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I think we should have $$f_\epsilon(N) \asymp_\epsilon \frac{N}{(\log N)^{\delta_0}(\log\log N)^{3/2}},$$ where $X \asymp_\epsilon Y$ means $X \le C_\epsilon Y$ and $X \ge c_\epsilon Y$ for some const …

Let $p_1,\dots,p_k$ be the first $k$ primes. Let $q = p_1\dots p_k$. By CRT there's some $m \ge 1$ so that $m+j \equiv 0 \pmod{p_j}$ for $1 \le j \le k$. Then $$\sum_{\substack{n=1 \\ (n,q)=1}}^m 1 = …

Ok, apologies if this is overkill, but this paper shows that for almost every (irrational) $\alpha \in \mathbb{R}$, only a measure $0$ set of $x \in \mathbb{R}$ satisfy $S_{N,\alpha}(x) = o(\sqrt{N})$ …

The same second moment argument, the one that shows almost all $n$ have $\bigl(1+o(1)\bigr)\log\log n$ distinct prime factors, shows that almost all $n$ have $\bigl(\frac{1}{\phi(b)}+o(1)\bigr)\log\lo …

Ford's methods provide lower bounds for this "asymmetric" multiplication table problem that match the lower bounds for the standard multiplication table problem. Define $H(x,y,z) := \#\{n \le x : \exi …

I think it's a standard exercise in summation by parts if I didn't make a mistake. We wish to show $$\lim_{\epsilon \to 0^+} \epsilon \sum_{n \ge 1} \frac{a_n}{n^{1+\epsilon}} = R.$$ The sum is the li …

The following human-verifiable proof is in collaboration* with Fedja. Lemma $1$: We have the following for $0 \le x \le 3$: $$\frac{6204}{6750}x^2-\frac{8429}{60750}x^4+\frac{4475}{546750}x^6\le x.$$ …