Riemann's hypothesis as an eigenvalue problem. III

Wfe!%b~ann’s Hypothesis as an Eigenvalue Problem. 111 Friedrich Roesler Mathematiches lnstitut der Technischen Uniwrsitit Arcisstra/3e 21 8000 Miinc...

Download PDF

2MB Sizes 0 Downloads 62 Views

Report

PDF Reader
Full Text

Wfe!%b~ann’s Hypothesis as an Eigenvalue Problem. 111 Friedrich

Roesler

Mathematiches lnstitut der Technischen Uniwrsitit Arcisstra/3e 21 8000 Miinchen 2, Germany

Submitted by Bicbard A. Brualdi

ABSTRACT We give conditional induction proofs for the existence of a small zero-free strip inside the critical strip of Riemann’s zeta function J(s). The starting point is some formulas for the eigenvalues A of certain matrices A, over the integers, whose determinants are connected with Riemann’s hypothesis by the equation det A, = N!C 1l-&.

0.

Ik I’RXWCTION The purpose of this paper is to prove the following conditional

tfNw =

estimates:

c -Cl(n)=O(N’“), l
n

and, what is equivalent (via partial summation; cf. [6, Theorem 411,

Rlv(l)=

c l
p denotes

the Wbius P(

for prime powers

p(n) -=O(N--“). ’

function, and p is multiplicative

P”) = ( - d”

l
(Pk

with

-1)-l

pm. ITS A~~~~ATZ~~~ 141:1-46 (1990)

0 Elsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas,New York,NY IOOPO

OO24-3795/90/$3SQ

FRIEDRICH

2

ROESLER

Estimates such as (0.1) and (0.2) imply the existence of a small zero-free strip inside the critical strip of Riemann’s zeta function l(s). In particular,

fN(O) = o( N- 1’2+E)

for every

e> 0

is equivalent to the Riemann hypothesis [8, Theorem 14.25(C); 5, Theo= rem I.]. The unproved conditions under which (0.1) and (0.2) are shown here are assumptions concerning the characteristic polynomials XJ x 1, and in normalized form

RN(X)=xN~~)2<~
. .

of the matrices

and assumptions on the position of the eigenvalues A,,. . . , A,_ r of the matrices A,. More precisely, these conditions refer to properties of x&) near x = 0 and/or the existence of small eigenvalues A of A,. Section 1 repeats results of [5 and [6] and establishes equations for the inductive proofs of the following sections. In Section 2 we assume the existence of small eigenvalues to prove (0.1) and (0,2). A typical result is, for sufficiently small e > 0 (Theorem 7): If for every N 3 No at least one of the polynomials X,&X), N G M G N -I- N1 -E, has a zero A such that - 0.09 < A < 1.04, then RN(O) = O( N’“) and RN(l) = OW9

In Section 3 we discuss the sequence of quotients

N=2,3,...

This sequence probably does not converge, elements should on average lie near

.

but Theorem

l( m) = 2.294856. . . .

&:= 1113 2

(04 8 suggests that its

RIEMANN’S HYPOTHESIS

3

Then we assume irregularities of this sequence and prove (0.1) and (0.2). A typical result (Theorem 9(l)): If - 13.706 Q yN < 0.482 sufficiently often, then RN(O) = C(N-E). In Section 4 we assume irregularities in the sequences

co)

RltW

KN

:= f&O)

’

cN := &,,(I)

and then do the same (Corollary If IK,I z 0.520 sufficiently If krNl >, 1.990 sufficiently

N= 2,3,...,

’

u-w

11):

of Theorem

often, then ,&(O) = O(N-&). often, then RN(l) = O( N-‘).

Mere again it seems unlikely should on average lie near

that

or (GN)N,

(K~)N,,

1

converges,

but

KN

1 -

= - 0.392073.

