 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-23/4/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

Put

D_{0} = ∣ D ∣

. The reference to part (c) in the printed hint is a reference to the positive-coefficient function from part (b). For

σ > 1

close to one, the preceding positivity and the supplied partial-fraction expansion give

0 \leq \frac{1}{σ - 1} + C lo g D_{0} - \sum_{β real near 1} \frac{1}{σ - β} .

All omitted zero terms have nonnegative real parts because their real parts are at most one. The

O (1)

zeta-pole remainder is included in

C lo g D_{0}

, increasing the absolute constant if needed; a nonprincipal primitive real conductor of a Dirichlet character is at least three.

Suppose there were two real zeros, counted with multiplicity, with

β \geq 1 - c / lo g D_{0}

σ = 1 + a / lo g D_{0}

lo g D_{0}

gives

0 \leq \frac{1}{a} + C - \frac{2}{a + c} .

Choose

C \geq 1

a = 1/ (4 C)

and

c = a /4

. The right side is strictly negative. Thus

at most one real zero, necessarily simple, in [1 - c / lo g ∣ D ∣, 1] .

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