The roots of trigonometric integrals

THE

ROOTS OF

TRIGONOMETRIC INTEGRALS

BY N. G.

DE

BRUIJN

1. Introduction. Concerning the roots of trigonometric integrals

G.

P61ya

(see

references at the end of the paper) has proved a number of results which

hederivedfrom properties of the roots of polynomials.

He

proved,forinstance, the reality of all the rootsofthe followingfunctions:

(1.1)

f

e-t2e

’zt

dt

(n

1, 2, 3,

...);

(1.2)

C(t)e

’

dt

(h

>

0),

where

C(t)

exp (-), cosh

t),

and

(1.3)

f_

exp

(--at

4"

-

bt2"

ct

)

exp iztdr,

wherea

>

0, breal, c

>_

_0, n 1, 2, 3,

....

(See

concerning

(1.1), [7], [8];

concerning

(1.2), [6],

[8];

concerning

(1.3), [8].)

Another important result of P61ya is the following one

(see [8]):

Suppose that the function

F(t)

of the realvariable satisfies

(1.4)

F(t)

integrable over

_<

_<

F(t)

(F(-t))*,

F(t)

O(e

-Itlb)

_for

--

,

b

>

2.

(The

* indicates the conjugate imaginary.)

Let (t)

be an integral function of genus 0 or 1, withreal roots only, and let the number

,

be

>_

0. If the function

F(t)

is such that all the roots of the integral

(1.5)

F(t)e

’zt

dt

arereal, then thesameholds for the function

_f:

F(t)q(it)e*t’e

t dt.

The function (it)e

"

is easily seen to be the limit of a sequence of poly-nomials, all of whoserootsarepurelyimaginary. P61ya’s result, statedinother

words,

is that these functions are universal

_factors:

which conserve the reality of the roots of any trigonometric integral whose integrandsatisfies

(1.4).

P61ya also proved that the functions (it)e

"

indicated above arethe only analytical functionswith this property. The latterresult will not beused in the present paper.

ReceivedJuly 16,1948.

198 N.G. DE BRUIJN

In

thesequelwecontinueP61ya’s researches.

Our

mainresults are

TEORE

1.

Let

f(t)

be an integral

_function

of

and such that its derivative

$’(t) is the limit (uniformlyin any bounded domain

_of

the t-plane)

_of

a sequence

of

polynomials, all

_of

whoseroots lie onthe imaginaryaxis.

Suppose

_furthermore

that

f(t)

is nota

constant,

and that

f(t)

f(-

t),

](t)

>_

0

_for

real values

_of

t. Then theintegral

_(R)

e-teTM dt has realroots only.

(The

conditions

(f(t)

f(-t), f(t)

>

0)

may be replaced by weaker ones, namely,

"f(t)

(f(-t))*, Ref(t)

>

0for o

<

<

oo

",

_{iff(z)isapolynomial}

orafunction of thetype

(1.6) (see

Theorems 19 and 20 respectively).

It

isnot easy toseewhetherthe latterset of conditionsissufficientinthe general

case.)

P61ya’s results

(1.1)

and

(1.2)

are special cases of this one, but

(1.3)

is not.

TEORE

2.

Let

N

beapositive integer andput

(1.6)

P(t)

p.e

"

(Re

pN

>

0; p.* p_.,n 0, 1, 2,

---).

Let

the

_function

q(x) beregular in the sector

-r/2N

N

-1

arg p

<

arg x

<

r/2N

N

-

arg prand on its boundary, with possible exception

_of

x 0 and x whichmay bepoles

(of

arbitrary

_finite

order)

_for

q(x). Furthermore suppose

(1.7)

(q(x))* q(1/x*)

in thissector(inother

words, q(x)

isreal

_for

x 1. Then all buta

_finite

number

of

roots

_of

the

_function

(1.8)

(z)

f:o

e-’("Q(t)e

’’t

dt

(Q(t)

q(e’))

arereal.

It

may be remarked that our method fails to give any useful information concerning the number and location of the non-real roots of

(1.8)

in thegeneral case,sothat this very peculiar result may be ofverylittle practical importance.

The special functions

(C(t)

exp( -hcosh

t))

(1.9)

(z)

C(t)

a.e

"’

e

"’

dt

(

_>

O,

a*.

a_.),

which have

N

pairs of non-real roots at most

(Theorem

21),

may be of some

interest since the Riemann

_-funetion

can be approximated by functions of this type

(see [8]).

It

will be worthwhile to determine classes of functions of this type with the property thut all the roots are real.

We

shall study these questionsin

_6.

(Those

readers who are mainly interested in considerations concerning the /-funetion may omit the proof of Theorem I and the relatedresults in

5,

and

inTheorem 2 need to considerthe ease

P(t)

(e’

₊

e

-t)

only.

In

that ease

the complicated

₄

is superfluous since the results of that section then reduce to well-known asymptotic formulas concerning the F-function.)

THE ROOTS OF TRIGONOMETRIC INTEGRALS 199

In

₇

we expose what progress has been made in this paper in the direction ofthe Riemann hypothesis, and also how small this progressis.

An

outline oftheproofs of Theorems 1 and2concludesourintroduction. Sections2and3 will furnishfunctions

S(t)

which arespecial universalfactors

in PSlya’s sense but which have stronger properties than those stated above.

A

function

S(t)

of the realvariable t, satisfying

S(t)

(S(--t))*

will becalled

a stronguniversal

_factor

if itjoins properties

(a)

and

(t) below,

foranyfunction

F(t)

satisfying

(1.4).

(a)

If the roots of

(1.5)

lie in a strip

Im

z

_<

A

(A

>

0),

then those of

f:.

F(t)S(t)e

‘t

dt lie in a strip

[Im

z

<

A1

where

_A1

_<

A,

A

independent of

F(t).

(t)

If

F(t)

is such that, for any

_>

0, all but a finite number of roots of

(1.5)

lie in the strip

_Jim

z

_<

,

then thefunction

_{f_ F(t)S(t)e}

it

dthas only

a finitenumber of non-real roots.

It

will be evident from

(a)

that any strong universal factor is a universal factorinPSlya’s sense.

A

function

S(t)

of thetype

(1.10)

S(t)

a,e’)’t

(k

>

O,

a,,

a*,,)

-N

is astronguniversalfactorifall itsrootslie onthe imaginaryaxis. (Conversely,

if

_S(t)

is auniversalfactor and ifit isof thetype

(1.10),

thenits roots lieonthe imaginary axis. This follows from PSlya’s result on universal

factors.)

This

resultisobtainedby generalizingatheorem of

J. L. W. V. Jensen

onthe location of the roots of the derivative of a polynomial with real coefficients

(2)

and applyingitto integral functions

(3).

The functionse

’,

,

>

O,

alsoturn out tohave property

(a),

butit isdoubtful whether they have property

(t).

The functions

(1.8)

willbe shownto have butafinitenumber of roots outside

anystrip

_Jim

z[

_<

e, e

>

0. This will becarried out by proving asymptotic formulas for

(z),

depending on the expansion

(5.7).

In

that formula an

auxiliary function

H(s)

occurs which is a generalization of the F-function. Asymptotic formulas for

H(s)

willbe derivedin

_4.

Now

let

P(t)

and Q(t) satisfy the conditions of Theorem 2; then also

P(t)

and Q(t)/(e

-t-

2

+

e

-t)

satisfy these.conditions.

From

what is said above it

is evident that the function

f_

e-()Q(t)(e

₊

2

+

e-t)-eit

dt has but a finite number of roots outside any strip

_Jim

z

_<

,.

