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 ]).

0, L.

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.

Proof. Firstletxbea

sequence

satisfyinglimkx_k L,and

suppose

>0. Then chooseN sothat k>N

implieslxk-Ll</2.

^Nowforany positiveintegerkands>lwe

see

that

.] ’11 x ^{;, /}

^{k) is}

bouby

’,’., ]x L M.

Since

5l, ]/k

^oc we^carlchooseN₂

>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 N_2,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

(s) ):% --- ^{+ (M+).}

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 29 Since

r(s) - ^as

^{s + we seethatifsis}sufficiently closeto onthe right, then

<1"

this implies that

>

^-1

+M+I M

SinceM>0waschosen arbitrarily,weconclude that

im V’ Xk oo

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

L

if

li,n._.,+ (s--l) E

xi

- ^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 ,-,.

Proof. Since

(w)

canbe expanded in the form(w 1)"1 +P(w 1),whereP(w 1)^isapowerseries in(w-1), (Hardy

[4],

p. 333), wehave

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)} cosafor k>N_1.

Let _k=l N

xk L,

M.

Since

(w)--oo

asw->1,wehave

l/(w)<

e/2Mfor w sufficiently closeto1.

Nowfor w

S*(a),

wehave

\- xk

^{< I(w)} ,, Ik"l ^{[xr 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 all}s>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

^andxn

O(tn).

^Definethe sequence xbyxn

(-1)n+lt

n, sothat

Xn )c+l

v E ^(-

n--I n=-I

Thisisa convergentalternating series, and its (positive)sumisboundedbyits firstterm

tl.

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 31 Hence,

lira

xn

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]. It^isan easycalculationtoshow that

Zl

^{satisfiestheconditious}of 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

n--N

=N

32

Now

,’,’, ^(1/t

^{(t(n)))divergesto}infinity 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 sectionCa denotes theCesarosummabilitymatrix 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 than}or equalto1,where

o

is givenby

log _xl

ro--

^{lira sup} _{log n}

(Hardyetal

[8]

orTitschmarch[9]). Since xisc

1-summable

^{toL,thereexistsa}positive 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.

The followingexample shows that theC method does not include theZ method.

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)>

andlimn

(t(n))

*,,,it iseasytoseethat

limn(Ztx)n

^{0. On}the 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 limz^._)a

Fx(z)

^{L,thenwesaythat"x is}F-summable to

L".

For agiven functionsequence

{fk(z)}

andagiven number

sequence

t,wecanalso formanassociated matrixF_t,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

T=

(t "limt(n)-- a}

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 xwhichisnotinSF.

It

follows thatlimz._>

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 Landxisin}the domain of theZ method, then x is

Z

summableto L.

COROLLARY. If the

sequence

xis

H2-summable

^to

^L

^{andxis in}the 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}

H2-summable

^{tozero. Since}

^(ClX)2k_ (-1)k+1k3/2/(2k-1)

^and

(CLX)2

k 0,we seethat the(odd)partial sumsalternate in sign after k 3"thus thepartialsumisnotgreaterthanthe lastterm,whichis

0(kl/2).

SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION 35 Therefore,upondividing by 2k-1toformC

(ClX)2k_

^wehave

H oX)k_:

2k-

=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+). Next^definex_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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Mathematics, Vol.XXIX,

No.

3,1977,

pp.

489-497.

CHU

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