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Statement 1 : (Robin) proved that if the R.H. is false then there exist constants $0<\beta <\frac{1}{2}$ and $c>0$ small , such that $\sum \limits_{d|n} d \geq e^\gamma n \ln \ln n+ n\frac{ c \ln \ln …

It is well-known that $|\pi(x)-\operatorname{li}(x)| \leq \epsilon(x)$, where $\pi(x) = \sum \limits_{p \leq x} 1$ is the prime counting function, where $\operatorname{li}(x) = \int \limits_{2}^{x}\fr …

For start, is $\int \limits_{2}^{x} \frac{e^{-0.3\sqrt{\ln(t)}}}{\ln^2(t)} dt \leq x\ln(\ln(x)) \frac{e^{-0.3\sqrt{\ln(x)}}}{\ln^2(x)}$ ? If the above is true, what is a better bound for the integral …

What is the best unconditional upper bound for $p_{n^2}-p_{(n-1)^2}$ such that $p_n$ is the $n$-th prime number? Asymptotics suggest it's somewhere near $4 n \ln n$, but how to prove this? Edit: it' …

A positive integer $n$ is called a Hilbert number if $\exists a,b,d \in \mathbb{N}$ such that $ 4ab-a-b = d n$ and $d|a b$. I ran an algorithm checking divisors for all $0\lt a,b\le500$, and the onl …