Summability methods based on the Riemann Zeta function

Internat. J. Math. & Math Sci. VOL. II NO. (1988) 27-36

27

SUMMABILITY

METHODS BASED

ON

THE

RIEMANN ZETA

FUNCTION

LARRY K.

CHU

DEPARTMENT

OF

MATHEMATICS

AND COMPUTER SCIENCE

STATE

UNIVERSITY OF NORTH

DAKOTA

MINOT

MINOT, ND

58701

(Received September 12, 1986)

ABSTRACT.

This

paper

isastudy of summability methods thatarebasedontheRiemannZetafunction. A limitationtheoremisproved which givesanecessarycondition forasequence xto be zeta summable. A zetasummabilitymatrixZ associatedwithareal

sequence

is introduced;anecessaryand sufficient conditiononthe

sequence

suchthat

Z

maps

11

to

11

is established. Results comparing the strength of thezetamethodtothat of well-known summability methodsarealso investigated.

KEY WORDS AND PHRASES. Zetasummability method, zetamatrixmethod,I-Imatrix,Cesaromethod, Euler-Knoppmethod.

1980AMS

SUBJECT CLASSIFICATION

CODE. 40C05, 40C15, 40D05, 40D25.

1.

INTRODUCTION.

Recall thatthe Riemannzetafunction is given by

(s)

k

(

!/k

’)

rot

>

Anumber

sequence

issaidtobezetasummable to

L

(or

-summable

to L)provided that

(T’schmarch [1 ]).

L.

0,

Thezetamethodisa"sequence-to4unction"summabilitymethod whose domain consists of those sequence xsuch that the Dirichlet’s sedes

;-1 (xk/k)

isconvergentfors>1.

Inthe second sectionit isshown that the zeta summability methodisregular andtotally regular (preserves finite and infinitelimits). Alimitation theorem is proved which givesa

necessary

condition fora sequence x tobezetasummable.

In

section3weintroducea zetasummability matrix

Z

associated witha realsequencet;a

necessary

and sufficientconditiononthe

sequence

such thatZ maps

11

into

11

is established. The final section contains results comparing the strength of thezetamethodto thatof well-known summability methods.

For

example, thezetamethod isstrongerthan the

Cesro

methodof order but does not include the

Ces.ro

method oforder 2; thezetamethoddoes not include andisnot includedintheEuler-Knoppmethod of order for 0< < 1.

2. BASIC

THEOREMS

THEOREM1. The-summabilitymethodistotallyregular.

bouby

’,’.,

]x

L

M.

Since

5l, ]/k

oc wecarlchooseN₂

>NlSOthat

-,.--1

1/k

>

(2M#)--1.

Nowchoose6suchthat0<8<log[l+(1/N2)]/IogN2.

Then for each k <

N2,

we

have

k’<’k

)(’’l’(;N")!/)u

N’

<_

+

(I/N2)

andif <s<l+8

(l/k)--(I/k’)

<

(k

-l)/k’ <

k

‘-

<

l/N.

Summingfromk to N2,weobtain

2M

Thus for <s< + 8, N2

.,) .>

v

k

k-l

2M

.--.

and

Hence,

Xk

Cs)

,%

<

M+

2M 2

Now assume xisareal numbersequencewhich divergesto

,,,,.

Then for each number

M

>0 there existsapositive intergerNsuchthat x_k>

M

+ for all k>N.

Suppose

s > and consider

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 29 Since

r(s)

-

as

s + _{we seethat}ifsissufficiently closeto onthe right, then

<1"

this implies that

>

-1

+M+I

M

SinceM>0waschosen arbitrarily,weconclude that Xk

im V’ oo

Aprevious definition of"zetasummability"wasgiveninDiaconis[2]. Inthatpaperthebounded sequence xis saidtobezeta summableto

L

if

xi

li,n._.,+

(s--l)

E

-

L.

This is equivalenttothe difinition of thezetamethod introducedinthis

paper.

