https://ift.tt/2NUC5Ew Last time we revisited Robin’s theorem saying that 5040 being the largest counterexample to the bound \[ \frac{\sigma(n)}{n~log(log(n))} < e^{\gamma} = 1.78107... \] is equivalent to the Riemann hypothesis. There’s an industry of similar results using other arithmetic functions. Today, we’ll focus on Dedekind’s Psi function \[ \Psi(n) = n \prod_{p | n}(1 + \frac{1}{p}) \] where $p$ runs over the prime divisors of $n$. It is series A001615 in the online encyclopedia of integer sequences and it starts off with 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, … and here’s a plot of its first 1000 values To understand this behaviour it is best to focus on the ‘slopes’ $\frac{\Psi(n)}{n}=\prod_{p|n}(1+\frac{1}{p})$. So, the red dots of minimal ‘slope’ $\approx 1$ correspond to the prime numbers, and the ‘outliers’ have a maximal number of distinct small prime divisor...