Research proof index

Let $f$ and its logarithmic derivative $f'/f$ be meromorphic on a set $R$, and assume the meromorphic order of $f$ is finite at every point of $R$. Then the poles of $f'/f$ in $R$ are exactly the points where the divisor of $f$ is nonzero, namely the zeros and poles of $f$.

Suppose $f$ is holomorphic and nonzero on a punctured neighborhood of $p$, and $f(s)=\frac{A}{s-p}+O(1)$ there with $A\ne0$. Then the logarithmic derivative has principal part $-1/(s-p)$: $\frac{f'(s)}{f(s)}+\frac{1}{s-p}=O(1)$ as $s\to p$. In particular, the residue of $f'/f$ at this simple pole is $-1$.

The logarithmic derivative of the Riemann zeta function is bounded in the half-plane Re(s) >= 3/2.

Let $0<r<R$, and let $B$ be analytic and nonvanishing on the closed disc $|z|\le R$. Then there is an analytic function $J_B$ on $|z|<R$ with $J_B(0)=0$, $J_B'(z)=\frac{B'(z)}{B(z)}\quad(|z|\le r),$ and $\operatorname{Re}J_B(z)=\log|B(z)|-\log|B(0)|\quad(|z|<R).$ Thus $J_B$ is a normalized analytic logarithm of $B/B(0)$.

Mellin convolution is commutative at positive arguments. For functions $f,g:\mathbb R\to\mathbb R$ or $\mathbb C$ and every $x>0$, $(f*_M g)(x)=(g*_M f)(x),$ where $(f*_M g)(x)=\int_0^\infty f(y)g(x/y)\,dy/y$.

Let $\nu$ be continuously differentiable and supported in $[1/2,2]$. For $\varepsilon>0$ and $\operatorname{Re}s>0$, the Mellin transform of the project's smoothed cutoff satisfies $\mathcal M(\widetilde{1_\varepsilon})(s)=\frac{1}{s}\,\mathcal M(\nu)(\varepsilon s).$