Now

applying property

()

with

S(t)

e* -t-

2

+.

e-t we obtainTheorem2.

In

order to sketch the proof of Theorem 1, let

P(t)

satisfy the conditions of Theorem 2 and suppose that

P’(t)

has purely imaginary roots only.

Let

A be the smallest numberwiththepropertythatthe roots of

O(z)

f:

e-P(t)e dr, which has but a finite number of non-real roots by virtue of Theorem 2, lie in

the strip

Ira

z

_<

A,

andsuppose A

>

0. The function

P’(t)

is a strong

200 N. G. DE BRUIJN

Im

_z[

_<

41

A1

<

A.

But

itis easilyseen from partial i,tegration that the latter integral equals

-z(z). It

follows that the roots of

h(z)

alsolie in the strip

Im

z

_< A

This contradicts the minimum property of A.

Hence

A 0and all theroots of

(z)

arereal.

It

will be relatively easy to extend thisto the integrals of Theorem 1 on

con-sidering

F(t)

asthelimitofa sequence of functions

P(t).

The followingnotations areusedthroughout the paper.

Re

aandImadenote the real and imaginary parts ofa:a

Re

+

i

Im

a;

_*

denotes the conjugate ofa. If

f(z)

isafunction of the complex variable

z,

then

f*(z)

is defined by

f*(z)

(f(z*))*. A

polynomial or integral function

f(z)

is calledrealif

_f(z)

_=-

_f*(z),

thatisto say if

f(z)

isrealforreal values ofz.

All the trigonometric integrals considered in this paper are real integral

functions ofz.

2. Theorems on polynomials.

We

shall deal with linear combinationsof the type

(2.3)

for a given polynomial

f(z)

with real coefficients; the simplest case is

_f

(z)

f(z

+

i)

f(z

i). Several propertiesoftheroots of

_fl

(z)

are

_known;

they all express in some way that the roots of

f(z)

lie closer to the real axis

than those of

f(z).

1. The number of non-real roots of

_fl

(z)

does not exceedthat of

f(z).

(This

is a special caseof Poulain’stheorem.

See [11;

Abschn.

VI,

Aufg.

63].)

2. If the roots of

f(z)

lie in the strip

Im

z

_<

1, thenfi(z) has real roots only. Namely,

f(z

+

i) #

f(z

/)]for

Im

z # 0. (Properties 1 and 2 arecontained asspecial cases inTheorem

9a.)

3. If a an are the roots of

f(z)

and

_f

n

those of

f(z),

then

1

Im

fl,

--<

i

Im

a,

].

_{(See [1;}

Theroem

5].)

These propertiesaremeant for illustration andwillnot beusedinthepresent paper.

We

shallnowderive a newresult ofthis type, Theorem3, whichforms the base of ourpaper.

It

is ageneralization of thesecond propertyabove.

We

first prove

LEMMA

1.

Put

z x iy

(x

andyreal), and

f(z)

z

+

A where A

>_

_0;

let Xbea positivenumber.

_If

is

_defined

by

(2

X2)i

(A

>

X)

and 0

(4

_

X),

then we have

f(z+iX)]

>

f(z

ih) i

_f

_(Izl

2)y

>

0 and

lf(z

₊

iX)[

<

f(z

iX) [/j"

(I

z

[

2)y

<

0.

Proof.

We

evaluate

f(z

+

iX)12

f(z

iX)12

{x

₊

i(y

+

X)

}2

₊

A212

{x

+

i(Y

X)}2

₊

₅₂₁₂

8yX(x

+

y2

₊

X

A2).

The assertion

directly follows.

Now

consider an arbitrary real polynomial

f(z)

of degree

>

0.

It

can be

writteninthe form

(2.1)

f(z)

A

{(z

a,)

+

A}

_l-I

(z-

b),

THE ROOTS OF TRIGONOMETRIC INTEGRALS 201 where

_a

b; real;

A

0,

A

0. Again, let Xbea positive number.

To

any

A

exceeding X we construct the circular region

C

defined by

(x

a)

y

_< /

X2;ifA

_<

Xwe take

C

to beempty.

ByS

S(f)

wedenotethe

sum ofall

C

and thereal axis.

We

now show

THEOREM 3.

_If

f(z)

is

_Of

the type

(2.1)

and X

>

O,

a complex number

O,

thenall the roots

_of

thepolynomial

(2.2)

f

(z

-

iX)

+

_*f

(z

iX)

(whichhasrealcoecients) lie in S.

Proof.

We

suppose to lie in the upper half-plane and outside

S.

Abbre-viating

(2.1)

we write

f(z)

A

I-I

g(z)

I

hi(z).

Trivially hi(i"

+

ix)[

>

[hi(

iX)

I,

anditfollows from

Lemma

1that also g,(

+

iX)

>

[g(

iX)].

Hence f("

-t-

iX)

>

f("

iX)

_I.

If

-

liesin the lowerhalf-planeand outside

S,

then

f(

-t-

iX)

<

f(

i},)[.

In

both cases we conclude that _i" isnot

a root of

(2.2).

We

remarkthat the limit caseof Theorem 3 forX

--

0leads toawell-known theorem of

J. L. W. V. Jensen

on the roots of the derivative of a polynomial.

(See [3]

and

[10;

Abschn.

III,

Aufg.

35].)

We

want to iterate the result of Theorem 3. Therefore, we firstdefine a set

S

S(f), N

1, 2, which isthesum of the real axisand the regions

C,i 1, ,n. If/

>

XN

we takefor

CN

the region

N-l(x

a)

y

_<

A

NX

",

which isbounded byan ellipse;if

_{A _<}

XN

1/2,

C

isempty.

It

is readily deducedfrom Theorem 3 that if the roots of the realpolynomial g(z) liein

S(f)

thenthoseof g(z

+

iX)

+

*g(z iX),

O,

liein

S+l(f).

THEOREM 4.

Suppose

that all the roots

_of

the polynomial q(u)

o

au

,

aN

O,

lie on the unit circle

_ul

1, that

f(z)

is areal polynomial, and that

X

>

O. Then theroots

_of

N

(2.3)

T-(T")I(z)

af{z

+

(2k

N)iX}

k--O

are contained in

SN(f).

Here

T

represents a translation operator

defined

by

T"f(z)

f(z

+

i).

Proof.

The function

u-(u)

canbe writtenin the form

N

U-%(U2)

Ol

y

(k

u

+

@k u-l)

(Ol

z

O, k

0)

kl

By

Theorem3, the real polynomial

(IT

*T-X)f(z)

has its roots in

S(f).

A

second application shows that the roots of

(T

-

*T-X)(IT

-

*T-X)f(z)

lie in

S2(f),

etc.,so that theroots of

T-Nq(T)f(z)

turntolie in

SN(f).

THEOREM 5.

Let f(z)

be areal polynomial whoseroots lie in thestrip

202 N.G.DE BRUIJN

of

the polynomial

(2.3)

satisfy

Im

z[ <_

A2

N

_{] if}

A

_>

hN

,

Im

z 0 if 4

<

N

.

Proof.

Follows directly from Theorem 4andfrom the definition of

S(f).

3. Application to integral functions. Strong universal factors.

Let

a real integral function be given of thetype

(3.1)

f(z)

Az"e

[I

(1

z/p,)e

z/p’,

where

A

isrealand 0,misanaturalnumber,aisreal, p 0,

Im

p

-

4, p -2

<

and therootsp

and

p* have thesamemultiplicity.

