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I am interested in the rate of decay of the sum $$\sum_{n=N}^{\infty}\frac{\phi(n)}{n^{s}}$$ where $\phi$ is Euler's totient function and $s>2$ (in which case the sum converges trivially). Of course …

I am looking for the tightest known bound for the sum $$\sum_{\substack{1\leq k\leq j^\alpha \\ k\mid j}}k^\lambda$$ where $j$ is a large positive integer, $\alpha\in(0,1)$ and $\lambda\geq 1$. I a …

This is a refinement (perhaps a simpler version) of a question I asked here before and couldn't get an answer for. Fix $\alpha \in (0,1]$ and a small constant $c>0$. For $x \in [0,1]$ and $N\in\mathbb …

Let $d(n)$ be the number of divisors of an integer $n$. Does there exists a bound for $\sum_{k\leq n}d^2(k)$? I saw in a paper of Barry and Louboutin that the asymptotics is $\frac1{\pi^2}nlog^3n$ but …

Here's a simplified version of a question I'm interested in. Given $p$ and $q$ distinct prime numbers, we consider sets $A\subset \mathbb{N}\cup\{0\}, 0\in A$ of size $pq$, which are uniformly distrib …