Exponential analogues oî a generalized Lambert series1. Introduction.

A N N A L E S SOCIETATIS MATHEMATICAL POLONAE Series I: COMMENTATIONES MATHEMATICAL X Y (1971)

G.

(Toledo), J.

(Milwaukee) and F. J.

(Saint Louis) Exponential analogues oî a generalized Lambert series

1. Introduction. The F -series

where p and q are positive integers and the an are complex valued con­

stants, was introduced by Garvin [1] as a generalization of previous work in the theory of Lambert series. If the transformation t — e~z is applied to Garvin’s F-series as a series in t, there results the series

If, in addition, the sequences {pn} and {qn} are generalized to {An}

and {/in} respectively, where {/n} and {/un} are sequences of real numbers which are strictly monotone increasing and unbounded, this last series becomes

hereinafter called simply the

-series. Kennedy [2] considered the special case of the $-series when Xn = [лп = lnw.

The purpose of this paper is to determine the regions of conver­

gence of the

-series, and the expansion and inversion relationships between the

-series and the associated Dirichlet series

00

CO

(1.1)

oo

71=1

A special $-series, the P-Q series.

(1.2)

104 Gr. K e r t z , J. M e y e r and F. J. R e g a n

resulting when Xn — p (Inn) and = q(lnn), where p and q are fixed positive integers, will also be considered and explicit formulas for relating ordinary Dirichlet series and P-Q series will be obtained. Finally, condi­

tions under which the axis of imaginaries is a natural boundary of the function represented by a P-Q series will be determined.

Hereafter, unless otherwise indicated, all summations will be un­

derstood to range from n = 1 to oo. Since all but a finite number of the elements of the sequences {/in} and {/un} are positive, we may assume that Àn and [лп are positive for all n.

2. Convergence of the $-series. The following are given without proof :

T

2.1. I f z is such that R(z) > 0, then the 8-series (1.1) (i) converges for all such z if Jfan converges;

(ii) converges and diverges at the same z points as the associated Dirichlet series (1.2) if ]Ian diverges.

I f z is such that R(z) < 0, then the 8-series

(iii) converges and diverges at the same z points as the series £ а пе~^п~ ^ г.

In order to simplify notation in the statements of the following the­

orems, set [in—Xn = vn and An— pn = gn.

T

2.2. (i) I f the 8-series converges at z' — x'-\-iy', x' > 0, the 8-series converges uniformly in each bounded region in the half plane R(z) > x'. (ii) I f the 8-series converges at z ’ = x'-\-iy', x' < 0, and

(a) if vn < vn+l -> oo, then the 8-series converges uniformly in each bounded region in the half plane R(z) < x '• I f i v J is a positive, monotonely increasing or decreasing, bounded sequence, then the 8 -series converges uniformity in each bounded region in —L < R(z) < x', where L is a positive constant ;

(b) if {gn} is a positive, monotonely increasing or decreasing sequence, then the 8 -series converges uniformly in each bounded region within {x' < R(z) < —H} u {R(z) > Щ, where H is a positive constant.

T

2.3. (i) I f ^ \an\ converges, then the 8-series converges abso­

lutely for all z for which R(z) > 0 .

(ii) I f £ \anI diverges, then the 8-series converges absolutely for those values of z for which the associated Dirichlet series (1.2) converges absolutely and for which R(z) > 0 .

(iii) Whether £

diverges or converges, the 8-series converges abso­

lutely for those values of z for which the series f?ane~enZ converges absolutely and for which R(z) < 0.

T

2.4. (i) I f the 8-series converges absolutely at z' = x'-\- iy',

x ’ > 0, then it converges absolutely and uniformly for all z with R(z) > x'.

(ii) I f the

-series converges absolutely at z' = ж'-f iy', x' < 0, and (a) if {vn} is a 'positive sequence, then the

-series converges absolutely and uniformly for all z with R{z) < x' -,

(b) if is a positive sequence, then the

-series converges absolutely and uniformly for all z with x' R(z) ^ —H, where II is an arbitrary but fixed positive constant.