It

ispossible to construct a sequence of polynomials

fl(z), f2(z),

all having their roots in the strip

Im

z

_<

4, converging uniformly to

f(z)

in any bounded region. Since the product

(3.1)

converges uniformly in any bounded region, it is ob-viously sufficient to prove it for the functions e

_(a

real),

(1

z/p)e/"

if

p is real and

(1

zips)(1 z/p*) exp (z/p z/p*) ifp is notreal.

In

the latter case

p71

_W

p.-1

is real, and thus it only remains to be proved that e

(a real)

istheuniform limitofasequence of polynomialswith roots onlyin the strip

Im

z]

<

A.

We

have indeede lim,

(1

W

az/n)’,

converging

uni-formlyin anyfinite region. Since also e-b"

(b

>_

0)

is the limit of a sequence of polynomials with real

roots, the

same applies to the function

e-b’f(z),

b

>_

O,

if

f(z)

satisfiesthe

con-ditionsmentioned above.

Conversely, it seems probable that, if a sequence of real polynomials with roots in the strip

Im

z

_<

A _{converges, uniformly}in any bounded region, to

an integral function, then this function will be of the type

e-bf(z),

where the genus of

f(z)

is either 0 or 1.

(The

corresponding problem for functions with real rootswas solved by Phlya

[4].) We

do not need the solution of this problemforourpresentpurposes. (Afterthispaperwas writtenthe conjecture stated above has been proved by

Mr. J. Korevaar.)

Namely, we are able to

restrict ourselves to integral functions of order

<

2; these functions satisfy

If(z)

<

exp

(I

z

i),

p

<

_2,for z sufficientlylarge. According toHadamard’s theory such a function can be expanded into a product ofthe type

(3.1)

with

or

1-2

<

o:

Ifwenowsuppose that the roots of

f(z)

lieinthe strip

Im

z

-<:

/x_{and that}

_f(z)

isreal for real

z,

itimmediately follows that

f(z)

has all the prop-ertiesmentioned inthe beginning of thissection.

We

thusobtain

THEOREM 6.

_If

the order

_of

the real integral

function

f(z)

is

<

2 and

_if

the roots

_of

f(z)

lie in thestrip

Im

z

-

4, A

_{>_}

_O,

_then there existsasequence

of

real

polynomials

f(z)

whoserootslie alsoin that strip, such that

f(z)

f(z)

uniformly in any

finite

region.

For

convenience we explicitly formulate the following well-known result

(see

[10;

123, Abschn. 3, Aufg.

201]).

THE ROOTS OF TRIGONOMETRIC INTEGRALS 203 THEOREM7.

_Iff

(Z),

_f

(Z),

_f

(Z),

areintegralfunctions,

f(z)

not identically

O,

with

f.(z)

f(z)

uniformly inany

finite

region, and

_if

theroots

_of

_f

(z), f(z),

allbelong to a givenclosedpoint-set

S,

then theroots

_of

f(z)

also lie in

S.

Theorems4and 5 cn nowbe applied to sequences of polynomials.

THEOREM 8.

_If

f(z)

satisfies

the conditions

_of

Theorem 6 and q(u) those

_of

Theorem4,then theroots

_of

(3.2)

T-(T)f(z)

(

>

O, Tf(z)

f(z

+

i))

satisfy

_IImzl

_<

(A

-

N)

1/2/fA >

N

,Imz

0/f0_<: A_<)N

.

Proof.

Let f(z)

f(z)

according to Theorem6.

It

is easilyseenthat

(3.3)

T-(TX)f(z)

--

T-(TX)f(z)

uniformlyin anyfinite region.

By

Theorem5, the polynomialsontheleft have their rootsinthe strip

Im

z

z

{Max (A

Nh

,

0)

}t.

Now

the desired result follows from Theorem 7.

THEOREM9.

Let

therealintegral

_function

f(z)

be

_of

order

<

2and supposethat

f(z)

has buta

_finite

number

_of

roots outside the strip

Im

z

-

A.

/f

furthermore

q(u)

satisfies

the conditions

_of

Theorem4, then all but a

_finite

number

_of

roots

_of

(3.2)

satisfy

_lIm

z

_

{Mx

(A

Nk

,

0)}

t.

Proof.

We

put

f(z)

g(z)h(z), where g(z) is a polynomial, and the roots of the integral function

h(z)

liein the strip

]Im

z

_<:

A.

It

iseasily seenfrom the arguments usedin the beginning of thissectionthat

_f

isthe limit ofasequence of polynomials of thetype

f(z)

g(z)h(z), where

h.(z)

hs no roots outside

Im

z

_<

5.

We

may actually tke for

h(z)

polynomials which have, apart fromunumber ofreulones, only rootswhich reroots of

h(z)

also.

According to Theorem 4, the roots of

T-X(TX)f(z)

lie in

S(f.).

It

fol-lows from the definition of

S

that

S(f)

S(h.)

+

S(g)

_

R

W

S(g), where

R

represents the set Ira

z[

_<

{Max (A

N),

,

0)}t.

_{Now (3.3)}

and Theorem 7 show that the roots of

(3.2)

belong to

R

-

S(g). S(g) consists

of a finite number of ellipses. Since each ellipse contains but a finite number of roots of

(3.2),

our proofiscompleted.

In

the special case

N

1, ),

>_

A we can obtain more complete information on the number of non-real roots.

THEOREM

9.

_If

thereal integral

_function

f(z)

of

order

_<

2hasexactly 2 roots

outside the strip

Im

z

_

,

and

_if

k

>_

A

>_

O,

O,

then the

function

fi

(z)

f(z

+

ih)

*f(z

ih) has2 non-realrootsat most.

Proof.

The function

W(z)

f(z

iX) has not more than ] roots in the

lower half-plane. The theorem nowfollows by

Lemma

2,

_6.

In

the general case ),

_<

A _{of Theorem 9 the analogue of Theorem 9a} is not true.

Even

the following statement is false’ if

f(z)

is a real polynomial with

204 N.G. DE BRUIJN

2k roots outside the strip

Im

z

_

1, then

fl(z),

0

_<

},

_<

_{1, has} at most 2/

roots outside the same strip. Taking i, ),

--

_0, we should inferthat

if(z)

has at most 2/ roots outside that strip. This is incorrect, for instance, for

f(z)

(z

+

4)(z q-

121/2),

f’(z)

6z(z

-

-(12))

.

In

order to be able to apply the preceding results to trigonometric integrals

wefirst state

THEOREM

10.

Let

b beanumber

_>

2, and lot thereal orcomplex

_function

F(t)

be integrableover

_<

_<

and satisfy

(3.4)

F(t)

(F(- t))

*

_for

allrealvalues

_of

t,

(3.5)

Then the trigonometric integral

F(t)

O(e

-’)

(t

----

).

(3.6)

f(z)

F(t)e

’

dt

representsareal integral

_function

_of

order

<

2.

A

simpleproofcanbe foundinPSlya

[8].

THEOREM

11.

Let F(t)

satisfy the conditions

_of

the preceding theorem and suppose that the roots

_of

the

function

S(t)

EM-M

aeXt,

a*

a_ aM

O,

>

O,

lieonthe imaginary axis. Then we have"

_If

theroots

(all

buta

_finite

number

of

the

roots)

of

(3.6)

lie in thestrip

Im

z

_

/_thentheroots

_(all

_buta

_finite

number

of

the

roots)

of

the real integral

_function

(3.7)

F(t)S(t)e

’z’

dt

lie in the strip

_Jim

z]

_<

{A

1/2Mh}

/fA >

(1/2M)

1/2,

and are real

_ira

<_

h(1/2M)

.

Proof.

