Let $\langle x\rangle$ denote the distance from $x$ to the nearest <prime power>. For $x\geq2$ not an integer and $T\geq2$, the truncated <Riemann–von Mangoldt explicit formula> is
$$
\boxed{
\begin{aligned}
\psi(x)={}&x-
\sum_{\substack{\rho=\beta+i\gamma\\0\leq\beta\leq1,
\ |\gamma|\leq T}}
\frac{x^\rho}{\rho}
-\log(2\pi)-\frac12\log(1-x^{-2})\\
&+O\left(
\frac{x}{T}(\log(xT))^2
+(\log x)\min\left\{1,\frac{x}{T\langle x\rangle}\right\}
\right).
\end{aligned}}
$$
Zeros are counted with multiplicity. The constant term comes from $s=0$, the logarithm collects the trivial zeros $-2,-4,\ldots$, and the finite sum contains the nontrivial zeros. Enlarging the implied constant covers $1\leq T<2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 150 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-150/4/a/solution