3. Relationship between the ^-series and general Dirichlet series.

Each term of the $-series can be written

e nZ vn ,

a --- — \ a e A,lZe 7fZpZ

i —0

and therefore

O O ___

? O O 00 e An^ \П vn

(3-1 ) 2 ^j l - ë - V ⁼ 2 ^j 2 ^j an e^ n+l"n)!>-

n = 1 n —1 i —0

From this double sum form a single series by summing the terms in ascending order of the numbers hn-\-jyn‘, гь = 1 , 2 , 3 , ...; j = 0 , 1 , 2 , . . . Should any two or more terms of the set {/n+ n = 1 , 2 , 3 , . . . ; j = 0 , 1 , 2 , . . . } be equal, factor out the common e”TfcS, where

^rii 11 j

Уп

^ns + j.4 and write its coefficient as the sum

S

hk = V a„ . m= l

This process utilizes all of the terms of the double series and gives a Dirichlet series

v

k = l

~Tkz

Assume the ^-series converges absolutely at z , R { z ) > 0, and con­

sider the array of absolute values of the double series (3.1). The r-th row of this array converges uniformly for all x > x' > 0 to

6- V

The sum of these “row-sums” is

106 G. K e r t z , J. M e y e r and F. J. K e g a n

which is the $-series in absolute values at a?; from Theorem 2.4 (i) this series converges absolutely and uniformly. Thus the double series (3.1) is absolutely convergent and can be deranged without affecting its con­

vergence. We have therefore

T

3.1. Given an 8 -series absolutely convergent in a region of the half plane R{z) > 0 , there is a Dirichlet series which converges absolutely and represents the same analytic function in that region.

Again by use of the double series (3.1) it can be shown that it is

OO

possible to express a given Dirichlet series Jdhke~TkZ as an $-series in a for- mal manner. The resulting 8-series will not be unique. If the ^-series so obtained is absolutely convergent in the half plane R,{z) > 0, it repre­

sents again the same analytic function as the given Dirichlet series.

Hence we have

T

3.2. A given Dirichlet series can be formally expressed as an 8-series. In its region of absolute convergence in the half plane R{z) > 0, the resulting 8-series represents the same analytic function as the given

Dirichlet series.

4. A special ^-series; its relation to ordinary Dirichlet series. The P-Q series

(4.1)

n= 2

П n

is the special case of the /З-series when Xn is taken to be p(\nn) and gn to be q(lnn), where p and q are positive integers. By Theorem 3.1 the P-Q series (4.1) which can be written

OO OO

(4.3) 2 2 aninlq+r,)-°

71=2 j’ = 0

may, in any region of its absolute convergence in the half plane R(z) > 0,

OO

be expressed as the series £ h kk~z by summing the terms of the double

k= 2

series (4.2) according to increasing values of niq+p, n = 2 , 3 , 4 , . . . ; j = 0 , 1 , 2 , . . . Then hk will be the sum of those an for which пт+р = kj that is, nd = к for some positive integer d = jq + P- Using the notation d =r(m od</)-f to indicate that d =jq- \-r, where j is a non-negative integer and q and r are positive integers, we can write

hk — ^ ar.

d=p(

r^=k

Conversely, a given Dirichlet series

CO

(4.3)

k —2

can be expressed as a P-Q series. Choose q such that q = cp, c a positive integer. Again we nse the double series (4.2) as an intermediate step where now the sequence {aw} is to be determined. Equating coefficients in (4.2) and (4.3) gives

/ lrin ,

> В (--- 1 Jis,

\ Ins

s^=n

where B(n) — 0 for n ф 1 (mod c) and for n = 1 (mod c), B(n) is defined recursively by

11) if ^ == 1 j У в щ =

é h , if n > i ,

n l(modc)

the symbol — — 1 (mode) indicating that Jc is a c-divisor of n) that к

is, Jc is a positive integer which divides n and Jc == 1 (mode). The function B(n) is an extension of Garvin’s inversion function [1]; if c = 1, B(n) reduces to the Mobius function.

It can be shown that if q does not satisfy the condition q = cp, where c is a positive integer, the representation of (4.3) is impossible by this method. The region in which the above expansion is valid is given in the theorem which follows.

The above relationships are expressed in

Th e o r e m

4.1. Any P-Q series

oo i

n pz Z йп l - n ~ qz

n —2

00

can be expressed as an ordinary DiricJdet series У JikJc~z, wJiere

k = 2

Jl>k (Ln •

nfi—k

d=p(mod3) + oo

Conversely a DiricJdet series Jr JikJc~pz can be expressed as a P-Q series,

k —2

wJiere p is determined from tJie given DiricJdet series, and q = cp for a

108 G. K e r t z , J. M e y e r and F. J. R e g a n

positive integer e. The coefficients an of the P-Q series are then given by

a„ =

Q^='»

/ 11Ш\

\ 1п$/

г

d = l ( m o d c ) +

Both representations are valid in the regions of absolute convergence of the respective P-Q series in the half plane R(z) > 0 .

