , n - , o, the convergence being again u n i f o r m in [a,/~].

Throughout, multiple zeros are counted according to t h e i r order of multiplicity.

Mean Motions and Values of the Riemann Zeta Function. 103 A m o n g t h e f u r t h e r properties of the J e n s e n f u n c t i o n we m e n t i o n t h a t , if a sequence of f u n c t i o n s

f l ( s ) , f 2 ( s ) , . . ,

almost periodic in [a, 8], none of which is identically zero, converges u n i f o r m l y towards

f(s)

in [a, ~], t h e n the J e n s e n func- tion r of f,,(s) converges u n i f o r m l y in [c~,/~] t o w a r d s ff.,.(a).

6. W e shall now establish connections between the J e n s e n f u n c t i o n a n d certain distribution functions.

L e t R~ be the complex plane with x = ~1 + i~2 as variable point. A completely additive, non-negative set-function it(E), defined for all Borel sets E in Rx, for which tt(Rx) is finite, will be called a

distribution fmwtio~

in Rx. W e shall n o t suppose t h a t tt(Rx)-- I.

F o r distribution f u n c t i o n s in this general sense we have a theory similar to t h a t of the case it(R~.)= I. W e briefly recall t h e parts of the t h e o r y which will be applied, r e f e r r i n g for details e. g. to the m o n o g r a p h by Cram~r [I] and to t h e s u m m a r y in Jessen a n d W i n t n e r [I].

Our n o t a t i o n for an i n t e g r a l with respect to a distribution f u n c t i o n tt will be

j" h

^,,

(d R,)

E

A similar n o t a t i o n will be used for o r d i n a r y Lebesgue integrals. The Lebesgue measure (in any n u m b e r of dimensions) ~vill be denoted t h r o u g h o u t by m. Thus t h e n o t a t i o n f o r an ordinary Lebesgue integral in Rx will be

E

A set E is called a c o n t i n u i t y set of ~ if

it(E') =

it(Ii ), where

;'" E'

denotes the set formed by all interior points of E , a n d

E "

the closure of E. ] f i t ( E ) = ~ , ( E ) or i t ( E ) ~

v(E)

for the common c o n t i n u i t y sets of it and v, t h e n I t ( E ) ~ - r ( E ) or

it(E) ~ v(E)

for all Borel sets E.

A sequence of d i s t r i b u t i o n f u n c t i o n s tt,, is said to be convergent if there exists a d i s t r i b u t i o n f u n c t i o n it such t h a t tt,,(E)-~ it(E) for all c o n t i n u i t y sets of the limit f u n c t i o n it, which is t h e n unique. The symbol tt,,--~ u will be used only in this sense.

W e have ,u,,-* it, if and only if the relation j" h,

(x)

^it,,

(,1 lC) f h

^,,

R x R x

holds for all bounded continuous functions h (x) in R~,.. I f it,-~ it, then lira inf tL,, (1'])

!t(E) for any open set E, and lim sup

!,,,(E)~= it(E)

for a n y closed set /~'.

104 Vibeke Borchsenius and Borge Jessen.

A d i s t r i b u t i o n f u n c t i o n tto d e p e n d i n g on a p a r a m e t e r a, which r u n s in an i n t e r v a l

(a, fl),

is said to d e p e n d c o n t i n u o u s l y on a, if tt~,~--~ tt~o w h e n an-+ a 0.

T h e n tt,(R~) is c o n t i n u o u s and t h e r e f o r e b o u n d e d in any closed i n t e r v a l (a < ) a 1

< a < a2 ( < fl). Moreover, try(E), c o n s i d e r e d as a f u n c t i o n of a, is semi-continuous f r o m below f o r any o p e n set E and s e m i - c o n t i n u o u s f r o m above f o r a n y closed set E . I n p a r t i c u l a r , fro(E) is a B a i r e f u n c t i o n f o r a n y open or closed set E and h e n c e f o r any Borel set E. 1 T h e i n t e g r a l

tl.,

~t(E) = f ~t.(E)da

r 1

is a g a i n a d i s t r i b u t i o n f u n c t i o n .

Suppose n o w t h a t go f o r every a is the limit of a d i s t r i b u t i o n f u n c t i o n tt,,,o, a n d suppose t h a t ~t,,,~ f o r e v e r y J~ depends c o n t i n u o u s l y on a; c o n s i d e r the distri- b u t i o n f u n c t i o n

t J=+

I,n(E) = fit,, o(E)da.

o |

T h e n tt will be t h e l i m i t of ttn, if

tt,.,,(R~)

is u n i f o r m l y b o u n d e d f o r all n a n d all a in a l < a < a 2 . F o r if E is a B o r e l set, and

E '

d e n o t e s the set f o r m e d by all i n t e r i o r points of E , and

E "

t h e closure of E , then, by F a t o u ' s t h e o r e m , we have

f f

lira i n f #,~ (E) > lim inf it,, ( F ' ) ~ lira inf it,,. o ( E ' ) d a

> tto(E') da -- ,u (E')

(~t O1

and

lim sup g . (E) < lim sup it. ( E " ) =< f lim sup it,,,. ( E " )

da < j it,, ( ~ ) d a ~ t' ( E ) ,

d | 0 l

so t h a t ~t.(/~)-+ i t ( E ) if E is a c o n t i n u i t y set of ~u.

A d i s t r i b u t i o n f u n c t i o n g is called absolutely c o n t i n u o u s if t t ( E ) = o f o r e v e r y B o r e l set E of m e a s u r e o; this is the case if a n d only if t h e r e exists in R : a Lebesgue i n t e g r a b l e p o i n t f u n c t i o n F ( x ) such t h a t

E

f o r a n y Borel set E ; we call F ( x ) t h e density of ft.

