{geni:about_me} http://www.ams.org/notices/199811/mem-fuchs.pdf

Wolfgang Heinrich Johannes Fuchs

(1915–1997)

J. Milne Anderson, David Drasin, and Linda R. Sons

1472 NOTICES OF THE AMS VOLUME 45, NUMBER 11

Biographical Sketch

Wolfgang Heinrich Johannes Fuchs was born in Munich

on May 19, 1915. He joined the faculty of

Cornell University in 1950, where he remained

through his retirement in 1985 until his death on

February 24, 1997. His life and career were characterized

by an unrelentingly positive and supportive

attitude. He read avidly (in many languages),

travelled widely, and was devoted to intellectual

dignity and the international mathematical community.

He wrote two important monographs and

more than sixty-five papers in complex function

theory and related areas. These achievements were

recognized by the award of three fellowships:

Guggenheim (1955), Fulbright-Hays (1973), and

Humboldt (1978).

Wolfgang graduated in 1933 from Johannes

Gymnasium in Breslau (WroclÃaw), where his

teacher, Hermann Kober (remembered for his Dictionary

of Conformal Representations), convinced

him to become a mathematician. Wolfgang’s obituary

of Kober, published in 1975, contains warm

memories of those times. Outside school hours he

studied Russian and Chinese.

Graduation occurred shortly after Hitler assumed

power in Germany. Wolfgang’s parents were

classified as Jews, and they recognized at once

that a normal life would be impossible in Germany.

They arranged for Wolfgang to enter St.

John’s College, Cambridge, in the fall term of 1933

and were able to join him before war erupted in

1939.

At this time the predominant figures in analysis

at Cambridge were G. H. Hardy and J. E. Littlewood,

and Wolfgang soon came under their spell.

He received a Ph.D. in 1941 under the direction of

A. E. Ingham [10]; we discuss this work below.

In 1938 Wolfgang received a fellowship to Aberdeen,

where he was fortunate to have W. W. Rogosinski,

another refugee, as a colleague. They had

known each other from Cambridge days and shared

an interest in summability. Their collaboration intensified

in the summer of 1940, when both were

interred on the Isle of Man as “enemy aliens”. He

would later describe this period as a “beautiful

summer vacation”: there was a rich mathematical

environment, and, since a chef of Buckingham

Palace was another detainee, the food was rather

good.

In 1943, while still at Aberdeen, he met and

married Dorothee Rausch von Traubenberg, another

refugee from Germany, who was a student

at the university. “His relationship with her was the

center of his being” [W. K. Hayman]. Dorothee

came from a distinguished academic family. Her

father was dismissed from a professorship at Kiel

in 1937 for his pacifist beliefs, but, above all, for

being married to a Jewish woman. He continued

his research in atomic physics without a proper position,

assisted only by his wife. However, his sudden

death in 1944 left her unprotected, and

Dorothee’s mother was soon sent to Theresienstadt,

where she was ordered to document this research.

In this way her life was spared until the

camp was liberated by the Red Army in 1945.

In time, Wolfgang’s work (in particular, [12]) attracted

the attention of R. P. Agnew, who was chairman

of the Cornell mathematics department.

Agnew invited Wolfgang to Cornell for 1948–49,

where he accepted a permanent appointment.

At Cornell Wolfgang carried out his research on

Nevanlinna theory (value-distribution theory). This

was the most sustained output of his career and

is discussed below. While Nevanlinna theory was

already well known among complex analysts, it was

usually viewed as a tool rather than as a subject

of its own. It had attracted almost no research in

the U.S. before the early 1950s.

Wolfgang was drawn to the subject by Albert

Edrei, of Syracuse University. Edrei had been trained

by G. Pólya and M. Plancherel in Zürich and thus

had a strong function-theoretic background. In

1950 Edrei, then at the University of Saskatchewan,

heard I. J. Schoenberg lecture about his conjecture

on the characterization of generating functions of

“totally positive sequences”. Within a few months

Edrei applied Nevanlinna theory to settle this problem

completely. This convinced him that many

important and beautiful results were ripe for harvest,

and in 1955, at a mathematics picnic at Fall

Creek Park in Ithaca, Wolfgang agreed to join him

in the chase.

