Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 25 / 2 / b / Solution

Write $\psi (x)={\sum }_{n\le x}\Lambda (n)$, where $\Lambda$ is the Von Mangoldt function. The Riemann–von Mangoldt explicit formula, in its symmetric limiting form for $x>1$, isHere nontrivial zeros are counted with multiplicity, the limit is taken symmetrically through admissible heights, and ${\psi }_0$ assigns half weight at a jump. Its difference from $\psi$ is at most $(\log x)/2$. The Euler product and part (a) exclude zeros with real part at least one. The Functional equation of the Riemann zeta function leaves only the trivial zeros of the Riemann zeta function at negative even integers outside $0<\Re \rho <1$; their already displayed logarithmic correction is $O(x^{-2})$ for large $x$. These terms and the constant are negligible in the requested asymptotic error, rather than literally absent from the exact formula.

A useful truncated explicit formula for the second Chebyshev function is, uniformly for $2\le T\le x$,One may first take a height in $[T,T+1]$ separated from zeros and then adjust to $T$ using the local count. The local zero count for the Riemann zeta function isIt includes multiplicity and is uniform in real $t$. It follows by subtracting the Riemann–von Mangoldt formula at endpoints, handling bounded heights separately and using conjugation for negative heights. Both closed endpoints change the count only by another local bound.