L e t By be t h e c o m p l e x plane with y = ~71 + i~72 as variable point, a n d let t h r o u g h o u t

x y

d e n o t e n o t t h e usual p r o d u c t of t h e two complex n u m b e r s , b u t

S i n c e t h e s y s t e m of s e t s E for w h i c h p a ( E ) is a Baire function c o n t a i n s t h e l i m i t of a n y d e c r e a s i n g or i n c r e a s i n g s e q u e n c e of s e t s from t h e s y s t e m .

Mean Motions and Values of the Riemann Zeta Function. 105 the inner product ~171 + ~ 2 of the vectors x =-(~1, ~) a n d y ~ (71, ~ ) . I f # is a d i s t r i b u t i o n f u n c t i o n in R~ t h e n the i n t e g r a l

]t x

defines in B~j a f u n c t i o n A(y;!,) which is uniformly continuous a n d bounded, the m a x i m u m of its absolute value being A(o;/~)----#(R~). W e call A ( y ; / ~ ) t h e

Fourier transform

of /~. I f A(y;/,) ~ A(y; r), t h e n /z ---- r.

I f /z,,-*#, t h e n A(y;/~,,)--*A(y;/z) holds u n i f o r m l y in every circle [y] ~ a ; conversely, if a sequence of F o u r i e r t r a n s f o r m s A(y;

t*,,)

is uniformly convergent in every circle ]y] =< a, t h e n tile limit f u n c t i o n also is the Fourier t r a n s f o r m A(y; ~) of a d i s t r i b u t i o n f u n c t i o n ~u, a n d /z,-* ,u.

I f the integral

fly I' I A (y; ,,,(d

Ry

is finite for an integer p >_--o, t h e n # is absolutely c o n t i n u o u s and its density

F(x)

: F ( ~ I , ~ ) , d e t e r m i n e d by the inversion f o r m u l a F ( * ) = (2 ~)-~

f e-'*"A

(y; u)m (d R,a)

Ry

is continuous a n d possesses in t h e case p > o c o n t i n u o u s p a r t i a l derivatives of order _--< p, which m a y be obtained by differentiation u n d e r the i n t e g r a l sign.

This is in p a r t i c u l a r the ease if for some e > o

oqyl-(-'+,'+')) as oo.

I f the estimate

A ( y ; , - ) = O ( e -~h'l) as l y l - ~ o o

holds for some c > o, t h e n

F(x)=

F(_~I, ~2) is a r e g u l a r a n a l y t i c f u n c t i o n of the two real variables ~l,~z. I f c m a y be t a k e n arbitrarily large, t h e n

F(x)

is an entire f u n c t i o n of t h e two variables _El, ~2.

7. L e t a g a i n

f(s)

be an a n a l y t i c a h n o s t periodic fullction in t h e strip [a, fl].

W e shall prove a t h e o r e m on t h e d i s t r i b u t i o n of the values of

f(s)

on vertical lines which is a special case of a general theorem on asymptotic distribution functions, to be f o u n d in Jessen a n d W i n t n e r Ill.

F o r a n a r b i t r a r y a a n d a n a r b i t r a r y interval (--oo < ) 7 < t < ~ ( < + ~ ) let ,u~;:,,a and ~o;~,a denote t h e distribution f u n c t i o n s of

f ( a + it)

a n d of f ( a +

it)

with respect to

If'(a + it)l 2

over t h e interval 7 < t < d, defined by

106 Vibeke Borchsenius and Borge Jessen.

uo;:,,,~,(E) -m(A~ 6 - - 7

a n d

v,,;~,,~(E) = 6 ~ - - 7 . I f If'(a + it) l~'dt'

"1o; 7, d (E)

where

Ao;~,,o,(E)

d e n o t e s t h e set of points in 7 < t < 6 f o r which

f ( a + it)

be- longs to E .

T h e n /~o; ~, ~ and v~; ~, ,~ c o n v e r g e f o r ( 6 - - 7 ) - + o o t t o w a r d s c e r t a i n d i s t r i b u t i o n f u n c t i o n s #o and

vo.

W e call these d i s t r i b u t i o n f u n c t i o n s t h e

asymptotic distri- bution functions

of

f ( a + i t)

and of

f ( a + i t)

w i t h r e s p e c t to I f ' (a + i t)[~.

T h e p r o o f is i m m e d i a t e . By t h e definition of t h e i n t e g r a l we have

5

(,) I . n d

7 5

_'~ f (o+it) y i t) ]*" d t,

A (y; vo; :,, ~') = 3- ei: i f ' (a +

7

where

f ( a + it)y

d e n o t e s t h e i n n e r product. Since t h e f u n c t i o n s e if(o+it)v and

e lf(a+it) y ]f'(a + it)] ~

are almost periodic f o r e v e r y y a n d f o r m a u n i f o r m i t y set of a l m o s t periodic f u n c t i o n s f o r [y[ =< a for a n y a, t h e m e a n values

d

I f ei.f(~

^{a n d}

M { e ' " ( ~ lira ~ - ' 7 J '

t (d-7) ~ 0r

7 f[

lim

I (df(o+,),./lf,(a +it)l~.dt M{eif(o+it) v l f ' (a + it)] ~}

d 7

exist u n i f o r m l y in every circle ] y ] < a. T h i s implies t h e t h e o r e m , a n d we o b t a i n

2)

A(y;tto)=M{eif(~

and

A(y;vo)=M{ei:(~+'t)vlf'(a+it)p}.

t t

F r o m t h e expressions (I) a n d (2) it follows t h a t t h e F o u r i e r t r a n s f o r m s are c o n t i n u o u s f u n c t i o n s of y and a t o g e t h e r . This implies t h a t t h e d i s t r i b u t i o n func- tions #o and v,, a n d #o; ~,, and vo;,:, ~ f o r fixed 7 and 6, d e p e n d c o n t i n u o u s l y on a.

