Research proof index

Algebraic correctness of one KZG opening for a coefficient vector.

Powers with exponents congruent modulo p agree in a group of prime order p.

Extract the exponent equation enforced by a successful KZG opening verification.

lim_{t→∞} L₁,₀(it)/(F(it)G(it)) = 1/2.

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 $x\to\infty$.

Given a set of strings all distinguishable by l (i.e., not related to each other by the Nerode congruence on l), the number of states in the DFA accepting l is at least the number of strings in the set.