Research proof index

A prime number theorem in arithmetic progressions, in Chebyshev-weighted form. If $q\ge1$, $a<q$, and $\gcd(a,q)=1$, then $\sum_{\substack{p\le x\\p\equiv a\pmod q}}\log p\sim\frac{x}{\varphi(q)}$ as $x\to\infty$.

Dirichlet's theorem on primes in arithmetic progressions. If $q\ge 1$, $a<q$, and $\gcd(a,q)=1$, then infinitely many primes satisfy $p\equiv a\pmod q$.

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 $x\to\infty$.

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.

Consecutive prime gaps are small relative to the primes themselves. If $p_n$ is the $n$th prime, then $p_{n+1}-p_n=o(p_n).$ Formally, there is a sequence $c_n\to0$ such that $p_{n+1}-p_n=c_np_n$ for every $n$.

For every $\varepsilon>0$, every sufficiently large real number $x$ has a prime $p$ in the short multiplicative interval $x<p<(1+\varepsilon)x$.