Prime Number Theorem and More

For positive parameters $A,B,C,R$, the classical error-bound function $A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ is nonincreasing once $x\ge \exp\!\left(R(2B/C)^2\right)$.

Let $f$ be analytic on an open set $s$ containing $0$, and suppose $f(0)=0$. Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$. Then the apparent singularity at $0$ is removable and $g$ is analytic throughout $s$.

Suppose $R>0$ and $f$ is analytic on the closed disc $|z|\le R$ with $f(0)=0$. Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$. Then $g$ is analytic on the entire closed disc, including at the removed singularity.

The L-series of d k equals ζ(s)^k for Re(s) > 1.

If g is a multiplicative arithmetic function, then for any $n \neq 0$, $\sum_{d | n} \mu(d) \cdot g(d) = \prod_{p | n} (1 - g(p))$.

The series ∑ f(n)/n · 𝓕ψ(log(n/x)/(2π)) is summable for x ≥ 1.

Every zero $s$ of the Riemann zeta function in the upper half-plane lies in the critical strip: if $\zeta(s)=0$ and $\operatorname{Im}s>0$, then $0\le\operatorname{Re}s\le1.$

The Riemann zeta zero-counting function has a crude polynomial bound: there is a constant $A>0$ such that, for every $T\ge2$, $|N_\zeta(T)|\le A T^{3/2}.$