Research proof index

Let $E_\psi(x)=|\psi(x)-x|/x$. If $E_\psi$ obeys the classical bound $E_\psi(x)\le A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ for every $x\ge x_0$, with $A,B,C,R>0$, then beyond any $x_1\ge\max\{x_0,\exp(R(2B/C)^2)\}$ it obeys the uniform numerical bound obtained by evaluating that expression at $x_1$.

Let $f:\mathbb R\to\mathbb R$. If the same function, viewed as taking values in $\mathbb C$, has derivative $u\in\mathbb C$ at a real point $z$, then $u$ is real and equals the ordinary real derivative of $f$ at $z$.

An integration-by-parts identity for real- or complex-valued $f$. If $a\le b$, $f$ is differentiable on $[a,b]$, and $f'$ is integrable there, then for every constant $c$, $\int_a^b(t+c)f'(t)\,dt=(b+c)f(b)-(a+c)f(a)-\int_a^b f(t)\,dt.$

For $0\le a\le b$ and a differentiable function $f$ whose derivative is continuous on $[a,b]$, $\int_a^b f'(t)(\lfloor t\rfloor+1)\,dt=(b+\tfrac12)f(b)-(a+\tfrac12)f(a)-\int_a^b f(t)\,dt-\int_a^b f'(t)B_1(t)\,dt,$ where $B_1(t)=t-\lfloor t\rfloor-\tfrac12$.

The summatory Liouville function has sublinear growth. Writing $\Omega(n)$ for the number of prime factors of $n$, counted with multiplicity, the theorem states $\sum_{n<\lfloor x\rfloor}(-1)^{\Omega(n)}=o(x)$ as $x\to\infty$.

A boundary Fourier identity for a Dirichlet series with a simple pole. Suppose $f\ge0$, the Dirichlet series $L(f,s)$ differs from $A/(s-1)$ by a function $G$ that extends continuously to $\operatorname{Re}s\ge1$, and $\psi$ is a compactly supported $C^2$ test function with nonnegative real Fourier transform. For $x>0$, $\sum_{n\ge1}\frac{f(n)}{n}\widehat\psi\!\left(\frac{1}{2\pi}\log\frac{n}{x}\right)-A\int_{-\log x}^{\infty}\widehat\psi\!\left(\frac{u}{2\pi}\right)du=\int_{\mathbb R}G(1+it)\psi(t)x^{it}\,dt.$