-

WI

..,

l

and UN near

c (l- mn> n &n))=-1.989548...

f1> 2

(Theorem 11). The definition (0.5) of KN and UN refers to RN(x) and its and (gNjN, 1 derivative at x = 0 and x = 1, so that irregularities of (K~)~,, can be explained by the existence or nonexistence of small eigenvalues of the matrices AN, for we have

KN

1

c

=

--

2
0, =

c 2
-n

1 c

-9

A A

1 -1

1 c h h-l’

where A runs through ah eigenvahres of ‘4,. It may be noticed that the investigations are concentrated at the points x = 0 and x = 1 of x&r). The reason is: A recursive exp$nsion of the determinant of the matrix XI - A:j yields ,&(x)- ,&_,(x)’ h,(r) for the normalized characteristic polynomial [S, Theorem 3(3)] with h ,jO) = p(n)/ n,

4

FRIEDRICH ROESLER

i;,cl> = phj/n [thus proving

the left equations in (0.1) and (0.211, and the functions h,(x) are multiplicative in N if and only if x = 0 or x = 1 [6, Theorem l(l)]. Moreover the point x = 1 seems to be more appropriate for investigations than x = 0, for x = 1 reflects in a simple way all the polynomials X&r):

dn) x~(n) =(-l)N-“N!~~~N,,l(l), [5, Theorem

(0.6)

4] and

S(2)]. The hmction

r(p”‘)

4n)

c

&(r)=(1-x) [6, Theorem

l,
=

&N/n](l)

r is multiplicative,

n

(l-

(0.8)

p-k)+,

I Q k < rn

and [6, Theorem

(0.7)

3(l)]

(0.9) p is related to T as p is to the constant

function

1:

(0.10)

66, Theorem 21, and hence

(0.11)

The conditions of the theorems in the Sections 2-4 can be regarded as generalizations of conditions on sign changes (for instance, yN < 0.177 in heorem 9(3) of Section 3: y,,, < 0 would imply that RN(O) and RN(l) have rent signs, i.e. that there is an eigenvalue of A, in the interval [O,l],

RIEMANN’S

HYPOTHESIS

5

which is a condition of the type in Section 2). The study of sign changes in connection with the Riemann hypothesis has a long tradition, starting with Gauss’s conjecture and Riemann’s statement in his famous paper [4] of 1859 that for x > 2, one has A,(r) := r(x)Ii x < 0 (cf. [2]). More intimately connected [for instance via (0.411 with our conditions are the sign changes of

M(x) = c

p(n)

and

M,(x)

=

ndx

c

PW

-

n
n

’

which have been studied by K&tai [l] and Pintz [3]. These papers indicate that the conditions here must be truly deep. This may suggest that examining x,(x) for small x is closely related to the classical study of the Riemann zeta function. Therefore it should be mentioned that it is by no means only the study of xw(r) near 0 that can exhibit appropriate information for proofs of the existence of zero-free strips for t(s). It certainly is unlikely that any knowledge about the large eigenvalues (as for instance in Theorem 6 of [6]) can furnish enough information to prove estimates like (0.1) or (0.2). But appropriate information on the real eigenvalues of A, in the neighborhood of fi can lead to proofs of (0.1) and (0.2): The “normal” situation is [5, Theorem 5] that each interval [n, n + l[, 1~ n < N - 1, contains exactly one eigenvalue of A,. But if, for sufficiently many numbers N, there are intervals [n, n + l[ near v% without eigenvalues of A,, then (0.2) is true. And the nonexistence of eigenvalues in [n, n + 1[ results from the existence of pairs of complex eigenvalues, which in turn can be caused by sign changes of the sequence (&,,(l))N,, (cf. the introduction of [61). The Appendix, which is partly due to H.-J. Toussaint, contains four tables with numerical examples illustrating the theorems of Sections 2-4. Numerical constants:

L:=

dn) n c --p-=

Pm :=

c

=2.294856...,

l(m)

m 3 2

n >, 1

s(m)

IPWI

n2

=

I

=

2.072956...,

n21

7,

:=

g(m)-1)=1.989548....

