Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 150 / 3 / b

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- Solution b

Solution

For

σ > 1

, both relevant Dirichlet series converge absolutely, so their product may be rearranged:

ζ (s) \sum_{n = 1}^{\infty} \frac{μ ( n )}{n ^{s}} = \sum_{n = 1}^{\infty} \frac{1}{n ^{s}} \sum_{d ∣ n} μ (d) .

The inner sum is one when

n = 1

and zero otherwise by Möbius inversion. Hence

\frac{1}{ζ ( s )} = n = 1 \sum \infty \frac{μ ( n )}{n ^{s}} (σ > 1) .

This is also the reciprocal of the absolutely convergent Euler product.

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