Primitive roots for every admissible integer base
Source: github.com/openai/math/tree/main/preprints/Primitive-roots-…
We prove the infinitude assertion in Artin's primitive root conjecture: for every integer a that is neither −1 nor a square, there are at least $c_a x/(\log x)^2$ primes in $(x,2x)$ with primitive root a , for some $c_a\gt 0$ and every sufficiently large x .
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Another result announced by OpenAI is an answer here - their proof of Artin's primitive root conjecture depends crucially on the version of quasi-RH for Hecke L-functions of cyclotomic fields: Theorem 1.2: Let $F$ be a cyclotomic field containing $\mu_{12}$ . For every finite-order Hecke character…
If the decimal expansion of $\frac1p$ repeats every $k$ steps, it means that $\frac1p = \frac{x}{999\dots 9}$ where the denominator has $k$ $9$ 's, or in other words $\frac1p = \frac{x}{10^k-1}$ . (This does not preclude $\frac1p$ repeating more often than every $k$ steps.) Cross-multiplying,…
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A Brief Digest of Some New OpenAI Results | Persiflagegaloisrepresentations.orgFrank Calegari
So again, this is an interesting type of argument which I don’t find particularly surprising. But no one was going to write this paper, which depends on the quasi-Riemann hypothesis for Dirichlet \(L\)-functions, without knowing anything about general Dedekind \(L\)-functions. And this brings us…