Nevanlinna theory remained the main focus of

both Edrei and Fuchs for the remainder of their careers.

They, their students and co-authors, and

other groups developing in China, England, Germany,

and parts of the former Soviet Union entered

a long golden age of one-variable value-distribution

theory. Almost all subsequent work that dealt

with functions of finite order used the formalisms

and computational techniques that Edrei and Fuchs

introduced. When Rolf Nevanlinna died, Wolfgang

was the obvious choice to deliver the address devoted

to Nevanlinna’s theory at the memorial conference

in 1981.

Edrei and Fuchs remained close friends until

Wolfgang’s death, and Edrei died the following

year, on April 29, 1998.

Wolfgang was also anxious to study new ideas

from others, and his monographs influenced a

generation of complex analysts. Extremal length?

In [20] P. Koosis writes, “The most accessible introduction

is in W. Fuchs’ little book” [16]. This “little

book” also includes Mergelyan’s solution to the

weighted approximation problem and an attractive

selection of ideas from potential theory.

He read

widely and

kept detailed

notebooks

with his own

derivations

and impressions

from

his reading.

Frequently

he would

contribute

these expositions

to the

original authors,

and

often this became

what

was published.

This

made him a

valuable editor

of several

mathematical

journals, notably the Proceedings of the AMS.

“My recollection is that [Wolfgang] always felt

that one-variable value-distribution theory …was

too narrow and needed infusion of the kind of

geometric ideas advocated by Ahlfors. …[The]

essence of his message was just to broaden the subject.

At first I thought he was merely making a

philosophical statement, but twice he wrote me

complimenting me on specific things I said in my

geometrical work. Although I believe he was unduly

impressed with something rather trivial with

which he happened not to be familiar, I came to

understand that this advocacy was more than an

empty gesture” [H. Wu].

“What impressed me is that he was still very

eager to learn what was happening …, so he would

take new papers and really work through them in

all details” [W. Bergweiler, who was at Cornell in

1987–89 under Wolfgang’s aegis].

Two of his many foreign contacts warrant special

mention. In fall 1964 he participated in an official

exchange with the USSR Academy of Sciences

and attended an international conference in Erevan

the following summer. There he intensified his

contacts with several mathematicians in the outstanding

schools of value-distribution theory and

approximation theory. Soon after, he worked out

Arakelyan’s construction of functions of finite

order with infinitely many Nevanlinna defects

(available at that time only in a short Doklady

note), and his 1967 Montreal lectures [18], with all

details, became the standard reference for this

work. Work by the Soviet mathematicians Keldysh,

Mergelyan, and Goldberg, in addition to Arakelyan,

occupy the major portion of [18]. His efforts to

bring important work to the attention of Western

Wolfgang Fuchs, 1949.

P{hotograph courtesy of Dorothee Fuchs.

1474 NOTICES OF THE AMS VOLUME 45, NUMBER 11

mathematicians continued all his life. This orientation

also led him to purchase gift memberships

in the American Mathematical Society for several

colleagues from abroad.

“Mathematicians of the

world can admire how in

the years of the cold war he

was building bridges between

the divided spheres

of mathematicians. Today

it is hard to imagine …what

was the common situation

of fifteen years ago. [In

those times] personal contacts

were the privilege of

only a very narrow circle.

W. Fuchs understood that

separation and mutual mistrust

would only be detrimental

to the science of

mathematics. Many people

understood this fact, but

only a few took an active

part in popularizing the

achievements of Soviet

mathematicians in the

West” [A. A. Goldberg].

