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

{c}

The <Riemann hypothesis> is equivalent to
$$
\psi(x)=x+O_\epsilon(x^{1/2+\epsilon})
$$
for every $\epsilon>0$. In fact, the hypothesis and the <Riemann–von Mangoldt explicit formula> give the stronger $O(x^{1/2}(\log x)^2)$ bound. Conversely, the stated error continues $-\zeta'/\zeta-s/(s-1)$ holomorphically to $\Re s>1/2$, excluding zeros there; the functional equation supplies the other half.

Riemann hypothesis equivalence for the second Chebyshev function

{c}

For $x>1$ away from a jump of $\psi$,
$$
\psi(x)=x-\sum_\rho\frac{x^\rho}{\rho}
-\log(2\pi)-\frac12\log(1-x^{-2}),
$$
where the sum over <Nontrivial zeros of the Riemann zeta function> is taken symmetrically. A finite-height version has an explicit truncation error controlled by $x/T$ and the distance from $x$ to the nearest prime power.

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 Incoming links: Riemann–von Mangoldt explicit formula