Since the roots of

S(t)

re purely imaginary, the roots of the

poly-nomial

(u)

_M

au-

lieon theunit circle

_u]

1; hence

(u)

satisfies

theconditionsofTheorem4

(2M

N). Now

ourtheorem immediately follows from Theorem 8 (Theorem

9)

and from the fact that

T

-

_

F(t)e

dt

F(t)e’e

TM

dr,

whence

f_

f_

M a

e*Xte

T-x/(Tx/)

F(t)e’"

dt

F(t)

E

dt. --U

A

slightly better result can be obtained if

S(t)

containsfactors

e

x -[-

_*e

TM.

For

instance, the function

S(t)

e

x*

-

2

-q-

_*e

-

gives rise to the strip

[Im

z

_

{A

1/2},}1/2,

but

S(t)

e

x’

-

*e

-x gives the strip

Jim

z

_

{/

),}t,

which follows from Theorem 8

(Theorem 9)

on taking

(u)

THEROOTS OF TRIGONOMETRIC INTEGRALS 205 THEOREM 12.

_If

S(t)

I]

(k

e’

+

*e-X),

where 1,

>

O,

k 1, 2,

N,

and the roots

(all

but a

_finite

number

_of

the

roots)

of

(3.6)

lie in the strip

_Jim

z

<_

/, then the roots

(all

but a

_finite

number

of

the

roots)

of

(3.7)

lie in thestrip

_Jim

z[

_<

{Max (A

h,

0)}

.

Theorem 11 proves the statements

(a)

and

()

madein the introduction

con-cerning strong universal factors. Although it will not be used in this paper,

we shall prove here that also the functions e

1/2x2t2,

),

>

0, have property

(a).

We

donotyetknow whether theyhaveorhavenot property

().

THEOREM 13.

_If

F(t)

satisfies

the conditions

_of

Theorem 10, and

_if

all theroots

of

(3.6)

lie in the strip

[Im

z

<-

A,

then alltheroots

_of

g(z)

_{f: F(t)e1/2X2’e}

TM

dt

lie in thestrip

(3.8)

}Imz] _<

{Max

(A:

,0)}

.

Proof.

By

Theorem 12, the roots of g(z)

f_ F(t) (cosh

kt/N)N’.e

TM

dt lie in the strip

(3.8).

Owing to Theorem 7 it is now sufficient to prove that g(z) --+ g(z) uniformly in any finite region. Now this follows from

(3.5)

and from the fact that for

>

},wehave

(3.9)

e-t’2’ (cosh

kt/N)

----)e

-t’’+t’,

uniformly in

-

_<

_<

o.

(3.9)

results from the inequality cosh y

_<

e

t’,

<

y

<

,

whence

(cosh kt/N)

<_

e

1/2’.

For completeness we mention the following theorem, a slight extension of Pdlya’s result on universal factors

(see

[8])

which dealswith thecase A 0.

THEOREM

14.

_{Let F(t)}

satisfy the conditions

_of

Theorem 10 and suppose that the roots

_of

(3.6)

lie in the strip

_Jim

z]

_<

A.

Let (z)

bea real integral

_function

of

genus 0 or 1

(that

is, a

function

of

the type

(3.1)),

with real roots only. Then the roots

_of

f

F(

t)(it)e

’’

dt

(3.10)

liein the strip

Ira

z

_

A also.

A

proof can be given by introducing quite trivial modifications in PSlya’s proof for the case A 0.

It

is,

however,

also possible to deduce Theorem 14

fromTheorem 12.

It

ishardly necessary to say that the function (it)e

’,

a

>_

O,

satisfies as a

"universal factor" aswellas (it), that isto say thattheroots of

(3.11)

F(t)(it)e""e

dt

lie in

_Jim

_z[

_

A for any

F(t)

satisfying the conditions of Theorem 14.

But

the roots of

(3.11)

evenlie inanarrowerstrip, which canbe shown by applying Theorem 13 to

(3.10).

20 N. G. DE BRUIJN

4. Application of the saddle-point method. siderations will beplayedby the function

An

important part in our

con-(4.1)

H(s)

e-g(U)u"_du,

where

Re

s

>

0 and

(4.2)

g(u) u

+

0/1u(N-1)/jV

+

0/2U(N-2)/N

+

+

We

are interested in the asymptotic behavior of

H(s)

for

Re

s

>

0,

_Is

large; the natural number

N

and the coefficients al 0/N remaining constant.

The 0/’sneed not be real.

For

0/1 0/N 0 we have

H(s)

sF(s),

so

that asymptotic formulas forthe F-functionwillappearasspecial cases.

In

the sequel, positive constants al a2 will occur, chosen sufficiently

large to suit some special purpose. These numbers may depend on

N,

0/1

av, butnot on s.

The integrand of

(4.1) has,

fors large,just one saddle-point in the domain argu

_<

r satisfying

(4.3)

;g’() s.

Namely, puttings z

,

w

(w

andzarepositive forsandu positive), we

obtainthe equation

wN

-t-

(N-

1)N-’0/,w

N-+

(N

2)N

-’a2w

-

-1-

-t-

N-10/-lw

z

r,

whence, for

w[

>

al the function

1/z

can be expanded into a convergent

power series z

-

w

-

_(N

1)N-0/lw

-2

_q-

....

Solving this equation by the Biirmann-Lagrange inversion formula we obtain w

-

z

-

-1-

z

-

-b

/3z-3

-

whence wN

z(1

₊

2

z-1

+

3

z-2

+

o)--N

z(1

_

lZ--1

+

2Z-2

+

"’’),

convergentfor

]z]

>

a. It

followsthat, for

Is]

>

aa, theonly

solutions of

(4.3)

are

(4.4)

where s

-lv,

s

-3/’r,

are derived from one and the same branch of the

func-tions-i/r.

We

shallrestrictourselves to the regions args

<

r,1 [arg

u

<

r,

andhencewe only have to consider thecasewheres/Nispositive fors 0.

We

henceforthdivide intotwocases,

Casea"

4r/9

<

[args

<

1

Case#:

args]

_<

4r/9,

andwe put

L

1/2

[

in

Case

0/,

L

1/2

[

1/2Nin

Case

f,

M

I

1/8N’

q

L

in both cases.

(The

constant

4r/9

is ofcourse not essential.

In Case

it may bereplaced byany other number

<

in

Case

0/however, itcannot be replacedby arbitrary small positive

numbers.)

Our

integration contour for the integral

(4.1)

will be

I.

A

straight line from 0 toq.

II.

Thestraight lineu

q-

y,

-L

_<

y

<

o _{we notice that arg}

t[ <

THE ROOTS OFTRIGONOMETRIC INTEGRALS 2O7 Themajor contribution to

H(s)

is furnished bythe integralalong

II

passing throughthe saddle-point

.

Its

valueis

(4.5)

e-aC")u’du

₁

e-x(’) dy,

L

where

K(u)

g(u)

sloguandu

_T

yt.

We

have

dK/dy

{’(u)

su

-

}

u-

lug’(u)

g’(O

(4.6)

u-11

t

_{(dlug’(u)l/dy)}

_dy

_u-11;;

_{(d{ug’(u)l/du)}

_dy.

It

is easilyseenfrom

(4.4) that,

for

_Is

_>

a4,

Re

s

>

0,y

>_ --L,

wehave

(4.7)

(4.8)

arg

(u/})

<

3r/8.