He was thrilled to make

an official visit to China in

1980. The Cultural Revolution had ended only a few

years earlier, and during that period no one in

China could enter a scientific career. Wolfgang enthusiastically

lectured and arranged for mathematics

students to come to the U.S. to help restore

mathematics in China. This was an important resource

and contact for Chinese mathematicians,

since function theory was one of the few active

areas of mathematical research in China at that

time (cf. [9]). “In simple words, the work done by

Zhang [Guang-Hou] and me in the seventies was

mainly based on the knowledge of French scholars

[active before 1940] and the influence of Edrei

and Fuchs’s papers” [Yang Lo]. Wolfgang publicized

the work of Yang, Zhang, and others and

arranged for many mathematicians to visit the

U.S. Thus he felt outrage at the massacre of June

1989 in Tiananmen Square and at once organized

and arranged that the letter [1] be published in the

Notices. “[It] was first suggested to me by [Wolfgang].

As a Chinese-American I would never have

done it alone” [H. Wu]. His concern continued for

the rest of his life: the letter [2] decried the reimprisonment

of the dissident Wang Dan. Because of

human rights considerations, he publicly declined

later invitations to China and, on other occasions,

to Israel.

Wolfgang was a charter member of the Ithaca

chapter of Amnesty International and served as coordinator.

“Our group had strong links with the international

scientific community, and Wolfgang

took an active role in establishing contacts with scientists

in Eastern Europe, the Palestinians in the

West Bank, and elsewhere, including such wellknown

dissidents as Andrei Sakharov and Yuri

Orlov. He continued to attend meetings and support

Amnesty activities long after the state of his

health would have justified slowing down” [Peter

Wetherbee].

He contributed a poem at the 1985 conference

to celebrate de Branges’s proof of the Bieberbach

conjecture. It is the closing item in the published

proceedings (1986), and the way he describes the

history of the problem and its solution displays and

preserves some of the charm his colleagues and

friends long appreciated. “My first encounter with

Wolfgang Fuchs changed my life. I visited Cornell

in 1965 to consider an offer from the mathematics

department. One evening in Wolfgang’s home

convinced me that it would be a privilege to live

and work in the same community as this wonderfully

wise, kind, and witty man” [Clifford Earle,

1997].

Some Mathematical Accomplishments

We describe some aspects of Wolfgang’s research

that display his breadth, insight, power, and influence

in analysis. A full bibliography and discussion

appear in a special issue of Complex Variables

[3] dedicated to him and Edrei.

Thesis

Many theorems about entire and meromorphic

functions are obtained by comparing growth rates

of appropriate increasing real-valued functions. If

f is entire, the most common such function associated

to f is the maximum modulus

M(r) = M(r; f )  max

 jf (rei)j;

but for given p > 0 we could as well consider

Mp(r; f ), the Lp-mean of f on fjzj = rg. Wolfgang’s

thesis [10] confirmed a remarkable conjecture by

Ingham: when p 6= 1; Mp itself is almost an analytic

function.

Let f and g be analytic in

fr1 < jzj < r2g and suppose for a fixed

0 < p < 1 we have Mp(r; f) = Mp(r;g)

for a sequence of r with limit point in

(r1; r2): Then Mp(r; f )  Mp(r;g) for

r1 < r < r2.

This theorem completely fails when p = 1: Hayman

considers this and [8], written with Erdo˝s, his

favorites. The theorem is not hard to prove when

f and g have no zeros, and “the proof of the analyticity

[in r of the Lp mean] across the modulus

of a zero is a brilliant and subtle piece of work”

[Hayman].

Nevanlinna Theory

Wolfgang’s greatest impact on American mathematics

came from his work on Nevanlinna theory.

Nevanlinna developed his theory in the 1920s as

Fuchs in Berlin, October 1978.

