The function $1/\zeta(x)$, whose graph is seen above, can be easily shown to equal the infinite sum

$\mu(1)/1^{x} + \mu(2)/2^{x} + \mu(3)/3^{x} + \mu(4)/4^{x} + \cdots$

where $\mu(n)$ is the Möbius Function which we encountered earlier. Recall that it is defined on the natural numbers as follows:

$\mu(n)$ equals zero when n is divisible by a square, and otherwise equals (-1)k where k is the number of distinct prime factors in n.

For example,
$\mu$(28) = 0, as 28 is divisible by 4 = 22
$\mu$(42) = (-1)3 = -1 as 42 = 2 x 3 x 7
$\mu$(55) = (-1)2 = 1 as 55 = 5 x 11.
$\mu$(242) = 0, as 242 is divisible by 121 = 112.

In this way, $1/\zeta(x)$ acts as a "generating function" for the arithmetic information associated with the function $\mu(n)$. Other functions constructed from $\zeta$ have similar properties. Three examples:

(i) $\zeta(x)/\zeta(2x)$ generates the sequence of values $|\{\mu(n)\|$ as follows:

$\zeta(x)/\zeta(2x)$ = $\mu$(1)|/1x + |$\mu$(2)|/2x + |$\mu$(3)|/3x + . . .

(ii) log $\zeta(x)$ generates the sequence of values {l(n)} where l(n) is defined to be 1/k when n = pk and to be zero otherwise:

log $\zeta(x)$ = l(1)/1x + l(2)/2x + l(3)/3x + . . .

(iii) $-\zeta'(x)/\zeta(x)$ generates the sequence of values $\{\Lambda(n)\}$ where $\Lambda(n)$ (the von Mangoldt function) is zero unless n is a power of a prime p, in which case it takes the value log p:

$-\zeta'(x)/\zeta(x)$ = $\Lambda$(1)/1x + $\Lambda$(2)/2x + $\Lambda$(3)/3x + . . .

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