8. By m e a n s of the d i s t r i b u t i o n f u n c t i o n s uo we o b t a i n f o r t h e J e n s e n f u n c t i o n

9f(a)

of

f(s)

the expression

(3) -- f l o g ,o(d R=).

Rx

L I. e. f o r a n y s e q u e n c e o f i n t e r v a l s 7 n < t < d n , w h e r e ( d n - - 7 n ) --> o o .

Mean Motions and Values of the Riemann Zeta Function. 107 For by the definition of the integral we obtain for every m > o

r~

I I i'lo

- ~ . log I,/(~ + i ~)I,. d t = . Ix

I,, .o; .,,, ~' (d ~.,),

"/ R x

whence

(4) M {log If(~ + it)I,,,} = .flog I~'lm ~o(d R,:),

t R. t

since we may replace log ] X[m by log 3I~ for Ix [ > M, where M denotes the upper bound of [ f ( a +

it)I,

and thus obtain a bounded continuous function under the integral sign. By w 5 the left-hand side of (4) converges for m-~ o towards 90:(a).

This implies the existence of the integral on the right in (3) and the relation (3).

Similarly, the Jensen function

~q,.-~(~) = M{log I f ( a +

it)--x[}

t

of the function

f(s) --x

is for any complex number x determined by the expression

(o)

s ^I

(d

B,~)

~.:-~

= j , o g l u . - x I

,,,

R u

9. We will now establish a connection between the Jensen functions

q~.r-~(a)

and the distribution functions v~.

For a fixed strip

(a<)al<a<a2(.~fl)

let

~,._~ (7, 6)

denote the number of zeros of

f ( s ) - - x

in the rectangle a l < a < a 2 , 7 < t < c ~ , and let us consider the distribution function

v~"~'(E)=.l- ~ - : r ,,(dR.,.).

E

By the area theorem we have

v . , ~ ( E ) =

' ~--7 ~1

I f f ,

(a+it) dadt,

is

3 7 , ~ (E)

where

A~,~(E)

denotes the set of points in the rectangle for which

f(s)

belongs to E. Hence, by Fubini's theorem,

v;,,~(E)= f v,;.:,~(E)d~.

108 Vibeke Borchsenius and Borge Jessen.

Also, v,; ~,, ~. (R~) is uniformly bounded for all 7 and cl and all a in al =< a _--< a~

(since f " ~ ~s,, is bounded in the strip al<=a<_%a2). Hence by w 6 v.,.~ converges for (d--7)--* oo towards the distribution function v determined by

(5) ,,(E)

⁼

f,,~(E)da. ^~

ot

By w 5 we have for every x the inequalities

(6)

_{- - -}

I (p:_.~

^{, : , % -} o) ^-- ~:-,~at +

o))

< lira inf

N:-~(7"

~)

< lira sup ^~- . N , . - , (7, a) < ~ ,

- - Z - - 2 - ~ ( r ( % + o) - r ;o,t - - oi).

By w 5 the convex functions ~.f-x(a) depend continuously on a~. This implies t h a t the left and right derivatives p}-~(a--o) and ~ - x ( a + o) considered as functions of x for every fixed a will be lower and upper semi-continuous, respectively.

Hence, the functions to the left and right in ( 5 ) a r e lower and upper semi- continuous, respectively. By Fatou's theorem we c o n c l u d e t h a t if E is a con- tinuity set of r(E), then

' l '

^" m d R , } < v ( E <

t :

" ' "

(z)

~-~. (~s-.(%-o,-~s-.,.,,,+o)) ( . . = ) = ~ (~.~_.~%+o,-~.-,,(,,,-o..).4dR,,).

E E

These inequalities must then hold for all Borel sets E.

I f we put E = R x , a l = a - - ~ , and a 2 = a q - ~ , then v(E) will by (5) approach zero for ~-+ o, whereas the first term i n ( 7 ) will converge towards the integral over Rx of (P'I-x (a + o) - - p':_~ (a - o))/2 z . This integral must therefore be zero.

Hence the two functions p}-~(a--o)

and p}_~(a+o)

will differ only in a null-set.

This implies t h a t the first and last terms in (7) are equal. Thus we have proved t h a t for an arbitrary Borel set E

f . , ,

^{' ' 9 , = ~}

f

( ~ _ , . ( % + o ; - ~ ) . _ , ( ~ l - o ) ) m ( d R , . ) .

(8) ~j(r162 + oj)m(dR,)=v(E)

E E

x Actually, the relation ~,./,d(E)---> v(E) holds not only for the continuity sets of v, but for all Borel sets E. This property depends on the fact that the densities N y - x ( y , d ) / ( J - y) of the distribution functions vT, d are uniformly bounded for ( J - y ) > I, say.

Mean Motions and Values of the Riemann Zeta Function. 109 I f in particular ~ for every a is absolutely continuous with a density

Go(x)

which is a continuous f u n c t i o n of x and a together, the relations (8) show (on a c c o u n t of the semi-continuity of the integrands) t h a t for every x

' ' a fGo(x)da<qD/-x(a, Wo)--qD.r-,(al--O).

9 ~ f - ~ ( a ~ - o ) - 9os-~( ~ + o) ~

2~.

^{= ' "}

o" I

The c o n t i n u i t y in al and a2 of the t e r m in the middle ~hen implies t h a t ~0.f-.<(a) is different,able, and we obtain

t ,' i i o'~

IjOf--, (0"2) -- IjOf--x (0"1) --- 2 iT S tCJ[is(X)dG,

OI

which shows t h a t

l,f-~.(a)

is twice differentiable with the second derivative

The Jensen Function of a Type of Generalized Analytic Almost Periodic Functions.