DECEMBER 1998 NOTICES OF THE AMS 1475

a potential-theoretic analysis of Picard’s theorem

(1879), which asserts that a nonconstant meromorphic

function in the plane cannot omit three

values. The obvious example f (z) = ez shows that

the theorem is sharp. For the next fifty years, Borel,

Valiron, and others attempted to find more insightful

proofs. Not only was Nevanlinna’s approach

the most successful, but his techniques

became standard in potential theory and the foundation

for a subject of its own. Thus, let f be meromorphic

in the plane. If 0 < r < 1 and n(r;a) is

the number of solutions to the equation f (z) = a;

with jzj < r, account being taken of multiplicities,

we set

N(r;a) =

Z r

n(t;a)t−1 dt

(this is slightly modified when f (0) = a). Nevanlinna’s

characteristic T(r) = T(r; f ) can be defined

as

T(r) =

Z

bC

N(r;a)d(a);

where  is the uniform distribution on the Riemann

sphere bC

, and the deficiency (a) = (a; f ), a 2 bC

,

is

(a) = 1 − lim sup

r!1

N(r;a)

T(r )

:

It is elementary that T(r ) " 1 and 0  (a)  1.

Of course, (a) = 1 if the equation f (z) = a has no

solutions. Nevanlinna’s famous Second Fundamental

Theorem implies that

(1)

X

a

(a)  2;

a deep generalization of the Picard theorem. Nevanlinna

theory in the plane asks for refinements of

(1), given other properties of f. Nevanlinna’s T(r )

plays the role of the maximum modulus M(r ) in

the special case that f is entire; we then simply have

(1) = 1. The standard references are [18] and

[19].

Nevanlinna’s key insight was his “lemma of the

logarithmic derivative”, which states that for most

large r,

(2)

Z 2

0

log+



f 0

f

(rei)



d = o(log(rT(r ))):

Thus, the left side of (2) is negligible when compared

to T(r ). Since in (2) we may replace f by f − a

for any complex number a, (2) indicates that if f

is very close to a complex value a on a portion I

of fjzj = rg, then f 0 must also be small on I.

Wolfgang’s first works revisited the expression

(2) in a direct manner, by estimating

(3) M(I; F) 

Z

I



f 0

f

(z)



jdzj;

where I is any subarc of fjzj = rg. This integral is

far more treacherous than (2); in fact, it diverges

whenever I contains a zero or pole of f. His estimates

appear in [14] and [15]; later Petrenko found

the sharpest bounds.

Wolfgang obtained two striking applications

from his estimates. To explain them, we need the

notion of the order  of a meromorphic function

f:

(4)  = lim sup

r!1

log T(r; f )

log r

:

For example, exp(zk) has order k. In [14] Wolfgang

showed that when  < 1, (1) alone does not describe

the full situation: in addition, we have that

(5)

X

1=2(a) < 1;

which confirmed a conjecture of Teichmüller. Wolfgang

considered this and the paper [8] (with Erdo˝s,

discussed below) his two best. Some years later,

Weitsman, following insights of Hayman, showed

that 1=3 is the optimal exponent in (5). That (5) reflects

special properties of functions of finite order

became clear when, with Hayman, Wolfgang

showed (cf. [18], Chapter 5, and [19], Chapter 4)

that Nevanlinna’s defect relation (1) is sharp among

all entire functions.

Paper [15] proved a conjecture that G. Pólya

posed in his famous paper [23]. Let f be entire and

of order  < 1 with power series development

(6) f (z) =

X

akznk :

If nk=k!1 as k!1, then

lim sup

r!1

L(r; f )

M(r; f )

= 1;

where L and M are, respectively, the minimum and

maximum modulus of f on fjzj = rg. Thus, in a

very precise sense, these gap series behave as

monomials for arbitrarily large values of r.

The first joint work of Edrei and Fuchs ([5] and

[6]) completely characterized entire functions f of

finite order for which

P

(a) = 2; i.e., equality holds

in (1). Not only is  a positive integer (this was

shown earlier by Pfluger), but the global asymptotic

behavior and Taylor expansion of f are completely

described. In particular, only a finite number of

nonzero terms can appear in the deficiency sum

(1). These conclusions were obtained as limiting

cases when the difference 2 −

P

(a) is sufficiently

small. One significant problem arising from this

work remains open to this day: If  is “close to”

an integer k and

P

(a) is “nearly” equal to 2, can

there be only finitely many nonzero terms in (1)?

