Research proof index

The summatory Möbius function has sublinear growth: as $x\to\infty$, $\sum_{n<\lfloor x\rfloor}\mu(n)=o(x).$ This is the Möbius-function form of the prime number theorem.

Fourier-transform decay from an integrable derivative: for integrable, differentiable g with integrable derivative, ‖𝓕 g w‖ ≤ (∫ ‖deriv g x‖) / (2π·|w|).

For $a>0$ and $1\le R\le B$, the damped sinc integral over the finite tail $[R,B]$ satisfies the uniform estimate $\left|\int_R^B e^{-ax}\operatorname{sinc}(x)\,dx\right|\le\frac{4}{R}.$

The oscillatory-integral form of the decay bound: for 0 < T, ‖∫ g y · exp(T·i·y)‖ ≤ (∫ ‖deriv g x‖) / T.

The |T| variant of the oscillatory-integral decay bound: for T ≠ 0, ‖∫ g y · exp(T·i·y)‖ ≤ (∫ ‖deriv g x‖) / |T|.

Let $f$ be integrable, $R>0$, and $K_T(u)=\sin(Tu)/(\pi u)$ for $u\ne0$, with the project's value $K_T(0)=0$. Removing the part of the convolution outside $[-R,R]$ incurs at most $\left\|\int_{\mathbb R}K_T(u)f(x-u)\,du-\int_{-R}^{R}K_T(u)f(x-u)\,du\right\|\le\frac{1}{\pi R}\int_{\mathbb R}\|f(u)\|\,du.$ The estimate is uniform in $x$ and $T$.