10. The Reimann zeta-function Ç is defined by

ς(2) = Σ

n=1

1. Σ

ηπ

and is used in number theory to study the distribution of primes.

(a) For values of r does converge? (i.e What is the domain of ?) Explain your reasoning.

(b) Leonhard Euler was able to calculate the exact sum of the p-series with p = 2 and p = 4

ζ(x) = Σ

1

(n + 2)²

1

II.

«Σ(3)

n²

n=1

Use Euler's results to find the sum of the following series:

1

n=π (n – 2)4

III. Σ

IV. Σ

n=2

1

1 πλ

$(4) = £ - -/- - 90

n=l

2