Past exam of the mathematics course of the University of Cambridge 2005 iii Paper 28 3 iii Solution Created 2026-10-03 Updated 2026-10-14
A fixed zero-free strip changes the zero contribution qualitatively. If every nontrivial zero satisfies , with fixed , the same zero-count estimate and partial summation giveThe truncated explicit formula for the second Chebyshev function then yieldsTake , or an admissible height of this size. The two contributions are bounded by the same order, proving the prime number theorem error from a fixed zero-free strip:In particular this is for every . It is a power saving, stronger than the decay obtained from the logarithmic zero-free region. The triangle-inequality estimate of the zero sum explains the two logarithms; removing them would require additional information or cancellation, not merely the asserted zero-free strip.