Their paper [7] introduced two major ideas that

have transformed much future research. First, they

made a serious study of “Pólya peaks” and showed

that these peaks gave a new, elegant, and unified

way to interpret the hypothesis that the order 

in (4) be finite. This led to the common principle

that a function f should be studied by comparing

its characteristic T(r ) to simple local comparison

functions defined intrinsically in terms of T(r ). Before

[7] authors were forced to create many ad hoc

comparisons between T(r ) and r, but after [7]

most of these notions were forgotten.

The derivation by Edrei and Fuchs in [7] of a key

inequality of Goldberg provided an essential foundation

on which A. Baernstein later could build his

-function. The -function continues to have a

major impact on symmetrization and geometric

function theory.

While the impact of [7] on later work was enormous,

its main conclusion should not be ignored.

The “ellipse theorem” gives a complete relation between

any two terms that appear in the deficiency

sum (1) and the order  of a function f when

0    1. Thus let a and b be fixed in bC

, and set

u = 1− (a); v = 1− (b). Any understanding of

the pair (u; v) sheds light on any two terms appearing

in (1). It is clear that (u; v) is always confined

to the square 0  u; v  1. Edrei and Fuchs

prove that, in addition, the point (u; v) must lie on

or outside the ellipse

u2 + v2 − 2uv cos = sin2 

and this condition is best possible in all cases. See

Figure 1.

This result is nearly forty years old. Except in

some trivial cases, when  > 1 there is no complete

description of the possibilities of f(u; v)g as f

ranges over all meromorphic functions of order :

For entire functions, the “trivial case” occurs when

 is an integer k, in which case (u; v) can lie anywhere

in the square, with fk(z) = exp(zk) being extremal.

Closure Problems

Although Wolfgang wrote fifteen papers during his

stay in Britain, he is probably best remembered

today for his paper [11] on the closure of the functions

fe−t ta g in L2(0;1). The subject begins with

Weierstrass’s theorem that polynomials are dense

in C[a; b]. H. Müntz proved in 1914 that if S is any

1476 NOTICES OF THE AMS VOLUME 45, NUMBER 11

subset of the positive integers, then the linear

span of ftm; m 2 Sg is dense in L2(0; 1) if and only

if

P

S m−1 = 1. This result suggested that approximation

of rather general functions might be

possible by using specific subclasses that have attractive

structures and led to a wide development.

A more refined analysis is needed in [11]; on the

finite interval (0; 1) it corresponds to a study of

the closure of f(log(1=x))a g. In addition, the possibility

that the a’s are positive but not necessarily

integers raises many technical complications. Let

(7) Ã(r) = 2

X

a<r

a−1



for r > a1. Wolfgang’s theorem shows that the

system fe−t ta g is complete in L2(0;1) if and

only if Z 1

a1

[expÃ(r )]r−2 dr = 1:

Roughly speaking, this says that if a   as

 !1, then the system is complete if   1=2 and

incomplete if  > 1=2:

The basic problem is that the integral may converge

without the fag satisfying the Blaschke

condition

P

a−1

 < 1. This necessitates consideration

of a function of the form

(8) H(z) =

1Y

=1



z − 

z + 

exp(−2z=a)



to cancel out the zeros of a certain function G(z),

analytic in the right half-plane. The paper, written

before the contribution of functional analysis to

closure problems in complex analysis was fully appreciated,

involves a highly sophisticated application

of the Ahlfors Distortion Theorem and

shows Wolfgang already at the height of his analytical

powers.

Although (8) is regular only in the right halfplane,

Wolfgang uses an inverse integral transform

to obtain the desired function orthogonal to

the family fe−t ta g.