I

follows from

(4.6),

(4.7), (4.8) that,

if

_{!1 >}

a

Re K(u)

decreases from y

-L

toy

--M. Furthermore,

for y

_>

M

we

have,

againby

(4.6), (4.7)

and

(4.8),

for

_I1

_>

aT,

(4.9)

I

follows that

(4.10)

Re

(dK/dy)

>

-

$/’u y

>

ly/

(1

-]-y)

>

1

-.

e-x(’’ dy

_<

_1/2l/i

1

exp(--K(/j

M/Ii))

I,

(4.11)

e-KC’) dy

<

81exp (-K(/I

+

M$t))

l-We

now consider the interval

-M

_<

y

_<

M.

Since

M

/jl/s,

wehave

u-’

1

q--

Y-i

<

asY

1-1,

Is]

>

ag and by

(4.6)

and

(4.7)

wehave dg/dy y

+

y%-][

<

alo

([

y-’/

+

Y%-’

l)

([

S

>

an).

Hence,

for--M y

M,

(4 12)

K(u)

K(O

_Y

W

Ya-i

<

a,o(I

y-’/

_I+

y’C

I),

(4.13)

-K()

208

.

G. DEBRUIJN

From (4.12)

we easily infer that the right members of

(4.10)

and

(4.11)

are

0(-’/e-(),

sothat

(4.5),

(4.10),

(4.11)

and

(4.13)

give

(4.14)

e-()u"du

(2r)1/2e-()

{1

-t-

O(-I/N)

In

the secondplacewe considerthe integral

(4.15)

e-()u" du

for which the

Cases

aand have to betreated separately.

-r

itiseasilyseen, on

Casea.

Sinceq

-

1/2]lttand4r/9

<

args

<

drawing a figure and carrying out some elementary trigonometric calculations, that arg q

>

arg

(e=’/

e

,1)

e

>

5r/9

for

_Isl

large.

It

follows that, if

Is

_>

_a.

andif u runsthrough the straight linefrom 0 to q, the maxi-mum value of e-(") is attained at u q. The same being true for

(since

Re

s

>

0),

we obtain

(4:.16)

e-’

d

_<lee

-"

I.

It

wasnoticedbefore that, if

_I1

_>

,

Re K()

decreasesfrom u q to M(

t,

soghag

Re

K(q)

>_

Re K(

M(1/2),

whence gheleft side .of

(4:.16)

is

O(-/%

-/’)

(see (4:.12)).

Case.

argsl

_< 4r/9.

We

putu qt, O

<_

<_

1, q

--

i/

Then wehave

(see (4.6))

(4.17)

dK/dt

q{g’(u)

s/ul

t-{ug’(u)

g’()},

and ugh(u) g’(ti)

_f’

{ug’(u)}’

du,

{ug’(u)}’

1

-+-O(U--I/N).

Hence aa

and

_a

canbechosen such that for u

>

a,,

Is

>

al,

(4.18)

ug’(u) g’()

-(1

+

)(

u)

(I

l<

sin

/20).

Now

it is easily seen from a figure that for

Is

>

a

arg

(

u)

arg

$1

<

9r/20.

So

itfollows from

(4.17)

and

(4.18)

that, for

_{sl >}

a,,

Re

K(u)

decreases if u runs through the straight line segment from u Uo

a,q/I

q tou q,whence e-("’

_<

e-() foru on thatsegment.

Let

u bethe maximumvalueof e-(") on the remaining segmentfrom 0 to

Uo. Ifu liesbetween0 anduo, 0

_<

_<

_aa

q

-,

wehave

I-’"’

_<

₁

‘(u/uo).-K‘.>l

<_

_I’"’>

_i

-‘>

!.

Sincetheconstant

a

tz e"(") isindependent ofs, we obtain

(.)

e-’u"

&

<

(1

+

.)I

e

-’’

THE ROOTS OF TRIGONOMETRIC INTEGRALS 209 THEOREM15.

_If

bisapositive constant and

H(s)

isgivenby

(4.1),

wehave

(4.20)

H(s)

(2r)1/2e-()’{

1

+

O(S

-l/N)

},

uniformly

_for

Re

s

_>

_-b, s

_]--oo.

Here s

+

"rs-i/r

-k

"rs-/

+

(absolutelyconvergent

_for

slarge)

satisfies

g() s.

(We

noticethat for

N

iweobtaing(u) 1, sand then

(4.20)

becomes the familiarStirling formula for

F(s

+

1).)

Proof.

For

Re

s

>

0 the result follows from

(4.14), (4.16),

and

(4.19).

In

order to be able to consider values ofs in the left half-plane, we continue

H(s)

bythe formula

(s

+

1)H(s)

H(s

+

1)

+

al(N

1)N-H(s

₊

1

N

-i)

(4.21)

.+-

-

aN_N-1H(s

₊

N-),

which canbefound bypartial integration:

(s

+

1)H(s)

f:

e-(")g’(u).u

+_du. Itfollows from

(4.21)

that

H(s)

canbe continued overthe whole plane but for

the points s -1, -1

l/N,

-1

2IN,

which arepossible poles of

H(s). It

alsofollowsthat

(4.21)

holds for all values ofsexcept for these points. The function

h(s)

(2r)te-()

’,

s

+

/1sl-1/N

+

is regular and

satisfies

(4.22)

sh(s

p)/h(s) 1

+

O(s

-/v)

uniformly for0_

p_

b

+

1,

[args]

<r,

sl >

a.

This can be shown by

some elementary calculations.

Now

suppose that

(4.20)

is true for the region

Re

s

>

_-k/N.

Then,

by

(4.21)

and

(4.22),

it can be verified for

Re

s

_>

-(k

+

1)IN. Hence

the theorem follows by induction.

In

the following Theorems 16 and 17 we shallcollect someresults concerning

H(s)

to be usedinthe nextsection.

THEOREM 16.

Put

s r

+

iv

(r

and real),andlet b andcbepositive constants. There are constants

A

and C

(A

and C may depend on a

ar)

such that,

for

-b

<

a

<

c

_"

I,

r

>

A

andanypositivenumber pwehave

(4.23)

[U(s

p)[

< Cl/!s

[-

U(s

[.

Proof.

Take

C

{1 +io/1 [(1

1/N)

+

_+lav-

I/N}(1

₊

c)

1/2.

It

follows from Theorem 15 and from

(4.22),

that

A,

A

>

C,

canbe determined such that for 0

_

p 1, a

_

-b,

[r

>

A,

we have

[sH(s

p)/H(s)

_<

(1

+

c)

<

C

+.

Now

the casep 1 can beproved by induction.

Suppose

that

(4.23)

is true for 0

_

p

_

k/N

(k

an integer, k

>_

N),

and that

k/N

<

pl

_

(k

+

1)IN.

Then we have

[U(s

pl

[o-t-_

1)[

<

C’

[H(s)[/[

s

[o,-1,

H(s

p, / 1

1/N)

<

C’//

H(s)

I/[

s

-/’/",

and so on, and

hence,

since

[rl

>

A

>

C,

IH(s

p,

+

1

k/N)

<

C

’

[sl

’-’ [g(s)[,k

1,2,

N

1.

It

nowfollows from

(4.21)

that

210

.

G. DE BRUIJN

_<

(1

q-

a,

I(N

1)N-’ -t-

_-t-

_a-

N-’)c’ls

I’-’

_H(s)

I.

Since

<

c r and p,

>

1, wehave

_is

p,

-t-

1

>-

Is

I(1

q-

c)

-

and

(4.23)

follows forp _m.

THEOREM 17.

_If

b, cand aregiven positivenumbers, thenpositivenumbers

A

and

C

canbe

_found

suchthat

H(s)

>

Ce

-’+)1+’"

s a

_-k

_it, inthe region

a

>

-b,

_{Isi >}

A.

Thisfollows from Theorem 15bysomesimple calculations.

