zeta function 535

π

10

––

93555

π

8

–

9450

π

6

–

945

π

4

–

90

π

2

–

6

The number 286 is the largest known power of 2

whose decimal expansion contains no digit equal to

zero. (Notice that 210 = 1024, for instance, has second

digit equal to zero.) All the powers of two up to

246,000,000 have been checked.

See also B

ABYLONIAN MATHEMATICS

;

EVEN AND ODD

NUMBERS

; G

REEK MATHEMATICS

; I

NDIAN MATHEMATICS

.

zeta function In 1740 the Swiss mathematician

L

EONHARD

E

ULER

(1707–83) studied infinite series of

the form:

The

INTEGRAL TEST

shows that this series converges if s

is a real number greater than 1. Euler called this func-

tion, defined for values s> 1, the zeta function.

This function is intimately connected to the distri-

bution of

PRIME

numbers. To see this, recall that the

FUNDAMENTAL THEOREM OF ARITHMETIC

asserts that

every number is a product of a unique set of primes.

Thus, in

EXPANDING BRACKETS

for the following infi-

nite product, selecting one term from each set of paren-

theses, every integer appears once, and only once, in

the infinite sum that results:

Although this argument is not mathematically pre-

cise, mathematicians have shown that it is valid to

perform this procedure on the reciprocals of all the

numbers involved, even when raised to the sth power.

Consequently:

The formula for a

GEOMETRIC SERIES

shows that this

equation can be rewritten:

thereby yielding an alternative formula for the zeta func-

tion as an infinite product over all prime numbers p:

Euler managed to compute the value of the zeta

function for certain values of s. He showed, for example,

that ζ(2) = , ζ(4) = , ζ(6) = , ζ(8) = , and

ζ(10) = . (He continued this list up to ζ(26).)

Below we show how Euler computed ζ(2). To this

day, extremely little is known about the values of the

zeta function on odd whole numbers.

In 1859 German mathematician G

EORG

F

RIEDRICH

B

ERNHARD

R

IEMANN

(1826–66) showed that the zeta

function is well defined even if the argument sis

a

COMPLEX NUMBER

. He showed the series

converges if the real part of sis greater than 1, and it is

possible to extend the definition of the function to incor-

porate all complex values of s. For this reason, the zeta

function is often also called the Riemann zeta function.

Riemann was particularly interested in locating the

zeros of the zeta function, that is, finding the values of

sthat yield ζ(s) = 0. He showed that the function has

no zeros if Re(s) ≥1, that its only zeros in Re(s) ≤0 are

at s= –2, –4, –6,…, and that it has infinitely many

zeros in 0 < Re(s) < 1. He called these the “nontrivial

zeros.” Riemann remarked that it is reasonable to

believe that all the nontrivial zeros lie on the line Re(z)

= 1/2, but offered no proof. This casual comment has

become one of the most famous unsolved conjectures

of all time. Mathematicians call it the Riemann hypoth-

esis, and proving its truth or falsehood would have

profound implications on the study of

NUMBER

THEORY

. (For example, a crucial part of proving the

PRIME

-

NUMBER THEOREM

relies on showing that ζ(s) ≠

0 for Re(s) = 1.)