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Is the set of all solutions $x > 0$ to the equation $\pi(x) = \operatorname{li}(x)$ unbounded? Is $\liminf_{x \to \infty} |\pi(x)-\operatorname{li}(x)|$ equal to $0$? Here, $\pi(x)$ denotes the prime …

A complete answer is at Asymptotic behavior of a "strange" arithmetic function. The sum $\sum_{n \leq x} a_n$ is not $O(x (\log x)^A)$ for any $A$, due to the precise asymptotics stated there for the …

One has \begin{align*} \sum_{n \leq x}\frac{n}{\operatorname{rad}n} & = (1+o(1)) \, x \sqrt{\frac{2}{\log x \log \log x}} \exp\left((1+o(1))\sqrt{\frac{8\log x}{\log \log x}}\right) \ (x \to \infty), …

I'm curious about what is known about the distribution of the function $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t) \in (-1,1]$, on a linear or logarithmic scale, where $\operatorname{Arg}$ i …

As all analytic number theorists know, iterated logarithms ($\log x$, $\log \log x$, $\log \log \log x$, etc.) are prevalent in analytic number theory. One can give countless examples of this phenome …

This is my last question, building off of Riemann-Von Mangoldt formula and Does $\frac{1}{\pi}\operatorname{Arg} \zeta(1/2+2\pi i t)$ have mean value $0$?. I apologize for taking a while to understan …

Let $x >0$. How can one find good $O$ bounds on the integrals $$\int_0^x\frac{\operatorname{li}(t)-\pi(t)}{t}dt$$ and $$\int_x^\infty\frac{\operatorname{li}(t)-\pi(t)}{t^2}dt$$ where $\pi(x)$ is the …

Let $\pi(x)$ denote the prime counting function and $$\pi_0(x) = \lim_{\epsilon \to 0} \frac{\pi(x+\epsilon)+\pi(x-\epsilon)}{2}.$$ I've seen noted in a few references the explicit formula $$\pi_0(x) …

I think I might now have an answer to my question, in that the approximation I gave should be within $O(1)$ of $t_n$, and it can be related to the function $N(T)$. However, I am not yet sure in what …

NOTE: I've edited the question one last time, to be much simpler, in the hopes of getting more responses. SETUP: Let $p_n$ denote the $n$th prime, let $p_x = p_{\lceil x \rceil}$ for all $x > 0$, let …

Let $h(x)$ be a function that is positive on $\mathbb{R}_{>0}$ and satisfies $h(x) = o(x)$ and $(\log x)^a = o(h(x))$ for all $a > 0$, as $x \to \infty$. Is it reasonable to expect under these condi …

For all positive integers $n$, let $$t_n = \frac{1}{2\pi} \operatorname{Im} \rho_n,$$ where $\rho_n$ donates the $n$th nontrivial zero of the Riemann zeta function in the upper half plane (listed in i …

I'm curious about the Dirichlet series $$F(s) = \sum_{n = 1}^\infty \frac{H_n}{n^s}$$ of the sequence $H_n = \sum_{k = 1}^n \frac{1}{k}$ of harmonic numbers. Its abscissa of convergence is $1$. Wha …

Let $$N(T) = \#\{\rho \in \mathbb{C}: \zeta(\rho) = 0,\, \operatorname{Im} \rho \in (0,T]\}$$ denote the number of zeros of $\zeta(s)$, counting multiplicities, with imaginary part lying in the interv …

A 1950 result of Tur'an establishes an equivalence between any prime number theorem of the form $\operatorname{li}(x)-\pi(x)= O(xe^{-C(\log x)^\alpha}) \ (x \to \infty)$ and a certain class of zero-fr …