5. An Abelian theorem and natural boundary theorems. We shall determine the behavior of a function represented by a P-Q series as the variable approaches a boundary point of the region of convergence, and shall establish conditions under which the axis of imaginaries is a natural boundary for such a function using both a “real” approach and a Stolz path approach; that is, along any analytic curve which terminates at a point z on the boundary of the region of convergence and lies entirely within the intersection of the region of convergence and the region bounded by two half-lines which originate at г such that they make non-zero angles with the tangent to the boundary at z and lie on the same side of the tangent as does the region of convergence.

Throughout the remainder of this paper square brackets shall desig­

nate the Greatest Integer function and M' = 0,

M ,

if r < 1, if r > 1.

In the proof of Theorem 5.1 we shall require the following lemmas:

L

5.1. I f к and n are natural numbers, k, n > 2, and kc+1 < n for a fixed positive c, then

ГГ Ь » ]-] t

1 1 Llnfc J ^{1 3 + c}

( [hm] c cln к ^ c{(ln%)—1}

L

5.2. I f (a j a n) converges absolutely, where 0 < an < an+1 -> oo, and lim f ij an = 1, then sn = — 1 n is a null sequence ([2], p. 453).

n - + OO f i n fc=1

T

5.1. I f the P-Q series (4.1), where q — cp, c a positive inte­

ger, converges absolutely for jR(z) > 0 , and if the series of constants

OO

(aJinn) converges absolutely, then for any approach to the origin in the

n—2

half plane R{z) > 0 along any Stolz path 1

lim cp 2->0

n

n —2

Y ^{_ "v\}

In n I

(5.1)

Proof. If in the left-hand side of (5.1) the P-Q series is replaced by its equivalent Dirichlet series and pz is set equal to t, the limit to be evaluated is

— lim

p 0

1 F 1

oo

У м - ^{= — lim}

fc = 2 fc= 1 J

By a theorem of Knopp ([3], p. 122), this last limit has the same value as

1 n

- lim { V A,/[In»]}

P n-s-oo j^{c_ 2}

which can be shown to be equal to

lim

n-*oo

Let s > 0 be given. For each n, let L{n) be the largest integer such that {L(n)}c+l < n for a fixed c. Let H be a fixed integer such that

OO

< spc

and let M' = Hc+1. Then for all n > M'

(5.2)

n S ® * k— 2

1 1— *1 1— 1 1— I p p ^ §

1 H1

1— . [ ! ____ !

O O 1 у 1 c

p [Inn] pc J lnfc |

ГПп» ] ] '

L{n)

a*

Lin h J

c

[Inn]

c(ln&) 1 1 3 + c vn

^ p c { ( ln n ) -l}

+ £ 2

k = L ( n ) + 1

K/lnfc|

by Lemma 5.1. Since lim {(Inn) —l}/lnn = 1

n-* 0 0

L (n ) n

£ K \ < Ы , and

(5.2) becomes arbitrarily small with increasing n by Lemma 5.2. The

theorem then follows.

п о Gr. K e r t z , J. M e y e r and F. J. R e g a n

In order to establish the axis of imaginaries as a natural boundary by “real” approach along lines parallel to the ж-axis, which is analogous to radial approach for the Lambert series, we shall require the following adaption of the Abelian theorem of Garvin ([1], p. 511) and the following lemma.

T

5.2. I f the coefficients of the series

x p n

are so chosen that the series £ a j n converges, then for real x

L

5.3. I f q, n, and к are natural numbers, n, к > 2 and n is not an integral power of k, y' = (27t)/(lnfc), then

2 |eW i«(in»)_№-a*| > 2/, {1

(1 +

Г 9'|й'1)}

for all x, 0 < ж < L, where k' — ± 1 , ± 2 , ± 3 , ... and L is an arbi­

trary but fixed constant. I f к — df for some integers d and f we further re­

quire that (k ' q , f ) = 1.

The following theorem represents the main development for the natural boundary theorem of the P-Q series.

T

5.3. Let z = ж + iy", where ж > 0 and y ” is a fixed element of the set {2?zk /(Ink} i k = A l , A2, A3, ...j к = 2, 3, 4, ..., where if к is of the form к = df for some natural numbers d and /, k'q and f are relatively prime for fixed q}. I f the coefficients an are so chosen that

ÿ v i n ( i + »-«'*■') converges absolutely, then

oo

lim <x У n

^

+ l ^ n=2 1 — П qz J q(lnk) r=1 a>k?lr -

Proof. Under the hypothesis ]?\an\ also converges and thus by The­

orem 2.3, the series

(5.3)

n 1

1 — n converges absolutely for B(z) > 0 .