10. W e shall now give an extension of some of the preceding results to a type of generalized analytic almost periodic functions. The f u n c t i o n s which we will consider will be supposed to be ~lmost periodic in a strip [a o,ri0] a n d con- tinuable n o t in a strip (a, fl), b u t in a half-strip, say

c~<a<fl, t>7o.

The t y p e of generalized almost periodicity with which we shall be concerned will be an extension to analytic functions of almost periodicity in Besicovitch's sense with index p; but while Besicovitch takes p ~ I, it is sufficient for our purposes to take p > o. No t h e o r y of this type of generalized almost periodicity will be needed, since all results follow directly from the definition.

I t will n o t be possible to m a i n t a i n the above definition of the J e n s e n function.

As m i g h t be expected, since we are dealing with generalized almost periodicity of t h e Besicovitch type, the limit has to be replaced by a limit, in which ~-~oc for fixed 7, b u t n o t necessarily u n i f o r m l y in 7.

The n o t i o n of a m e a n value will be t a k e n t h r o u g h o u t in this sense, i. e. a f u n c t i o n

H(t)

defined on a half-line t > 70 will be said to possess the

mean ~:alue

i]lt {H (t)} = e_ oo .d I - - l i r a

~, ]_ H (t) dr,

7

if the limit on the r i g h t exists for a fixed 7 > 7o (the i n t e g r a l need n o t exist for 7 =-7o). Evidently the existence of the limit for one

7>7o

implies its existence for

110 Vibeke Borchsenius and Borge Jessen.

all ~'>7o, a n d t h e value is i n d e p e n d e n t of 7. I f H ( t ) i s a real f u n c t i o n defined on a half-line t > 70, the upper mem~ value is defined by

M I H ( t ) } = lira sup - H(t) dt,

t ~ r

7

w h e r e 7 > 70 is a g a i n a r b i t r a r y , b u t fixed.

A similar c h a n g e will have to be made in t h e definitions of t h e m e a n motions, t h e f r e q u e n c i e s of zeros, a n d t h e a s y m p t o t i c d i s t r i b u t i o n functions.

T h e u s u a l n o t a t i o n f o r strips will be m a i n t a i n e d f o r half-strips w i t h o u t change.

T h u s , if we are d e a l i n g w i t h f u n c t i o n s defined in a half-strip a < a < fl, t > 70, a s t a t e m e n t is said to h o l d in [a, fl], if it holds in the p a r t of t h e half-strip b e l o n g i n g to an a r b i t r a r y r e d u c e d strip (a < ) a : < a < fll ( < fl).

Suppose t h a t p > o, a n d t h a t f ( s ) , f:(s), f~(s), . . . are f u n c t i o n s defined in t h e half-strip ~ < a < f l , t > 7 o . T h e n we shall say t h a t f,(s) c o n v e r g e s in t h e mean, w i t h i n d e x p, t o w a r d s f ( s ) in [a, fl] if

B, 1

f o r a n y r e d u c e d

strip (a <)ax

< a <

#:(< ~).

Since t h e left side decreases as p decreases it is plain t h a t c o n v e r g e n c e in m e a n w i t h index p implies c o n v e r g e n c e in m e a n with i n d e x p:, if o < p : < p .

11. W e shall ])rove the following t h e o r e m s .

T h e o r e m 1. Let - - c o <~ a < ao < ~o < fi X 4- co and - - c o < 7 o < + c ~ , and let f:(s), f2(s), 9 9 9 be a sequence o f fi~nctions almost periodic in [a, fl] conve~ying u~i- formly in [ao, flo] towards a function f ( s ) , which is lhen almost periodic in [ao, flo].

Suppose, that none of the functions is identically zero. Suppose further, that f ( s ) may be continued as a regular ./'unction in the half-slriIJ a < a < fl, t > 70, and that f~(s) converges in mean with an i n d e x p > o lowards f ( s ) in [a, fl].

Then the Jensen function

~.,-(a) = M { l o g { f ( a + it)I}

t

exists uniformly in [a, fl], i.e. the function

: floglf(a+it)let

~(~; 7, ~ ) -

^{~ _ y}

7

M e a n M o t i o n s a n d V a l u e s o f t h e R i e m a n n Z e t a F u n c t i o n . 1 1 1

converges for ~-+ oo for any fixed 7 > Zo uniformly in [a, fl] towards a limit func- tion 9f(a). The Jensen function 9fn (a) orris(s) converges for n-+ oo uniformly in [~, fl] towards 9s(a).

The function qDf(a) is convex in (a, fl), and, for every a in (a, fl), the four mean motions defined by

,7 (~)1

~; (a) ]

and

inf a r g - . f ( a + i6) -- a r g - f ( a + i 7) -- lim

sup ~ ^-- 7

f f ~ o o

~_:~ ( a ) / = lira i n f a r g + . f ( a + i~) -- a r g + f ( a + i~,)

sup ~ ^-- 7

satisfy the inequalities

(9) ~0~(a -, o) =< , r (a) _-< I d (~) ] ^_+

L ~,=- (~ --< ~s (~,) =< ~;(a + o/

Further, for every strip (61: as). where c~< 61 < as< fl, the two relative frequencies of zeros de.fined by

H f (al, 62)

~ inf

^{N f} (61, as; 7, (l) H f (61, as) j = lira sup 6 -- 7 '

({ ~ o o

where N f (61, 62; Z, (l) denotes the number of zeros of f ( s ) in the rectangle 61< a < 62, 7 < t < ~, satisfy the inequalities

