246 Hardy, Godfrey Harold

Hardy, Godfrey Harold (1877–1947) British Anal-

ysis, Number theory Born on February 7, 1877, in

Cranleigh, England, eminent mathematician Godfrey

Hardy is remembered for his significant contributions

to the fields of

NUMBER THEORY

, inequalities, and to

the study of the Riemann

ZETA FUNCTION

and the Rie-

mann hypothesis. Hardy also encouraged the Indian

mathematician S

RINIVASA

A

IYANGAR

R

AMANUJAN

to

come to England, and collaborated with him for five

years to produce a number of significant results, most

notably on the theory of

PARTITION

s. In 1940 Hardy

published A Mathematician’s Apology, which remains,

to this day, one of the most vivid and eloquent descrip-

tions of how a mathematician thinks.

Hardy entered Trinity College, Cambridge, in 1896

to pursue an advanced degree in mathematics. After pub-

lishing a number of papers on the topics of

INTEGRAL

s,

SERIES

, and general topics of

ANALYSIS

, Hardy published,

in 1908, the undergraduate textbook A Course of Pure

Mathematics, which explained, in a rigorous manner, the

concepts of function,

LIMIT

, and the elements of analysis.

This work was very influential and is said to have trans-

formed the entire nature of university teaching.

In 1911 Hardy began a 35-year-long collabora-

tion with English mathematician John Littlewood

(1885–1977), leading to a change of focus in the field

of number theory. Together they produced over 100

joint publications. This work also tied in nicely with

the research Hardy was conducting in 1914 with

Ramanujan, also on topics in number theory.

Hardy was recognized as an important figure in

mathematics. In 1910 he was elected a fellow of the

R

OYAL

S

OCIETY

of London, received the Royal Medal

of the Society in 1920, the De Morgan Medal of the

Society in 1929, and the Sylvester Medal of the Soci-

ety in 1940 for his work in pure mathematics. Seven

years later he was awarded the Copley Medal of the

Society for his contributions to the field of analysis.

Hardy died in Cambridge, England, on December 1,

1947. He is generally recognized as the leading English

pure mathematician of his time. His 1932 book The The-

ory of Numbers, cowritten with E. M. Wright, is consid-

ered a classic text and is still used by graduate students

and researchers in the field of number theory today.

harmonic function A function defined at a discrete

number of locations is said to be harmonic if the value

of the function at any one location equals the average

of the function values at its neighboring locations. For

example, if 10 students sit in a circle, then their “age

function” would be harmonic if the age of any one stu-

dent equals the average age of his or her two neigh-

bors. (A little thought shows that this is only possible if

the age of each student is the same. A harmonic func-

tion for points arranged in a circle must be constant.)

Harmonic functions play an important role in the

study of

RANDOM WALK

s and calculating odds in gam-

bling. Imagine, for example, a gambler playing a simple

game of tossing a coin to either win a dollar or to lose a

dollar. We ask: with $3 in hand, what are her chances of

reaching the $10 mark before going broke? To compute

this, let P(N) denote the

PROBABILITY

of achieving the

goal starting with Ndollars in hand. Clearly P(0) = 0

and P(10) = 1. We wish to compute P(3).

The key is to note that P(N) is a harmonic function

on its 11 values zero through 10. This follows because

there are equal chances for the gambler to lose or win a

dollar and thus to next play with either N– 1 or N+ 1

dollars in hand. Consequently:

and so each quantity P(N) is indeed the average of the

values just preceding and succeeding it. Some thought

shows that the values P(0) = 0 up to P(10) = 1 must be

strictly increasing by equal intervals of one-tenth. The

values P(N) are thus P(N) = . In particular, P(3) = ,

showing that there is only a 30 percent chance that the

gambler will achieve her goal before losing all her cash.

harmonic mean See

MEAN

.

harmonic sequence (harmonic progression) A

SE

-

QUENCE

of numbers of the form , , ,… with

integers a1,a2,a3,… forming an

ARITHMETIC SEQUENCE

is called a harmonic sequence. The numbers a1,a2,a3,…

have a constant difference between them. For example,

1, , , ,… is a harmonic sequence, as is , ,

,… and , , , ,…. The word harmonic is

1

–––

100

1

–

75

1

–

50

1

–

25

1

–

13

1

–

10

1

–

7

1

–

4

1

–

4

1

–

3

1

–

2

1

–

a3

1

–

a2

1

–

a1

3

–

10

N

–

10