5. Proof of Theorems 1 and2.

In

ordertocompletetheproof forTheorem2, which was outlined in the

introduction,

it is sufficient to show

(Theorem 18)

that the integral

(1.8)

has butafinitenumber ofroots outside any strip

_IIm

z

<

e,

>

0. Theorem2follows

(for

Q(t)) by applying Theorem 18 to the function

Q(t)

Q(t)/(e

-t-

2

-1-

e

-)

whichalso satisfies the conditions of Theorem

2,

andusing Theorem 12or 11.

We

shall establishaseriesexpansion (formula

(5.7))

for

(z)

whichgeneralizes

aformula ofP61ya

(see [6;

formula

(11)])

for thefunction

(1.2).

Theoccurrence

of Q(t) in our

(z)

makes it very difficult to carry out P61ya’s method in the present case.

We

therefore develop a new method which uses contour inte-gration.

Let

thenumbers a a a be defined by

py

-t-

pr-yr-1

-t-

-t-1/N N 1/N N--1

1/2po

=--

(Pr Y)

-t-

a(pr y)

-I-

-t-

a and define g(u) by

(4.2).

Then

we

have,

by

(1.6),

(5.1)

P(t)

g(pe

r’)

-t- g*(p*e-’).

Let B

>

0besuchthatQ(t) isregularfor

Re

>_

B.

Considerthe following paths

W,

W,

W.

inthecomplext-plane.

W

consists of ahalf line from 2ri

q-

up to the point 2ri -{-

_B,

the line segmentfrom2ri

--

B

to

B,

and the real axis from

B

to

-t-W.

isthe contour ofarectangle, taken in positivedirection, withvertices

B,

2ri

+

B,

2ri

B,

--B.

Wa

consists of the segments -o _to

-B,

-B _{to 2ri} B _{and 2ri}

B

_to

2ri

Considering

e-’(*)Q(t)e

’

as ourintegrand, we immediately find that

(5.2)

,q-L.q-L.

f:i’_+:

(1

e

-2,’))(z).

For

on

W1,

andBsufficientlylarge,we havean expansion

THE ROOTS OF TRIGONOMETRIC INTEGRALS 211

(5.4)

convergingabsolutely and

_

uniformlywithrespecttot.

Hence

,

(exp

(-(r’)))e

-’’/’"’

d.

The integral

f.

(exp

(--g(pe’)))e

’

dt

(5.5)

convergesforany complex value ofz.

To

evaluate it, firstsuppose

Im

z

<

0,

then shift thevertical part of the path

W

infinitely to the left.

It

follows that

(5.5)

equals

(1

e

-)

f(R)

(exp

(--

g(pve)))e

’

dt andhence

(5.6)

(exp

(--g(pe2C)))e

’’

dt

N-(1

e-")pT’/nH(izN

-

1),

w

where

H

is the function introduced in

(4.1).

Since

H(s)

is regular over the whole plane, with exception of simple poles at s

-k/N,

k 1, 2, the

right side of

(5.6)

isan integralfunction, and

(5.6)

holdsfor allz.

We

candealwith

.

in thesame way,and after that

(5.2)

and

(5.3)

leadto expansion 1

,,+,,/H(.iz--v

)

(5.7)

+

#’*P*(’

N

--1

T

1--e

fw.

e-"t)Q(t)e

’t dt---

l(z)

+

3(z)

-

.(z).

Thisformulaholds for all values of

z,

with theexception of the points z ik, k 0, :1, :t:2,

We

are nowable to prove

THEOREM

18.

_If

P(t)

and

Q(t)

satisfy theconditions

_of

Theorem2,and

_if

weput

(5.8)

(z)

N-l_:p(’/K)lNH((iz

+

K)N

-

1),

thenwehave,

for

O,

(5.9)

(z)

(z)(1 -t-

o(1)}

uniformlyinthe half-plane

Im

z

_

-,

and

(z)

hasbuta

_finite

number

_of

roots

212 N. G. DE BRUIJN

Proof.

We

first consider the point-set

R

defined by the inequalities

(5.10)

_-IRe

z

_<

Im

z

_<

-e

_(I

z

>

A).

(On

generalizing Theorem 16 it is possible toextend our considerations to the regions defined by the set of

_inelualities

IRe

z

>

1,

Im

z

_<

-e, z

_>

A.

The power of the Phragmt!n-Lindelhf theorem applied below however enables

us to restrict ourselves to the smaller regions indicated

above.)

Here A

is

chosen sufficiently large to suit some conditions indicated in the followihg.

By

Theorem 16, thefirst seriesof

(5.7)

satisfies

(5.11)

l(z)

(z)

_<

_"P.-KPN-I

(K+)/iVfl+v/1VZ--(K+v)X.Zl

/(Z)

-K+I

<

C

[z

1-1/2V

((z)I,

if

A

is sufficiently large and z

R.

(C1

C2,

may depend on

N,

K,

p

;,

but not on

z.)

Analogously

(5.12)

a(z)

<

z

Furthermore wehave. again by Theorem 16, taking p (iz

iz*)/N,

N

1

H

N

1

<C4]z[

-2’/N

H

iz+K

N

-1

for z

R

and

A

sufficiently large.

Hence,

by

(5.11)

and

(5.12)

(5.13)

&(z)

+

&3(z)

(z){1

+

o(1)}.

We

now turn to the third part of

(5.7),

i.e., the function

&2(z).

It

follows from the regularity properties ofQ(t)

(see

Theorem

2)

that the path

W2

canbe reducedto the path

W:,

consisting ofarectanglewithre,ices

B

_W

(

T

3 argpN)i/N;

B

₊

2i

+

(--

arg pN)i/N;

--B

+

2i

+

(--argp)i/N;

B

(

+

argpN)i/N for a certain positive number The rectangle being independent of

z,

wefind, for

Im

z

-,

_<

_C

exp

{B[Imz[

(1/2r

+

i)

[Re z/N[

+

argp

Re z/N}.

From

Theorem

17,

with c

_>

BN,

<

,

we noweasily derivethat

(see (5.8))

(5.14)

(z)

(z)o(1)

([

z

[--)

uniformlyinthe region considered.

It

follows from

(5.13)

and

(5.14)

that

(5.9)

holdsuniformly for z

R.

In

the regionR, defined by

Im

z[

<

[Re zl,

zl >

A,

we have, by Theorem 10,

(z)

0(exp

[z

Ix),

<

2.

On

the boundary of

R,

we have

THE ROOTS OF TRIGONOMETRIC INTEGRALS 213

(z)

(z){1

-t-

o(z)]

andfurthermore

I(z)

>

1 for z in

R

providedthat

A

has been chosen sufficiently large.

It

follows, by a well-known theorem of Phragmdn-LindelSf, that

(5.9)

holdsuniformlyin

_R

Since

H(s)

has only a finite number of roots for

Re

s

>

-b, which follows from Theorem 17, now Theorem 18 is completely proved.

Hence

Theorem 2

istrue

(see

Introduction).

We

nowturn to the proof of Theorem 1.

In

the first place we deduce from Theorem2and Theorem 11, by an argumentexplained in theintroduction,

THEOREM

19.

/f

P(t)

_

pe

(N

>

O, Re

p

>

O,

p* p_) and

_if

all the.roots

_of

itsderivative

_P’(t)

arepurely imaginary,then the

_function

dthas real roots only.

We

easily infer

THEOREM

20.

_If

all the roots

_of

the derivative

_of

the polynomial

f(t)

qt

(N

positive and

even)

are purely imaginary, and

_if

qzv

>

0

_(so

that

f(t)

is

real

_for

purely imaginary), then the

_function

(z)

f_

e-()e

_’

dt has real roots only.