I

-- '~ - ' ' "-- - ' -o~I,

(io) _{2 ~} i (gs,a~ o, - ~ s ( ~ l + o))=<_H/~l,~s)=</Ts(a~,o2)= < _{2 ~}

(9)(as

^{+ o) 9s(ol}

As a corollary we have t h a t if r is differentiable at t h e point a, t h e n the left a n d r i g h t mean motions

c 7 (a) -- lim a r g - . f ( a + i(~) -- a r g - f ( a + i7)

~ ~ 6 - - 7

and

e + ( a ) = lira a r ~ + ' f ( a + i ~ ) - a r g + f ( a + i7)

.r o ~ 6 - - 7

both exist and are d e t e r m i n e d by

' 1

'7 (~ = e/(o) = ~j

(o).

t ( \

1 T h i s m e a n s t h a t a r g - f ( a + i t ) a n d a r g + f ( a + it) are b o t h --ct+ o(t), w h e r e c = (pj.,aj.

1 1 2 V i b e k e B o r c h s e n i u s a n d B o r g e J e s s e n .

S i m i l a r l y , if ~f(a) is d i f f e r e n t i a b l e a t a 1 a n d a2, t h e n t h e r e l a t i v e f r e q u e n c y o f z e r o s H f (aa, a2) = l i m N.l (ff_L,_a~; 7' 6)

a ~ 6 - - 7 e x i s t s a n d is d e t e r m i n e d b y t h e J e n s e n f o r m u l a

I t P( ,

9 - - 9 J " ~0"lJ)"

H r (fiX, 62) : ~ ( 9 . f ' a 2 ' ' " '

T h e o r e m 2. The ./iowtion .f(a + i t) possesses for every a in (a, fl) an asymptotic distribution funelion uf, ~, i.e. the distributio, function #j, o; .~., J of f ( a + i t) in the interval 7 < t < 6, defined by

m. ,,; .., ~ ( E ) = m (Aj. o: ~,. ~ (E~)

' 6 - - 7 '

where A f, ,~; :, ~ (E) for an arbitrary Borel .~.et E de~wtes the set oat" points of 7 < t < 6 for which f ( a + it) belongs to E , eolwerges for 6--> oo for any .fixed 7 > 7o towards a distribution function /~.0. The asymptotic d&b'ib, dion function /%,. o oj'.f,,(a + i t ) converges for n -~ oo towards tq. o.

T h e r e a r e o f c o u r s e s i m i l a r t h e o r e m s f o r f u n c t i o n s f ( s ) w h i c h m a y be c o n t i n u e d in a h a l f - s t r i p a < a < fl, t < 6 o . T h e l i m i t s m u s t t h e n b e t a k e n f o r Z ~ - - ~ 1 7 6 a n d fixed 6 < 60. I f b o t h p a i r s o f t h e o r e m s a r e a p p l i c a b l e f o r t h e s a m e s e q u e n c e f l ( s ) , , f z ( s ) . . . 4 h e J e n s e n f u n c t i o n 95(0") a n d t h e a s y m p t o t i c d i s t r i b u t i o n f u n c t i o n /~f.o will b e t h e s a m e in b o t h cases, s i n c e in b o t h c a s e s t h e y a r e t h e l i m i t s o f ~f",(a) a n d /~f,, ,, r e s p e c t i v e l y .

W e s h a l l n o t g o i n t o t h e e x t e n s i o n o f t h e r e s u l t s o f w 1 6 7 8 - - 9 , s i n c e t h e i r e x t e n s i o n is n o t n e e d e d f o r t h e t r e a t m e n t o f t h e z e t a f u n c t i o n .

12. L e t a 1, ill, a2, flz b e c h o s e n s u c h t h a t a < cg I < {2' 0 < •2 < if2 </~0 < E1 < ~, a n d l e t 6 > o be c h o s e n so s m a l l t h a t a < a l - - 8 6, /~1+ 8 6 < / ~ , e~0< ct2-- 3 6, a n d f12+ 3 6 < f l 0 ; w e m a y s u p p o s e 6_--< 2 x. D e f i n e t h e r e c t a n g l e s R,,(to) f o r o ~ r ~ 8 b y

lt,.(to) : a l - - v 6 _-< G _-< #1 + v 6,

a n d t h e r e c t a n g l e s S,(to) f o r 0 ~ v =< 3 bv S,.(to) : u 2 - - v 6 ~ a ~ ~2 + v6,

] t - t o l ~ , ~ ( I +~),

I t - t 0 1 _ - < 89 + ~ ) . 1

a N o t all of t h e s e r e c t a n g l e s will be u s e d in t h e p r o o f s of T h e o r e m s I a n d 2. T h e r e m a i n d e r are b e i n g k e p t in r e s e r v e for t h e p r o o f s of T h e o r e m s 3 a n d 4.

M e a n M o t i o n s a n d V a l u e s of t h e R i e m a n n Z e t a F u n c t i o n . 113

Since ~ =< 89 the distance between t h e frontiers of two successive rectangles

R.(to)

or

S.(to)

is & The rectangles

B.(to)

are contained in t h e half-strip a < a <,8, t > 7 o for t o > 7 o + ~ .

We shall begin with some lemmas, which may be proved w i t h o u t difficulty from the assumptions of Theorem I.

( I I ) NOW,

L e m m a 1. I f for t o > y o + ~ we put g(to) = m a x I f ( s ) l , KL,(to) = m a x J.r

R:

^(t.)

t';

^(tol

the f u n c t i o n s K(to) v, A;,(to) v,

L,,(to) v

possess mean values, a n d

M{L,,(to) v} -~ o and M{h,,(to) v} -~. ) f ! K ( t o ) v}

as

to It, t o

P,'oof I f !l(s) is r e g u l a r in I s - s o l =< d, the mean value

'(,

M (el = 2-~7~ g (So + q e i o) p, d 0

o

is, according to H a r d y [I], increasing for o < e ---<- $- H e n c e d

' f f

I :~(,o)1,' = M(o) _-< M ( e ) e d e = ~ ,

I.q(,,,)l"d,dr.