There equivalence isan immediate

consequence

ofthe fact that

lims_l

+

(s)

(s 1) 1.

Recall thataStoltz domain of angle, where 0< a< /2, isa complexnumbersetofthe form

S(a)

{w

A,’g (,,’--)

<,

d w

<}

(Powell etal

[3]).

We

shalluse avariantof thisconcept,whichweshall calla"reflected Stoitz domain of angleo"

S’(a)-- {w

Arg(w-1)

<

and

Re(w)>l}

Thisconceptisnowusedtoextend thezetamethodtooneusingacomplex-valued function limit, andwe establishthe regularity of this extension.

THEOREM2. Let S

*()

beareflected Stoltzdomainofangle x;ifthe

sequence

x

converges

toL then

The proof of Theorem2thatweshall give needs the following preliminary result.

LEMMA

1.

Forw

o + t,we

S*((z),

andw sufficiently closeto1,wehave

_<2

s ,-,.

7T

+

P(a-l)

+

(w-l/

Since the limitvalue

Iw

11

o

11

-<

sec z for w

eS*(cz),

thisprovesthe assertion. Now we proveTheorem2.

Proof(ofTheorem2).

Let

e>0. Sincex

converges

toL, we canchooseN 9-

Xk

I_ < e/4)

N

cosafor k>N1.

Let

_k=l

xk L,

M.

Since

(w)--oo

asw->1,wehave

l/(w)<

e/2Mfor w sufficiently closeto1.

Nowfor w

S*(a),

wehave

\- xk

<

,,

[xr

L

I(w)

Ik"l

N,

IXk-LI

+

P

IXk-LI

Ik’l

k>%,

Ik*l

M

+

COSO

E

k--,

Ik

<

7;

+

(’os

a,)

’2 see a,

Next

weprove alimitationtheorem whichassertsthat the -summabilitymethodcannotsum a

sequence

that divergestoorapidly.

THEOREM3. Ifa complexnumber

sequence

xis

-summable,

then for each s>1,_xn

o(nS).

Moreover,theterm o(n

s)

isthe best possibleinthesensethat the conclusion failsifnsisreplaced byany real

sequence

tosuch that

tn/n

sdecreasestozero.

Proof.

For

x-tobei-summable, x mustbe in the domain of the-summabilitymethod. Therefore

xn

/

n

converges

for alls>1, which implies that

limn(xn/nS

0. Ifnsisreplacedby n,where

tn/nS

decreasedto0, thenwe assertthat itwillnotbetruethat_xn

O(tn)

wheneverxis

-summable.

This is equivalenttoshowing that there isa

sequence

xsuch thatxis

-summable

and_xn

O(tn).

Definethe sequence xby_xn

(-1)n+lt

n, sothat

Xn

)c+l

v

E

(-n--I n=-I

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 31 Hence,

xn

lira

i.e.,xis

-summable

to0.

But

_xn

=

O(tn)

because for eachn,

xn/tn

1. 3.

ZETA SUMMABILITY

MATRICES

Definition.

Let

bea

sequence

ofreal numbers such thatt(n)> forevery nand

limnt(n

1. Thenthezetamatrix Z

[Znk

associated withthesequence is definedby

forn,k ’,2,3,

z,,k-(.(,,))k(.)

Inthissectionwemakeuseoftwo well-known theoremsinsummabilitytheory,whichweshall subsequentlyciteby nameonly;they areSilverman-Toeplitz Theorem([5]and [6])and the

Knopp-Lorentz

Theorem[7]. Itisan easycalculationtoshow that

Zl

satisfiestheconditiousof the Silverman-Toeplitz Theoremfor regularity. Moreover,

Z

istotallyregular because all of its entriesarepositive real numbers

([3]

p. 35). Wesummarizethese observationsinthe following theorem.

THEOREM4. Thezetamatrix

Z

associated with the

sequence

istotally regular.

