Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 3 / b

Table of contents
- Solution b

Solution

Put

t = s - 1

. The Laurent series of the logarithmic derivative at the simple pole of

ζ

has the form

\frac{ζ ^{'} ( s )}{ζ ( s )} = - \frac{1}{t} + c_{0} + O (t),

where in fact

c_{0} = γ

. Hence

(\frac{ζ ^{'}}{ζ} (s))^{2} = \frac{1}{t ^{2}} - \frac{2 c _{0}}{t} + O (1), \frac{x ^{s}}{s} = x (1 + t (lo g x - 1) + O (t^{2})) .

The coefficient of

t^{- 1}

in their product, and therefore the residue, is

x (lo g x - 1 - 2 c_{0}) = x lo g x + A x,

with the constant

A = - 1 - 2 c_{0}

; equivalently,

A = - 1 - 2 γ

.

Solved by gpt-5.6-sol high.

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