2 *

o Is-~ol~t

Consequently

I f,(

L,,(to)" <--_ ~fi~

if(s)--./;,(s)l,'dadt.

lt~ (ti,!

The mean convergence therefore implies t h a t

M{L,,(to)P}-'.o

as n ~ o o .

1o

Hence ~

K(to) <= ~3,(to) + L.(ro)

.,,d

hL,(to) ~ K(to) + L,,(to).

L,,(to)- ,na, I,r(.~t-f,,(~)/,

1,'; (t,,)

5 d d

, I 2 '

]i~--II'A'(to)P(I,o] P ,

^[(j

'=i: ! __< m ;: j

7 7 ;'

where P----I if p < I a n d

I ' = I / p

if p > I. F r o m the ahnost periodicity of./;,(.,') in [a, fl] it follows t h a t

K,(to)

is almost periodic. H e n c e

K,(to) p

possesses a mean value. The inequality ([2) t o g & h e r with (I I) t h e n shows, t h a t

K(to) I~

possesses

' See e.g. Hardy, Littlewood and P61yaii~. Theorems 28, J98, and I99.

8

114 Vibeke Borchsenius and Borge Jessen.

a mean value, and t h a t

M{K,,(to) ~'} ~M{K(to)~' I

as u - ~ c ~ .

to to

value of

L,,(to)~

will exist, since

f,,(s)--fi,(s)

converges in

.f(s) --/;,(,,:)

f o r m -+ cx~.

Finally, the m e a n t h e mean t o w a r d s

L e m m a 2. K(to) I ' = o(to) as to~OO.

Proof

Since ].f(s)] takes t h e value

K(to)in

some p o i n t of R;(t0), t h e r e will (for t0> ~'o+ !~) exist an i n t e r v a l of l e n g t h > I c o n t a i n i n g to in w h i c h

K(t)>K(to).

H e n c e

t I, + 1

K ( t.),' < .1" K ( t)" at,

t o - - 1

and the r i g h t h a n d side is o(to) , since

Jll{K(t)vl

exists.

t

L e m m a 3. T h e r e exists a c o n s t a n t k > o such t h a t f o r all to max I.f(.,.)l > k a n d m a x ]fi,(.s')] > k f o r all n.

E, (to~ .% it,,)

Proof.

Since the f u n c t i o n s are a h n o s t periodic in [uo,,3o] a n d n o t identically zero, a n d since fi,(s) c o n v e r g e s u n i f o r m l y in [ao,/~o] t o w a r d s

f(s),

t h e r e exists a c o n s t a n t h > o a n d a b o u n d e d closed sub-set S of t h e strip (do, rio) such t h a t f o r e v e r y f u n c t i o n

f ( s + ito)

or

f,(s+ito)

the absolute value is > h in some p o i n t of S.

I f the l e m m a were false, we could e x t r a c t f r o m t h e system of f u n c t i o n s

f ( s + ito)

a n d f,,(s + ito) a sequence c o n v e r g i n g u n i f o r m l y t o w a r d s zero in So(o).

Since the f u n c t i o n s are u n i f o r m l y b o u n d e d in [%, ~o], this sequence would con- verge u n i f o r m l y to zero in any b o u n d e d closed sub-set of (%, rio), in p a r t i c u l a r in S, and this is impossible.

13. W e shall also use c e r t a i n g e n e r a l f u n c t i o n t h e o r e t i c lemmas. ~ L e t F ( s ) be a r e g u l a r f u n c t i o n in R3(o) f o r w h i c h

,~o (0,

where k is a given positive number. "~ T h e l e m m a s will give estimates d e p e n d i n g on t h e n u m b e r

~ = n,ax I F (.~') I (--> ~'1.

R~ (o)

i S o m e of t h e s e l e m m a s are well k n o w n , b u t for t h e convenience of the reader l h e p r o o f s are given.

*- W h e n t h e l e m m a s are a p p l i e d k will be t h e n u m b e r of L e m m a 3.

Mean Motions and Values of the Riemann Zeta Function. 115 By A we denote constants (not necessarily the same a t each occurrence) d e p e n d i n g on t h e rectangles involved a n d on k (but n o t on

F~s

In one of the lemmas t h e c o n s t a n t depends on a parameter m and is t h e r e f o r e denoted by A (m).

Besides R3(o), t h e rectangles R~(o), Rl(O), and So(O) occur. An)" sequence of f o u r rectangles, each of which contains tl~o. n e x t in its interior, would do. L a t e r on we shall sometimes use the lemmas for other sets of f o u r r e c t a n g l e s .

L e m m a 4. The n u m b e r N of zeros of l"(.v) in Rz(o ) satisfies an inequality

N < = A l o g ( K + I).

Proofl

L e t So be a point of So(o) in which

I F ( s ) [ ~ k .

L e t

z = z ( s )

be a regular f u n c t i o n in //3(0) which maps //3(o) on the circle ]z I ~ x so t h a t

Z(So) ---- o,

and let

s = s(z)

be its inverse function. ~ The image of Rz(o) will depend on s o, b u t will for all possible so be contained in a circle { z l ~ Q < l, where q is in- d e p e n d e n t of So. A p p l y i n g J e n s e n ' s inequality to the f u n c t i o n

H ( z ) =

F ( s ( z : ) w e obtain

k, < K ₌ or N--< _{_} I log K log -

whence the desired result. ~ e

The n e x t two lemmas will be proved together.