Thenextresult isacharacterization of thosesequences for whichZ isan I-Imatrix, i.e.,Z maps

11

into

THEOREM5. ThematrixZ isan I-Imatdxifandonlyif isin

11.

Proof. Since eachrow

sequence

of the matrix

Z

is decreasing, thesetof the

sums

of column sequencesof the matrix

Z

isbounded by thesumofitsfirstcolumn entries. Thereforebythe

Knopp-Lorentz

Theorem,it isenoughtoshow that the first columnsumis finitewhenever

V

(t(n)

1)isconvergent. Thisisa

consequence

ofthe inequality

Y

< =,(()-)’

o=

(-))

whichfollows immediately from the fact that fors>1, s-1

<

()

<-

(*)

Hence

Z

isan

I-I

matrix.

Conversely,

assume Z

maps to

I1.

Sincet(n)> andlim_{n t(n)} forevery n, we canchoosea positive integerNsuch that0<t(n) < forn> N.

Suppose

isnotin

11

then

32

Now

,’,’,

(1/t

(t(n)))divergestoinfinity because of the inequality

1/

(t(n)))>_ ₍₁ 1/t(n)) asin(*). Therefore, by the

Knopp-Lorentz

Theorem,

Z

isnotan

I-I

matrix. Thiscompletesthe proof of the theorem.

4. INCLUSION THEOREMS.

Inthis sectionwe comparethe strength of the zeta method and the zetamatrixmethodstoseveral well-known summability methods. Throughoutthis sectionC_{a denotes the}Cesarosummabilitymatrix of order andE theEuler-Knoppsummability matrix of orderr.

LEMMA

2. Ifxisasequencethat isC

1-summable,

thenxis inthedomainof the-summability method, and hence, x isinthe domain ofevery

Z

method.

Proof.

Assume

thatxisC

1-summable

to L:lim_n

(Xl

+- +

Xn)/n

L.

To

gettheconclusion it is enoughtoshow that the abscissa of

convergence Go

ofthe Direchlet series

n.__=

xn/n

sisless thanor equalto1,where

o

is givenby

log xl

ro--

lira sup

log n

(Hardyetal

[8]

orTitschmarch[9]). Since xisc

1-summable

toL,thereexistsapositive integerNsuch that ifn>_

N,

then

<

ILl

+1.

Thisimplies that

=-:,

=

_<

(

L

+

),

,

log

[n(ILI + ])].

Therefore

log

rkl

x

cr lim sup logn

log n

(ILl

+

<_

lim.__.sup

logn

THEOREM

6. The

Z

method includes theC method.

Proof. This inclusion is equivalenttothe regularity of the matrix

ZtC1-1

which

can

be verifiedby directcalculation using the Silverman-Toeplitz Theorem.

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 33

Since

EXAMPLE.

’.,et x

{(-

l)lk}

l.hen

(tx),,

N

’

(-

])kk

7,

kt(n)_

t,(n))

for

L(n)>

andlim_n

(t(n))

*,,,it iseasytoseethat

limn(Ztx)n

0. Onthe other hand,wehave

(C,x)n

1

(_l)kk

Ilk=

-,if

uiseven

]-1

ifnisodd

Thus

limn(ClX)n

doesnotexist,so xisnot

Cl-summable.

By

a"continuousparametersequence-to-function transformation",we mean asummablility methodFthat is determinedasfollowsby afuction

sequence {fk(z)}

--i

foragivensequence xform the function

Fx(z)--

E

fk(Z)Xk

(*:)

k=l

if lim_z._)a

Fx(z)

L,thenwesaythat"x isF-summable to

L".

For agiven functionsequence

{fk(z)}

andagiven number

sequence

t,wecanalso formanassociated matrixFt,whichis givenby

Ft[n,k

fk(t(n))

Thenextlemma, whichwillbe usedto

compare

theC method and the

t

method,isacomparision of the method ifand the associatedmatrixmethod

F

LEMMA

3.