Proof.

We

put

f’(t)

NqN

I-

(t

ip,), p,real, and for

>

0,

N-1

Nqz

I

ksinh

(t-

ip,))-’

x(t),

f

Cx(r)dr

hx(t).

zo

cx(t)

hasthe form

N-1

(5.15)

ce’/x.

--N+I

Since

N

iseven wehaveCo 0;hence

Cx(t)

also has the form

(5.15). It

iseasily verified that

Cx(t)

is realfor purely imaginary vlues oft, and furthermore

x(ht)

2N(N

1)-l(1/2}k) Nque-"’/x

e.v-,)t

₊

....

hr-1

It

follows,

that,

for

>

ho 2r

-

I,

p.

1,

the function

Px(t)

satisfies the conditions of Theorem 19, so that

x(z)

f-

e-t()e"

dt has real roots only.

It

remains to beproved that

(see

Theorem

7)

(5.16)

lim

x(z)

(z)

,-+

uniformly in anyboundeddomainofthe z-plane.

We

candetermine positive_{constants i}

<

r/2N

and

i

suchthat

(5.17)

(sinh

w)/w

>

1/2,

arg (sinh

w)/w

<

1/2N,

for any value of w satisfying argw

_<

_{i and}

Im

w

_<

i. Furthermore wecanfix positive numbers

A

and

A

such that the numbers w

(t

ip)/h lie in that region for all vlues of

>

A,

k

> A

and u, 1

_

u

_

N

1.

It

214 N, G. DE BRUIJN

nowfollows from

(5.17)

that a positive constant c existssuch that

_{Re (t)}

_>

ctN-lfort>

A1,X

>

A.. Hence

(5.18)

lim lim sup

e-*(t)e

zt

dt =0

uniformly in any bounded domain ofthe z-plane. The same

holds,

of course, for

f:.

Since qn

>

0,

N

even, wealsofindthag

(5.19)

lira

e-te"

de 0

uniformlyin any bounded

domain,

andtheamefor

J’;.

For

any fixedvalue of

A

wehave

(5.20)

lim

e-*X(’)e

’’

dt e-f(’)ez’_dt

-co A A

uniformly in any bounded domain, because

(t)

--

f(t)

uniformly in

--A

<

t<A.

From (5.18), (5.19)

and

(5.20)

we infer

(5.16)

andourtheoremisproved.

Proof

of

Theorem 1. Suppose that

f(t)

satisfies the conditions of Theorem 1. These conditions can also be expressed in the following form

(see [4])"

f’(t)

is of thetype

f’(t)

aet*t*+

H:I

(1

+

2vt2

)

wherea

>

0,b

>

0, kaninteger

>

0,

.

_>

0, r 1, 2,

---,

₀

<

o.

Now

let

f.(t)

be defined by

f’(t)

a(1

A- bt2/n)"t

+’

I-’

(1

-I-

it,t2),

f,(0)

f(0)

andput

(z)

f:

e-’(t, eTM dr,

n(z)

f:(R)

eTM eTMdr.

It

iseasy tofindpositive numbers

A,

and no, such thatforn>

no,t

>

Aort

<

-Awehavef(t)

>

d

andfn(t)

>

d

.

Since

f,(t)

f(t)

forn

--.

,

uniformlyinanyfinite t-interval, we noweasily inferthat

(see

proof of Theorem

20)

(5.21)

lim

4,=(z)

q,(z)

uniformlyin anybounded domain of thez-plane.

It

is easily seenthat the polynomials

f=(z)

satisfy the conditions of Theorem 20,whence itfollowsthat

(I).(z)

has real roots only. Application of

(5.21)

and Theorem 7completestheproofofTheorem 1.

6. Functions of the type

(1.9). We

considerthe functions of the type

(1.9).

In

connection with the Riemann hypothesis

(see 7)

it may be important to investigateclassesof functions of thistype.

In

the following we shall prove some results concerning these functions.

Very

little of the preceding sections will be needed

here,

since the reality of the roots of

_f:

C(t)e

TM dr,

C(t)

exp

(-h

cosh

t),

(special case of Theorem 1 or

19)

was already proved by P61ya

(see (1.2)).

Only Theorem 25 requires asymptotic expressions and stronguniversal factorsas itstools.

THE ROOTS OF TRIGONOMETRICINTEGRALS 215

LEMA

2.

_If

U(z)

and

V(z)

are real polynomials such that

W(z)

U(z)

"F

iV(z) hasnrootsinthelowerhalf-plane,then

U(z)

hasnpairs

_of

conjugatecomplex roots at most.

(The

case n 0 is the well-known theorem of Hermite-Biehler

(see [10;

Abschn. 3, Aufg.

25]).)

Proof.

We

may assume that

U(z)

and

V(z)

have no real roots in common,

so that

W(z)

has no real roots.

We

also assume that the degree m of

W(z)

satisfiesm

>_

2n, for otherwise the result is trivial.

Now

if zrunsthrough the real axis from -oo _{to oo, the} argument _of

W(z)

_{increases by} an amount of

r(m

2n). Hence

there areat least m 2n different points on the real axis where

W(z)

ispurely imaginary. If z 0o isnotone of these points, wethus

find m 2n real roots of

U(z)

at least, and otherwise at leastm 2n 1.

But

in thelastcase thedegreeof

U(z)

must be

_<

m, so that

U(z)

has atmost 2ncomplexrootsinbothcases.

THEOREM 21. The

_function

(1.9)

(which has but a

_finite

number

_of

non-real rootsonaccount

_of

Theorem

2)

has

N

pairs

of

conjugatecomplex roots at most.

Proof.

Thefunctions, where

C(t)

exp

(-

cosh

t),

>

0,

(6.1)

(z)

C(t)e

’z’

dt

(6.2)

q(z)

(C(t)

cosh

t)e

dr, q(z)

(C(t)i

sinh

t)e

’

dt have real roots only (Theorems 1 and

11),

whereas the roots of q(z

)

ql(Z) iq.(z) lie in the upper half-plane.

By

partial integrationit iseasilyseenthat, for k 0, 1, 2, the functions z

q

(z)

andz q.

(z)

are of the form

k+l

z(z)

C(t)

=.e

e dt

(1

1,

2)

-k--1 ()

1, 2,are 0. Further-where

a

().

a

()_.,

andthehighest coefficientsa+.,

-()

isrealifk isodd,and purely imaginaryifk

+

iseven.

It

follows

more an,

thatthe function

(z),

given by

(1.9),

canbeexpressed as alinear combination of

(z),

i(z),

z(z),

z-(z),

(z),

z-(z),

with real coeffi-cients. Since

z(z)

-h(z)

wehave

(6.3)

z(z)

A (z) (z)

₊

B(z) (z),

where

A (z)

and

B(z)

arereal polynomials ofdegree

N

at most.

Now

if

F(z)

and

G(z)

are real polynomials, of arbitrary degree, with the property that the roots of

F(z)

iG(z) all lie in the upper half-plane, then

A(z)F(z)

+

B(z)G(z)

has at most

N

pairs of conjugate complex roots. This follows by application of

Lemma

2 to the function

U(z)

iV(z)

(A(z)

₊

216 N.G. DE BRUIJN

iB(z))(F(z)

iG(z)).

On

approximating

(z

i)

l(z)

i2(z)

by poly-nomials

F(z

i) whose roots also lie in the upper half-plane, we find that

(6.3)

has

N

pairs of conjugate complexroots at most.

In

connection with the Riemann hypothesis it appears to be important to findlarge classes of functions

N

(6.4)

ane"

(a*

a-n)

-N

withthepropertythatthe integral

(1.9)

has real rootsonly.