FRIEDRICH

6

1. INDUCTION

ROESLER

FORMULAE

The equations

f,(O) =

c

0.2)

IgndN

[6, Theorem 41, which connect the values of the characteristic polynomials x&r) at x = 0 and x = 1, can be interpreted as limits of similar equations which arise when we pass step by step from

c l-44 = 4,”

W)

cl I n to

[6, Theorem

21, via the functions

p.(‘)(n):=p( n),

&p+y

n)

c

:r

ptk)(a)p(“( 6)bsk,

u , Ii b 1, ub o- n

T(‘)(n) := 1, Induction

+-+I)(

n)

c

:zz

rtk)( u)T(“( 6)6-“.

on k, starting with (1.3), shows immediately

(1.5)

and we have lim p (k) = P* k-+CO

lim P)

k+m

= 7;

(l-6)

RIEMANN’S HYPOTHESIS

7

hence (1.4) is the limit of (1.5) for k -00. To prove (1.6) we notice that by definitio-n of JL(~),

c

nal

which yields by induction

c

ptk+‘)(n) nS

n2l

n

=

J(s+m)_’

0.7)

Odmdk

and, by (0.9) and (o.ll),

c (pJI/&))cn,)fi I-I 5(s+nq’= c m&O

nbl

nbl

f”@. ns

This implies equality for the coefficients cf the Dirichlet series, which is the first assertion in (1.6). The second one follows in exactly the same way. THEOREM1.

For all k >, 1,

c

0)

l
c

(2)

IgntN

and

(1. l), (1.2)

Proof.

-TTyR[N,“]( n

1) =

c l
R[N,n](O)~ “’

PW ~&N,nJf”) n

result from

Equation

dk)( n)

W

=

these

(1.1) shows

c I
equations

for k --) 0~.

FRIEDRICH RBESLER

8

and

since

-1

rs r>l =

44

iE

--i;-PC4

mn=r

=

n

n

I

l(s -I- m)-’

[by (0.9) and (O.ll)]

O
=

4 c dk’( rs

bY

(WI

l

fB1

The second equation

n

follows in exactly the same way.

TII~WWM 2 (Interpolation theorem). Suppose tb~t A,,. . ., A, are zeros of ,y,Jx), which need not be difirent. Then for S = 0 and S = 1,

i[N,n](

l)

’

Proof. Lagrange interpolation of the polynomial xN(x>13~ JX - Aj)-’ of degree N - M-l at n= M+l,M+2,...,N shows M

(X j-1

XNW Ai)-’

=

Ml-l,
n

lcj
(n_Aj) M+

RIEMANN’S HYPOTHESIS

and by (0.6),

Hence

&,,(x) =( -l)N-l

l? 0j-4

\j=l

and, by definition of RN(X),

XN(o) = ( -l)N-*N!&,(o),

THEOREM

3.

xN(l) =(

-I)“-‘(N-l)!fN(l).

For all x in a &neighborhood of 0, 6 < 1, where xN(x) has

no zeros,

with KN,k

=

c 2GndN

n-k

-

CA-k, A

and A runs through all zeros of XN(x)# Proof.

2GlnGN

(x-m,) c 29rn~NX-m

J-,

FRIEDRICH

10

and hence, by definition

of RN(x),

= 2,(m
Now integration

hence

yields

ROESLER

RIEMANN’S

HYPOTHESIS

11

and the coejFficients m-e determined by the equation

In particular, with K,, FN,Q

= KN,k + 1, =

F N.l =

1, KN,,,

F N.2 = 2‘@i.,

Proof.

+

I(,.d,

On one hand we have by (0.7)

On the other hand, by Theorem 3, iNtX) =

l-x

(

iN( 0) exp log

efinition of H;iv. k

an

,

FRIEDRICH ROESLER

12 In particular,

the first equation

of Theorem

4 gives for z; --) w

COROLLARY. limk ._, FN k = RNW/&JO). REMARK. By definition

c

of the coefficients

c

FN,kXk =eXp

k>,o

$(l+KN,J)xk

kal

=

=

j-&exp( z,

iKN.kxk)

-

( c xk)(c kao

=

fN,k in

=

.frv,k)xk

k,
the same manner

F N.K

frv.kxk)

k>,o

c (c

KaO

when we define KN,j + 1. Hence

FN_k we have

c O=zk
as FN,k but with

KN,~

instead of

fN,kT

and, by the corollary,

Cf

k a 0

2.