Consider now the subset of terms of the series (5.3) for which n = V ,

where n, к and r are natural numbers with к fixed. Designate the sum

of these terms by the sum of all other terms denote by £ 2.

includes all those terms and only those terms for which 1 — n~as vanishes at г = iy".

If we set е~^ЫК)х = k~x — then

=

У ,4 '

r = l

i - i qr

which is Garvin’s series in real variable. The convergence of £ \ a n\ implies

OO

that of

and hence by Theorem 5.2 O O

lim 1(1— I) V akr

§-+1- l ;

t r \ 1 v ,

= — >, akrjr 1 - 1

or oo

) = ^ 2 a^ l r -

I _{r= 1} ^{v^i (1 k~m}

Since a? (In k) is asymptotically equal to (1 — k~x) as x -> 0, this last limit can be rewritten to give

Ж lim -Э-0 + dn

n-VZ 1 — n~QZ

OO

Now 2 1 = S i + S i i hence if

(5.4) lim {x%s) = 0,

X - + 0 +

the theorem is complete. In order to verify (5.4) it suffices to prove the uniform convergence of in some closed interval 0 < x < M, where M is a finite constant.

Since £ \ a n\ converges, by Theorem 2.3 JT2 converges for all x > 0.

To show uniform convergence in Ü < x < M consider

n - p z

l - n ~ QZ

K \

\ l —n~qz\

S :

K1 ln{l + n- q[k'{) by Lemma 5.3. By hypothesis this latter series of constants converges -r hence by the Weierstrass M -test the series У

converges absolutely and uniformly in 0 < x < M , and the theorem follows.

A set of integers will be called dense if it is unbounded above and

below and there is a positive number l such that every interval of length l

contains an integer of the set.

1 1 2 Gr. K e r t z , J. M e y e r and F. J. K e g a n

T

5.4. I f to each positive integer к of an infinite set there cor­

responds a dense set of h' for which the hypotheses of Theorem 5.3 are ful­

filled, and if for each such h 1

ha

00

У akrjr ф 0,

r = l

then x = 0 is a natural boundary of the function represented by the P-Q series.

The proof consists in showing that the set of singularities {i2nkr/Ink}

is everywhere dense on the axis of imaginaries.

A result similar to Theorem 5.3 can be obtained for approach to the singular points along any Stolz path by the variable z of the P-Q series with the restriction that q = cp, where c is a natural number.

T

5.5. I f the coefficients of the P-Q series, where q = cp for some natural number c, are so chosen that

1 l n ( l

+ n - cplk'{)

converges, then for B{z) > 0 and for an approach along any Stolz path to z ’ = iy", where y" is a fixed element of the set {2izk'jink : k' =

= ± 1 ? ± 2 , ± 3 к = 2 , 3 , 4 , . . . ; and when к = dtf, where d and f are natural numbers, (cpk',f) — 1}? then

<5.5) lim

Z->S' 1 A - ю v . n

■n cp(hxk)

Proof. In the limit on the left-hand side of (5.5) let pz = t and pz' = t'. Then this limit is

<5.6) — lim

P t->i'

n 1 I l - n - ct)

Under the hypothesis the P-Q series converges absolutely for B(z) > 0 and hence may be replaced by its equivalent Dirichlet series. Further for B ( t —t') > 0

/ 00

t - r = 1 / £ ( t - t ’+ l )-”

' m = l

so that limit (5.6) can be written

— lim p t->c

00

£ _m=2 ^hmm~

>

If in this last limit we set t — t' = r and hx = 0, the conditions of the theorem of Knopp ([3], p. 122) referred to above are fulfilled so that we can consider the limit

— lim {—- — V* hm m P s^XlhiS] Z

m J

It is sufficient then to show that for any e > 0 there exists an N such that

P [In s]

m-t

pc (Ink) £ a*J m

for all s > N. Now this absolute value is

p(lnk)

ln& wi [In*] "

1 n

akJ m e

_j--- 2

for all n greater than some integer N'. The theorem follows from the fact that for n sufficiently large

m=2 m= 1

and

ln& d' [ln&n] m—

kn+1

hv„ m

can be shown to be each less than e/4 for any natural number d' such that fcn+ l < d' < kn+l- l .

References

[1] M. C. G a rv in , A Generalized Lambert Series, Amer. J. Math. 58 (1936), p. 507-513.

[2] E. S. K e n n e d y , Exponential Analogues of the Lambert Series, hidem 63 (1941), p. 443-460.

[3] K. K n o p p , Grenzwerte von Dirichletshen Meihen bei der Annaherung an die Konver- genzgrenze, Journal fiir Mathematik, 138 (1910), p. 109-132.

R oczn ik i PTM — F r a c e M atem a ty czn e X V 8