For

instance, this followsfrom Theorem,ll, ifall the roots of

(6.4)

lie on the imaginary axis; but this result does not help much in the direction of the Riemann problem, which seems to berelatedto functions of thetype

(6.5)

C(t)

(,

₊

cosh

t)e

’

dr,

n=l

where p are real and 1.

In

the following we establish some new

results, which do not help either, but which may serve s material for obser-vation.

THEOnnM

22.

_If

_>

O,

O,

theintegrals

(z)

_f

C(t)(

+

cosh

t)e

TM

dt and

(z)

f:

C(t)(

+

cosh

t)e

dthave realroots only.

Proof.

(The

reality of the roots of

(z)

can also be proved as follows.

It

follows from Theorem 21 that

(z)

has t most one root in the upper half-plane. Since

_T

(z)

is real andeven, this possibleonemust be purely imaginary.

But

since the integrand is positive forz purely imaginary sucha root does not exist. The same argument is used in the proof of Theorem

23.)

By

partial integration we easily express

T

and in terms of the functions

(z)

and

(z)

definedby

(6.2)"

(6.6)

z,(z)

z,(z)

x,(z),

(6.7)

z2(z)

z(z)

+

{z

2(1

₊

)}2(z).

The roots of the polynomials z

ih

andz

+

i{z

h(1

₊

t)}

liein the upper half-plane and hence, by the lemma,

(6.6)

and

(6.7)

have real rootsonly

(see

the proof of Theorem

21). We

noticethat the same can be said if -1

_

p

_

0, for thenp cosh and

-

cosht

areuniversal factors.

On the other hand it is not difficult toshow that both integrals have a pair ofpurely imaginary roots if

-

ispositive and sufficiently large.

The following theoremisobtainedbyageneralization of themethod employed above. Theresult, however,seemsto be too complicated for application to any wide class of polynomials.

THEOREM23.

Let

be

_>

0 andletf(y) beareal polynomial, suchthat

~(z)

:

C(t)f (cosh t)e

i

THE ROOTS OF TRIGONOMETRIC INTEGRALS 217

fl(Y)

Yf(Y)

T

f(Y)

X-’f’(y)

₊

X-2f"(y)

-

_if

>_

O.

<

O,

providedthat

_fl

(y) does not change sign

_for

y

>_

1.

Proof.

Putting

q,(e)

(C()f(eosh )eosh

)e’"

d,

It

isalso true

_for

(C(t)f(cosh

t)isinh

t)e’"

dr,

we findby partial integration

(6.8)

z

C(t)fl(cosh t)e

’’

dt

z4(z)

X(z).

Furthermore, wenoticethat the roots of

-(z

i)

(z)

i:(z) all have imaginary part 1.

For

>_

0 the function

A (z)

iB(z)

z

iX has no

roots inthe lower half-plane sothat theargument used inthe proof of the pre-ceding theorem shows that the function

(6.8)

has real roots only.

If

<

0 we inferthat

(6.8)

has onepair of conjugate complex roots at most. Since that function isan odd function of

z,

these possible roots must bepurely imaginary.

But

if

_f

(y) does not change sign for y

>_

1 and if z is purely imagi-nary, then

C(t)f(cosh t)e

TM doesnot change sign for

<

<

.

It

follows that the integral does not vanish.

The following applicationmay be of some interest. The functionf(y) satisfies the conditions of Theorem 23, for

(6.1)

has real roots only and

yN

(cosh t)

N

isa universalfactor. ConsequentlyTheorem 23 showsthat, forX

>

0,

(6.9)

C(t)

1

+

X cosh

+

2

+

+

N

hasreal roots only.

THEOREM 24.

Let

the polynomialf(y)

of

degree

N

have negative roots only, and letX beanumber

>_

1/2N.

Then the

_function

~(z)

has realroots only.

It

may be surmised that the condition X

>_

1/2N

can be replaced by a much weakerone.

Proof.

We

may assume thatf(y)has noroots for -1

_<

y

_<

1, the factors y a cosh a

_(-1

_<

a

_<

₁₎

being universal factors

(Theorem 11). We

now proceed by double induction, in the first place with respect to the degree

N

off(y)

(the

theorem is true for

N

0)

and secondlywithrespecttoanumber n,the smallestpositive integer with thepropertythatf(y) has atleast oneroot exceeding -1

n/X. We

shall reduce thecase

(N, n)

either tolower

N

or to lowern;the case

(N,

0)

will always be reduced to lower

N.

Suppose

thatf(y)

(of

degree

N)

has negative roots

<:

-1 only, the largest of which, p say, satisfies-1-

n/X

<

p

<_

-1-

(n- 1)/X,

n

>_

1.

Now

218 N.G. DE BRUIJN

consider the polynomialfl(y) f’(y) Xf(y). Sincef(y) has real roots only, the same applies tof(y).

Now

f(y) has no roots for y

_>

1 and exactly one

rootpsatisfyingp

-t- -1

<

p

_<

1. Namely,f’(y)/f(y) decreasesmonotonically for y

>_

p.

For

y p

-t-

h-i

we havef’(y)/f(y)

>

(y

p)-I

h, and, since

theroots off(y) aresupposed to be

<

-1, theinequality

f’(1)/f(1)

<

1/2N

_

holds.

Now

ifp

<

-1,the polynomial

_f

(y) belongstoacaseoflowernand hence

(6.10)

f_

c(t) f(cosh t)e

’zt

dt

issupposedto have real rootsonly. If,

however,

-1

_<

p

_<

1, the polynomial f.(y)

f(y)/(y

p)is ofdegree

N

1, anditfollowsthat therootsof

(6.10),

with

_f2

instead off, arereal.

But

sinceinthat case cosh p isa universal factorthesameapplies to

(6.10)

itself.

Again applyingauniversal

factor,

we findthat

i

C(t)f(cosh

t)sinh e’zt dt z

C(t)f(cosh

t)e

’zt

dt

hasreal rootsonly. This completes ourinduction.

We

have notyetbeen able to generalizeTheorem 24 to cases where

f(cosh t)

isaninfiniteproduct of factors 1 c,cosh

t,.

c

>

0. Suchanextensionmight be given perhaps by carrying outasuitable reductionprocessand using Theorem 25below.

In

Theorem 26 weshallusesuchamethodin adifferentcase, where

areduction processcanbe found indeed. Theorem 26 may be of some interest sinceitgivesa result ofa typeweshould like tohave for functions of the form

(6.5).

THEOREM25.

_If

O,

0 8

.,1

and

_if

n is anatural number, then there

existsapositive number

A(h,

8,

n)

withtheproperty that theroots

_of

(6.11)

(z)

f_

C(t) f(cosh t)e

’zt

dt

lie inthestrip

Im

z

<-

A

(,

8,

n)

for

any real polynomialf(y)

_of

degreenwhose roots lie inthesector

1/2

<_

arg y

_<

i.

(It

is possible toprovethesame for the region consisting of the real axisand circle

_zl

_<

A(),,

8,

n).

This can be done by introducing a denominator

(e’ -t-

2

-

e

-t)

_(see

theproofofTheorem

2).)

Proof.

Thefunction

(6.1)

canbedevelopedasfollows:

(z)

(- 1)

r(iz

)

+

2.

(- 1)

r(-iz

)

,--0

.

(1/2x),z-,

-o

!

_(1/2x)

(see (5.7)

or

[6]). It

follows that for

Im

z

<

-1 wehave

(z)(1/2x)’/r(iz)

1

<

z

I-’A

(),