CONDITIONS

RN(l) N.k = RN(O)

*

ON SMALL EIGENVALUES

THEOREM 5. If E > 0 is sujjkiently small, and if for every N a No at least one of the polynomials xJx1, N G M G N + N1 -‘, has a zero A such that 0.305 < A < 1.342, th

,&J(1) =

(N-')

for

N -+m,

RIEMANN’S HYPOTHESIS

Proof. XL&)

13

The interpolation

Theorem

IZ*W=(~-1)

2 with 6 = 1 yields for zeros A of

44

c

R[iv,n]( 1) 2

2
and a numerical

calculation

IA-llnE2

nCn-

(2.1)

A)

shows

n;n(l\,

if

0.304
Hence there exist positive er, e2 such that

]A-llnF2

r(n) n,_F,(n_A)

<1-sE2

We choose E < sr, N,, arbitrary,

if

0.305~A~1.342.

and Nr large enough that (2.3)

N, 2 N,,, r(n)

c

(2.2)

< 2&N’-’

for

N> N,,

(2.4)

N
which by the Lemma in the Appendix of [6] is possible, and (l+

N;1)E<(1-~2)-1.

(2.5)

c 2 X&4!,

(2.6)

Then we choose C such that

and Ca(N+l)Ei&,(l)I

for

N= 1,2,3 ,..., N,.

(2.7)

Now we prove by induction 1&(1)(~C(N+i)-~

for

N>,P.

(2.8)

FRIEDRICH ROESLER

14

This is true by (2.7) for N < N1. Suppose now that N > N1 and that (2.5) is true for all M < N. Then N > No by (2.31, and by assumption there exists a zero A E [0.305,1.342] of x&j for some M E [N,N + N'-"1. We have

by (2.1), and [M/2] hypothesis gives

< N, since

G

M < N + N1-’

If

1-. &2)

mr’(

2 N. Thus the induction

44

CM-"[A-11 c

n’-“(n-h)

2
<

by (2.2) and E
M = N,then (2.8) follows from this with (2.5) and N 3 N,. If M > N,then PW c N
IRnr(lkMl)I= G

c

dn)

Ncn6M

bY

(0.2)

by definition

of

p

and 7

n

C dn) NC rtc M

~2?~~(N-t-l)-”

by(2.4)

and

Hence we get

1~1~(1)I~IR,(l)I+2~~(N+l)-’ GCM-‘(~-E,)+~&(N+~)-’
by (2.6).

M< N+ N"".

RIEMANN’S HYPOTHESIS REMARK.

completely

15

The proof shows that the admissible by the inequality

h-11

‘(‘)

c

nln-A/

,,a2

A-interval is determined

All induction proofs here will be performed according to this method, and the assumptions of the theorems will be the same in principle. Therefore we formulate them more pointedly-and less precisely-in this way: If sufhciently many polynomials x&x) have zeros in [0.305,1.342], then RN(l) = O(N-&).

THEOREM 6.

If

sufficiently many polynomials x&l

have nonreal zeros

A such that IA -0.4261~

1.016,

then RN(l) = O( NV”).

Proof.

Suppose A to be a nonreal zero of x&r).

Then, by (2.11,

’

?b)

c

R[N,“)(

I
1)

= O¶

n(n-A)

and we also have the same equation with x instead of A. The difference of these, multiplied by II- A12(A- i)- ‘, is

-RN(l) = b-At2

c

2,
Hence the induction condition

proof

II-

Al2

of’ Theorem

w 2R[N,n]W*

nln

-

Al

5 can be copied

Riemann's hypothesis as an eigenvalue problem. III

Riemann's hypothesis as an eigenvalue problem. III

Recommend Documents