Let F

beacontinuousparameter sequence-to-functiontransformationasin

(**)

and define the

sequence

sets

:r

{x

lira

l".[z)

exists}

SF,

{x

t;’tx

is convergent and

L.K. CHU then

Proof. Weshow that each of

SF

and

I’It(TSF,

containsthe other. Since

Ft

includesFfor inT, we have

t.T

To provethereverseinclusion,weconsidera sequence xwhichisnotin_SF.

It

follows thatlim_z._>

aFx(z)

doesnotexist.

By

thesequentialcriterion for function limits(Almsted [10], p. 73),there is a sequencet\

inTsuch that

limn(Ftx)n

doesnotexist. This implies thatxisnotintheset

Sv,

Hence xisnotinthe set

r"i.

THEOREM

7. The -summabilitymethodisstrongerthan theC method.

Proof.

By

Lemma3,wehave

S

r’lt(TSz,.

Since theZ method includes theC method for all inT,we have

Sc

C

,t?

S.,,

$..

NowifxisasequencethatisC -summable toL,thenxisZ summable toLfor all inT. Therefore the sequential criterion for function limitsensuresthatxis

t-summable

to L. Hence,the method includes theC method. Itiseasy to seethat the

C

method doesnotinclude the

t

method becauseC method does not include theZ method.

As

a consequenceof Theorem 6,we caninfer that

Z

includesanymethod thatisincludedbyC_1. Forexample,Z includes the divisor methodD for >0. (Fridy[11]).

Let H

_{2 denote the Holder method of order}2.

By

arguingasintheproof of Theorem 6,wecan

prove

THEOREM8. If the

sequence

xis

H2-summable

to Landxisinthe domain of theZ method, then x is

Z

summableto L.

COROLLARY. If the

sequence

xis

H2-summable

to

L

andxis inthe domain of the-summability method, then xis

t-summable

toL.

The conclusion of thepreceding Corollary doesnothold ifxisnotinthe domain of the method. Thisisshownbythe followingexample.

EXAMPLE.

Letxl)etilese(lueneedefinedI)y

(-1)kk iln---2k,k--1,2

Xn

(-1)"

/k

il"n=2k-1, k--l,2

Ifx< 3/2, then the series

3

__,,___

(xn/n

s)

is divergent becauseitsnth term doesnot approach0. Therefore x isnotinthe domain of the method, and hence,xis not

-summable.

Now weshow thatxis

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 35

Therefore,upondividing by 2k-1toformC

(ClX)2k_

wehave

H

oX)k_:

2

k-=o().

whichprovesthatxis

H2-summable

tozero.

Since the Holder method of order 2isequivalenttothe

Cesaro

method of order 2 (Hardy[4], po 103), we canimmediatelygetthefollowingtheorem.

THEOREM9. Ifxisa

sequence

which is

C2-summable

to

L

andxis inthedomainof the summability method, thenxis

-summable

to L.

It

iswell known that for each number satisfying 0< < and

any

nonzerorealnumber x,

Ero

By using these facts,wehavethe following result.

THEOREM10. The methodisnotincluded inE for0< <1.

The followingexampleshowsthat the method doesnotincludeE for0< <1.

EXAMPLE. Given between 0 and choose > 0 satisfying <2 (2+). Nextdefinex_k (-1

_)k.

Then

(.x)

E

()

,

(-r)

-

(--’)

[(

--

),.

+ (I-r)]

Since0< <2 (2+), wehave-1<

(-2-)r

+ < 1. This implies that

--0,

i.e.,xis

Er-summable

to 0. But xisnotinthedomainofthe method because the series v’

(-]-)"

k:=l

isnotconvergentforany s,whencexisnotinthe domain of the

t

method.

ACKNOWLEDGEMENT.

This workisaportion of the author’s doctoraldissertation, writtenunder the supervision of ProfessorJ. A.Fridyat

Kent

StateUniversity,Kent, OH,1985.

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CHU

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Arch.

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No.

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