The <Truncated Perron formula> says that if $F(s)=\sum_{n\geq1}a_nn^{-s}$ converges absolutely for $\Re s>\sigma_a$, then for $c>\sigma_a$, $T\geq2$, and $x$ not an integer,
$$
\sum_{n\leq x}a_n
=\frac1{2\pi i}\int_{c-iT}^{c+iT}F(s)\frac{x^s}{s}\,ds
+O\left(\sum_{n\geq1}|a_n|\left(\frac xn\right)^c
\min\left\{1,\frac1{T|\log(x/n)|}\right\}\right).
$$
Take $a_n=\Lambda(n)$ and $c=1+1/\log x$. The <logarithmic derivative> identity gives $F(s)=-\zeta'(s)/\zeta(s)$. Since $x-1/2$ is an integer, $|x-n|\geq1/2$ for every integer $n$. In the range $x/2<n<2x$,
$$
|\log(x/n)|\asymp\frac{|x-n|}{x},
$$
and hence the contribution there is
$$
\ll\frac{x\log x}{T}\sum_{x/2<n<2x}\frac1{|x-n|}
\ll\frac{x(\log x)^2}{T}.
$$
The ranges $n\leq x/2$ and $n\geq2x$ are bounded by the same quantity using absolute convergence and $-\zeta'(c)/\zeta(c)\ll\log x$. Therefore
$$
\sum_{n\leq x}\Lambda(n)
=-\frac1{2\pi i}\int_{1+1/\log x-iT}^{1+1/\log x+iT}
\frac{\zeta'(s)}{\zeta(s)}\frac{x^s}{s}\,ds
+O\left(\frac{x(\log x)^2}{T}\right).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 150 / 2 / a / Solution

Put $t=s-1$. The <Laurent series> of the <logarithmic derivative> at the simple pole of $\zeta$ has the form
$$
\frac{\zeta'(s)}{\zeta(s)}=-\frac1t+c_0+O(t),
$$
where in fact $c_0=\gamma$. Hence
$$
\left(\frac{\zeta'}\zeta(s)\right)^2
=\frac1{t^2}-\frac{2c_0}{t}+O(1),
\qquad
\frac{x^s}{s}
=x\bigl(1+t(\log x-1)+O(t^2)\bigr).
$$
The coefficient of $t^{-1}$ in their product, and therefore the <residue>, is
$$
x(\log x-1-2c_0)=x\log x+Ax,
$$
with the constant $A=-1-2c_0$; equivalently, $A=-1-2\gamma$.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 150 / 3 / b / Solution

{c}

The classical zero-free region asserts that for some $c>0$,
$$
\zeta(\sigma+it)\ne0
\quad\text{if}\quad
\sigma\geq1-\frac c{\log(|t|+3)}.
$$
Together with bounds for the <logarithmic derivative>, it permits contour arguments with exponentially small errors in $\sqrt{\log x}$.

Zero-free region of the Riemann zeta function

{wiki}

The logarithmic derivative of a nonzero differentiable function $f$ is $f'/f$. For $\Re s>1$, the <Euler product> for the <Riemann zeta function> gives
$$
-\frac{\zeta'(s)}{\zeta(s)}
=\sum_{n\geq1}\frac{\Lambda(n)}{n^s}.
$$

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