The product (8) itself has many uses. It provided

a key ingredient for [12], which was so admired by

(1, 1)

v

u

(1, 1)

v

u

Figure 1. The

possible values of

(u; v) must always be

inside the first

quadrant. According

to the “ellipse

theorem”, they are

also limited to the

portion outside the

ellipse u2 + v2− 2uvcos = sin2.

The shaded regions

here show the set in

question for  = :33

and  = :7.

DECEMBER 1998 NOTICES OF THE AMS 1477

Agnew. Let f be of exponential type k (we write

f 2 Ek): this means that logM(r) = O(kr) as

r !1; with M(r ) equal to the maximum modulus

as above. Let us call a sequence fag of positive

real numbers a determining sequence corresponding

to Ek if the conditions a+1 − a > c > 0 for

all , f 2 Ek, and f (a) = 0 for  = 1; 2; : : : imply

that f  0: The most famous theorem of this type

is due to F. Carlson: If k < , then f;   0g is a

set of uniqueness for Ek, and the example

f (z) = sinz shows that this bound on k is exact.

The contribution of [12] is to give a condition both

necessary and sufficient for any k: If à is constructed

as in (7) from the fag, then

lim sup

r!1

Ã(r )r−2k= = 1:

Several other papers are in this vein and are extensively

discussed in the monographs of Mandelbrojt

[22] and Boas [4]. These problems, with

more general weights than e−t, were also considered

in the thesis of Malliavin, who related them

to “Watson’s problem”. In 1967 Wolfgang showed

that his original approach led to an elegant solution

to one result in Malliavin’s thesis.

In the 1950s Malliavin carried the ideas of (8)

much further and deduced the converse to Pólya’s

maximal density theorem concerning gap series.

In [23] Pólya had proved that if a power series of

the form (6) has radius of convergence one and the

(Pólya) maximal density of the nonzero coefficients

in (6) is D, then every arc of fjzj = 1g of

length greater than 2D contains a singularity of

f. This richly amplifies the well-known fact that the

circle of convergence of any power series has at

least one singularity.

While the precise definition of maximal density

is too complicated to be reproduced here, any subset

of integers does have such a density (this density

is defined in terms of a lim sup). Thus it is natural

to ask if the Pólya density is the precise notion

needed to guarantee Pólya’s theorem. Malliavin

was influenced by [13] to develop an extensive

theory, which among other things showed that

Pólya’s notion of density was exact. In [20], Chapter

9, Koosis uses Malliavin’s arguments to establish

this converse directly from [13]. This discussion

also provides an exhaustive explanation of the

significance of products such as (8).

Additive Number Theory

Erdo˝s shared Wolfgang’s enthusiasm for their joint

paper [8]: “[It] certainly will survive the authors by

a few centuries” (quoted in [24]). An excellent exposition

is in [21], Chapter II.

Thus, let A = fakg be a nondecreasing sequence

of nonnegative integers, and for n 2 Z let r (n;A)

be the number of solutions to the inequality

ai + aj  n with ai; aj 2 A, using any consistent

method of enumeration. Special techniques are

available when A = Q = fm2; m  0g; in this case

r (n;A) is simply the number of points of the integral

lattice in fjzj  n1=2g, and so r (n;Q)  n:

In classical work dating back to Hardy in 1915, it

was shown that this asymptotic relation cannot be

attained too rapidly: when A = Q and c = ; then

(9) lim sup

n!1

jr (n;A) − cnj

Ø(n)

> 0;

where Ø(n) = fn log ng1=4.

These arguments were heavily based on the interpretation

of r (n;A) when A = Q. The contribution

of [8] is that such limitations are, in H. Halberstam’s

words from 1988, “a law of nature.” In

fact, if A is any such sequence, then there is a universal

Ø(n) " 1 such that (9) must hold for any

c > 0. Of course, if we allow c = 0, then a sufficiently

sparse A allows that r (n; a)n−1 ! 0 as

rapidly as desired. Erdo˝s-Fuchs show that Ø(n) =

n1=4 log−1=2 n gives (9) for any A.