Lemm& 5. I f s l , . . . , s.v are t h e zeros of F(.~) in //2(o), and we p u t

N

F(8)

^{= I F [} ( " -

t h e n in //1(o) ~='

log {Fl(s)] a - - J log ( K + I), i . e . 1 1"~(~)1 >----

(K§ i),,"

I

Lemm& 6. The left or r i g h t variation V of the a r g u m e n t of F(s) along an arbitrary s t r a i g h t s e g m e n t in /t1(o ) satisfies an inequality

]V I=<Aiog(K+ I).

Proof

The f u n c t i o n

l"l(S)

is regular i n / / 3 ( o ) a n d 4=O in//2(o). I f d denotes t h e d i a m e t e r Of B2(o), we have

[I"l(so)] :> kid ~',

Where s o is the point introduced in t h e proof of L e m m a 4. On the f r o n t i e r of R3(o ), and hence in Ba(O), we have

I

=<

Kl<i.".

I T h u s S=S(Z) is c o n t i n u o u s in {z I _-< I a n d r e g u l a r e x c e p t in f o u r p o i n t s on Izl = I corre- s p o n d i n g to t h e v e r t i c e s of R3(o).

Since K / k ~ K + i w h e n k ~ I, a n d K/k_-<IK%I) l/t" w h e n k < I . T h e e x p r e s s i o n l o g ( K + I ) is i n t r o d u c e d for t h e s a k e of u n i f o r m i t y t h r o u g h o u t t h e lemmas~

116 Vibeke Borchsenius and Borge Jessen.

L e t z t = 2'1(8) be a regular f u n c t i o n in R2(o) which maps B2(o ) on t h e circle l Zl[ ~ I such t h a t zt(so) = o, a n d let s = s ( z i ) be the inverse function. The image of Rl(O) will be contained in a circle ]zil ~ Qt< I, where 01 is i n d e p e n d e n t of so.

A p p l y i n g Carath60dory's inequalities ~ to a branch of the f u n c t i o n log H i ( g 1 ) =

= log ]HI(~'I)[ -{- i arg HI(Zl), where H i ( z 1 ) = Fl(S(~l~), we o b t a i n for 12'1] ~ q1 the inequalities

k 2 o t K

log IH~(zl) l

->-- x + ~ log dX " log 6-~

I - - q i I - - 01

a n d

(i3)

and

I a r t H , (2'1) -- a r t H 1(O) 1 ----< l ~ log ~ -- log il x . H e n c e in R t ( o )

9 K I + Q1 l o g ~7~

log = > - _2gl

l o g &v

I - - (J1 I - - ~I

I a r t ib I (.~) --- a r t 1~1 (,e0) I < 2 ~1 l o g K d -v The l a t t e r inequality gives the estimate

('4)

I I'l < 4Q, _{= ~ log} K d : " _{+ N,w,.}

I n (13) and ('4) we may by L e m m a 4 replace N by A log (K + I), since d > I and d < I. W e t h e n o b t a i n the desired results.-"

L e m m a 7. There exists a horizontal s e g m e n t a i ~ a ~ : l 1, t-=--t* ill 1~o(O) on which F(s) =~ o and

( ~ arg l"(a + it) <-- A log- ( K + I).

Proqf L e t .,',,=an + i t , be the zeros i n t r o d u c e d in L e m m a 5 a n d let t* be chosen in the interval It*l__< 89 such t h a t min I I t * - - t,,l/ is as large as possible.

B y L e m m a 4 t h e distance of the s e g m e n t rhNa<=fll, t = t * f r o m the zeros s, and t h e f r o n t i e r of R l ( o ) is >=I/A log ( K + I), a n d we m a y therefore find a rectangle

=< =< + r, I t - t*[ --< r belonging to Rt(o), where r > , / A log ( K + I), in which F ( s ) 4 ' o . I t follows f r o m L e m m a 6 t h a t , if s* lies on the segment, t h e n

See e~g~ C~rath~odory if], w 74- T h e i n e q u a l i t i e s as t h e r e given m u s t be a p p l i e d to t h e f u n c t i o n f ( z i ~ = a log H i ( z 1 ~, + b for s u i t a b l e v a l u e s of a a n d b.

F o r t h e first e s t i m a t e we h a v e to a p p l y t h a t , s i n c e K ~= k, a c o n s t a n t d e p e n d i n g on t h e r e c t a n g l e s and on k~ will be ;< an e x p r e s s i o n A log ~K-k f .

M e a n M o t i o n s a n d V a l u e s of t h e R i e m a n n Z e t a F u n c t i o n . 117

a branch of arg F(s) satisfies in this rectangle, and a f o r t i o r i [,~" -- s* I < r, an inequality

l arg I"(.,')- arg F(.,.*)I ~ A log ( K + I).

in the circle

By the familiar inequality ~

(o, o) I _<- 4 , v

~ 1 "

for a harmonic function u(x, y) in the circle x ~ + y~ =<- r-', for which [u (x, Y) I =< J l , we obtain the desired result.

L e m m a 8. I f we put

I F ( s ) ] , , = max {]F(s)[,ml f o r o < m ~ I, the integral

r)'

l - - j ' ( l o g IF(a +

it) Ira--

Iogl/"(a +

it) t)dt

7

satisfies for ~ < 7 ~ cT ~ 89 :and ~l < a =< fit a,n inequality

(IS) ] ~ A(m) log ~ (K § ,),

where A (n 0 -~ o as m -~ o."

(=> o)

Proo[: Since m < I

log ] F ( a + i t ) I . , - log I F ( a + it)] <= - log- ] F ( a + it)]."

