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Use $1-x \ge e^{-\frac{x}{1-x}}$ for $x \in (0,1)$ to establish $$\ln \prod_{p \in S}\bigg(1-\frac{1}{p+1}\bigg) \ge -\sum_{p \in S} \frac{1}{p}.$$ This is tight since $1-x=e^{-x + O(x^2)}$ in a nei …

Show that $$\left( \frac{n}{\phi(n)} \right)^a =\prod_{p \mid n} \left(1 + \frac{1}{p-1} \right)^a \le \prod_{p \mid n} \left(1 + \frac{C}{p} \right) = \sum_{d \mid n} \frac{\mu(d)^2}{d} C^{\Omega(d) …

One can improve on $X^{2n-a}$ as long as $(a,n)\neq(1,1)$ (for $a=n=1$, the sum grows like $X/\zeta(2)$ so there's no room for improvement). Let us introduce $$f_n(m) := \# \{ (x_1,\ldots,x_n) : \math …

The circle method was an inspiration behind the solution of the toroidal semiqueens problem by Eberhard, Manners and Mrazović ("Additive triples of bijections, or the toroidal semiqueens problem", JEM …

Joshua gave nice argument, let me provide my perspective on 1, with several arguments. The bottom line is that the constant $k^{\ell}$ comes from $f(1)=k^{\ell}$, and that very little information on $ …

After enough searches, I've found the answer in the literature, at least for $y$ which is not too small w.r.t $x$. I would still appreciate input on smaller $y$, or on older references. As is often th …

You basically ask about the sum $$ \sum_{n \le x} \alpha(n)$$ where $\alpha$ is a completely multiplicative function with $\alpha(p) = \mathbf{1}_{p \notin \mathcal{P}}$. This is addressed by Wirsing …

GH from MO's answered your question in full. Some possible points of interest: Estermann was the first to evaluate $\sum_{m \le N} r(m)r(m+k)$ in "An asymptotic formula in the theory of numbers" (Pro …

Let $k(n)=\prod_{p \mid n}p$ be the kernel of the integer, so that $f(n)=n/k(n)$. As indicated in p. 7 of Finch's article, $$\sum_{n\le x} \frac{1}{k(n)} = \exp\left( \left( \frac{8\log x}{\log \log x …

The authors seem to claim that if a real quantity $X$ satisfies $X \le N$ and $$\tag{$*$}\label{483297_star}X^2\le N+\frac{2N^2}{q}+4(N+q)\log q$$ then $$X\le 2\frac{N}{\sqrt{q}}+\sqrt{q}\log q.$$ Her …

R. W. K. Odoni has his own proof of the theorem, see this paper on the problem, "On norms of integers in a full module of an algebraic number field and the distribution of values of binary integra …

In function fields, the Gauss Circle Problem, at least in its usual formulation, is much simpler than in number fields. In a number field $K$, one studies the difference $\sum_{n \le x} r_{K}(n) - C_ …

Effective versions of Wirsing's theorems are worked out in a recent paper of G. Tenebaum: ``Moyennes effectives de fonctions multiplicatives complexes'', published in Ramanujan J. 44 (2017), no. 3, 64 …

Balasubramanian, Ivić and Ramachandra ("An application of the Hooley-Huxley contour", Acta Arith. 65 (1993), no. 1, 45–51) prove the asymptotic result found by Lucia and Terry Tao. Their error term is …

Your function $\beta$ is a completely additive function, in the sense that $\beta(nm)=\beta(n)+\beta(m)$ for all $n,m$. There is a vast literature on the statistical behavior of additive functions, e. …