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The Riemann hypothesis is true, if primes are random in certain ways.

There are two main sources of repeated logs. (These sources can be further refined into natural subcategories, but I'll only mention a couple of those subcategories.) Those two main sources are: Typ …

I think that there is indeed some possibility to lower the bound, and this is something I've looked at seriously a few times. I spent a semester (in 2019) with the Computational Number Theory Group h …

Let $f$ be a monotone decreasing, continuously differentiable function with $\lim_{x\rightarrow \infty}f(x)=0$. Let $\chi$ be a non-principal Dirichlet character. It is standard to show that $\sum_{ …

Suppose that $p$ is a prime number that divides $2^{2n+1}-1$. This means that $2^{2n+1}\equiv 1\pmod{p}$. Consequently, the order of $2$ modulo $p$ must be an odd number dividing $2n+1$. On the othe …

An update on this problem: I found out how to compute effective (and asymptotically accurate) bounds for $\sum_{p\leq x,\, p\equiv a\pmod{k}}\log(p)/p$. Basically it boils down to the usual analytic …