H e n c e by L e m m a 5 f o r a l < a < { / 1 and I t [ < 8 9

log [ F (a + i t)1,,, -- log ] F (a + i t)[ ~ -- log [ I"I (.") ] -- ~ log [ s -- s,, ! A"

< A [ o g ( K + I ) ~ l o g I t - t,I.

Since ,,= 1

-flog lt--t,,ldt

< - - f l o g - J u } , t , , J = e

*' - - 1

for any t,,, we find t h a t

I < . 4 1 o g ( K + ] ) + . ~ , and hence, on using L e m m a 4, t h a t

(I 6) ] =< A log (K + I).

I Cf. S e h w a r z [[I, w 6.

I Instead of log'-(K+ I) w e might use ally 'positive function which does not take arbitrarily small values and which tends to infinity m o r e rapidly than log ( K + I).

a In analogy to the notation l o g + x for the function m a x {log x, o}, x > o, w e denote by log- X the fimelion rain {log x, o}, x > o. T h e function -- iog:-x is non-negative and decreasing;

moreover, if x = x 1 . . , x N, w e have - - l o g - x ~ - - I o g - x t . . . I o g - x N.

118 Vibeke Borehsenius and Borge Jessen,

This estimate does n o t depend on m and implies (I 5) for every m. I t r e m a i n s to prove t h a t if ~ > o is given, then I ~ t l o g s ( K + I) for - - 8 9 < ~ 8 9 and

~1 ~ a ~/~1, provided t h a t m is sufficiently small. From (I6) it follows t h a t I ~ r log ~ ( K + I) for all m, provided t h a t K ~ (some) K o depending on r. Thus, to complete the proof we m u s t show t h a t I ~ r l o g ~ ( K + I) for K ~ < h o, when m is sufficiently small. I t will be sufficient to prove t h a t I X ~ log ~ (k + r).

W h e n K=</~o, L e m m a 4 shows t h a t

N~No=A

log (K0+ I), and L e m m a 5 then shows t h a t when s belongs to RI(O) a n d all Is--s~]-->---r, where o < r < I, then

r.V r.~'o

[F(s)[=>(K+ I) a ~(K o+I) A"

Thus, if we p u t

rNo

(Ko+ P'

the total length of those sub-intervals of } tl =< 89 in which the i n t e g r a n d in / is positive, is at most N 2 r . Consequently

3"r

9 z a logIK + f l o g I-I t"

--.Vr .V. r

=<

a log (K0+ ,) N o f log-I,, I

- X o r

The last expression tends to zero as r - ~ o. H e n c e I < r log ~ (k + I) when r is sufficiently small, i. e. when m is sufficiently small.

Connected with L e m m a 8 is the following lemma, which will be used later on in the p r o o f of T h e o r e m 3.

L e m m a 9. The integral

3

J== flog

[ F ( a +

it)[ dt

7

satisfies for - - 89 _--< 7 < ~ ~ 89 and al ----< a ~ fll an inequality [ J [ =< A log ( K + I).

Proof

W e have

d d

J = f log+[F(a+ it)[dt f log-[F((r+it)[dt.

7 7

The first integral on the right is < log* K < log ( K + 1). The second integral is the integral I of L e m m a 8, for r e = I , which by (i6) is = < A l o g ( K + I).

Mean Motions and Values of the Riemann Zeta Function. 119

Remark.

All the p r e c e d i n g l e m m a s r e m a i n valid if in t h e estimates on t h e r i g h t we replace log ( K + I) or log ~ ( K + I) by

Kq,

w h e r e q is a given positive n u m b e r .

14. W e now t u r n to t h e p r o o f of T h e o r e m I. On a c c o u n t of L e m m a 3 we m a y apply the lemmas of w x3 to t h e f u n c t i o n s f ( s +

ito)

and f n ( s + i t 0 ) a n d h e r e b y o b t a i n estimates on t h e f u n c t i o n s

f(s)

and

f~(s)

in

Ba(to).

I n s t e a d of t h e n u m b e r K we m a y use t h e n u m b e r s

K(to)

a n d

K,(to)

i n t r o d u c e d in L e m m a I.

F r o m L e m m a 81 it t h u s follows t h a t f o r aa _--< a _--< fll

a n d

to+ 89

f (log If('~ + i t ? l . , -

tu- 89

log I f(,~ + i t;: I)dt <=

A

(m)K (to) v

to+89

f (log If. (,7 + i t~ I,,, - log If,, ::~ + i t~l) ,t t __< A (.,) K,, (to)".

to-

L e t us now s u p p o s e , as we m a y a c c o r d i n g to w zo, t h a t p _--< I. T h e n if ul a n d u~ are b o t h ~ m

l i o g ~,~ - l o g , , ~

I ~

l o g (,,, + I , , . - ,,~

I)

- l o ~ , , , = < . I . . - ~,~

I",

where a is the (finite) u p p e r b o u n d of (log (m + x) - - log

m)/x ~

f o r x > o. H e n c e

log If(s)I-,

- l o g

If.(s)I,,, I ~ a If(s)I.,

-

I j;,(8)Im

I p ~ a

If(s)

- f . ( s ) I p

a n d c o n s e q u e n t l y f o r al =< a _--</~t to+ ~,

f l o g

!f(a + i t)I,..-- log If. (a + it)I-I d t

<__ ~ L .

(to)".

tQ- 89

I t f o l l o w s t h a t f o r t~ 1 ~ o" <: ~1

_ 7 l o g l f . ( o + i t l l d t

7 ' 7

_--< A ( m ) ~ I

K(to)"dto+ A(m) K,,(to)Vdto+ a L,,(to)Vdto .

7- 89 " z - t " / - t

z W i t h 7 = - - ~, 6 = 8 9 and w i t h A(m) K p on the right instead of A / m ~ : l o g ~ ( K + I ~ .