References

[1] W. ARVESON et al., Letter, Notices Amer. Math. Soc. 37

(1990), 263.

[2] R. ASKEY et al., Letter, New York Rev. of Books 44

(March 27, 1997), 50.

[3] K. F. BARTH and D. F. SHEA, eds., Complex variables,

Special Issue Dedicated to Albert Edrei and Wolfgang

Heinrich Johannes Fuchs, vol. 12, 1989.

[4] R. P. BOAS, Entire functions, Academic Press, New

York, 1954.

[5] A. EDREI and W. H. J. FUCHS, On the growth of meromorphic

functions with several deficient values, Trans.

Amer. Math. Soc. 33 (1959), 292–328.

[6] ———, Valeurs déficientes et valeurs asymptotiques des

fonctions méromorphes, Comment. Math. Helv. 33

(1959), 258–295.

[7] ———, The deficiencies of meromorphic functions of

order less than one, Duke Math. J. 27 (1960),

233–250.

[8] P. ERDO˝S and W. H. J. FUCHS, On a problem of additive

number theory, J. London Math. Soc. 31 (1956),

67–73.

[9] W. FEIT, A mathematical visit to China, Notices Amer.

Math. Soc. 24 (1977), 110–113.

[10] W. H. J. FUCHS, A uniqueness theorem for mean values

of analytic functions, Proc. London Math. Soc. 48

(1945), 35–47.

[11] ———, On the closure of fe−t ta g, Proc. Cambridge

Philos. Soc. 42 (1946), 91–105.

[12] ———, A generalization of Carlson’s theorem, J. London

Math. Soc. 21 (1946), 1057–1059.

[13] ———, On the growth of functions of mean type, Proc.

Edinburgh Math. Soc. Ser. 29 (1954), 53–70.

Ph.D. Students of Wolfgang Fuchs:

Tseng-Yeh Chow (1953)

Alan Schumitzky (1965)

Linda R. Sons (1966)

David Drasin (1966)

Virginia W. Noonburg (1967)

M. A. Selby (1970)

I-Lok Chang (1971)

Subinoy Chakravarty (1975)

1478 NOTICES OF THE AMS VOLUME 45, NUMBER 11

[14] ———, A theorem on the Nevanlinna deficiencies of

meromorphic functions of finite order, Ann. of Math.

68 (1958), 203–209.

[15] ———, Proof of a conjecture of Pólya concerning gap

series, Illinois J. Math. l7 (1963), 661–667.

[16] ———, Topics in the theory of functions of one complex

variable, Van Nostrand, Princeton, 1967.

[17] ———, Théorie de l’approximation des fonctions d’une

variable complexe, Université de Montréal, 1968.

[18] A. A. GOLDBERG and I. V. OSTROVSKII, Distribution of values

of meromorphic functions (Russian), Nauka,

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[21] H. HALBERSTAM and K. F. ROTH, Sequences, Vol. I, Oxford

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[22] S. MANDELBROJT, Séries adhérentes, régularison des

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[23] G. P´OLYA, Untersuchungen über Lücken und Singularitäten

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[24] C. POMERANCE, Paul Erdo˝s: Number theorist extraordinaire,

Notices Amer. Math. Soc. 45 (1998), 19–23.

J. Milne Anderson is professor of mathematics at University

College, London. His e-mail address is

ros@math.ucl.ac.uk.

David Drasin is professor of mathematics at Purdue University.

His e-mail address is drasin@math.purdue.edu.

Linda R. Sons is professor of mathematics at Northern Illinois

University. Her e-mail address is sons@math.

niu.edu.

David Drasin coordinated the writing of this article. The

authors thank Luchezar Avramov and Paul Koosis for their

assistance. All quotations were obtained in 1998 unless

otherwise cited.

DECEMBER 1998 NOTICES OF THE AMS 1473