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- W2767928218 abstract "For $r in mathbb{R}, r> 1$ and $n in mathbb{Z}^+$, the divisor function $sigma_{-r}$ is defined by $sigma_{-r}(n) := sum_{d vert n} d^{-r}$. In this paper we show the number $C_r$ of connected components of $overline{sigma_{-r}(mathbb{Z}^+)}$ satisfies $$pi(r) + 1 leq C_r leq frac{1}{2}expleft[frac{1}{2}dfrac{r^{20/9}}{(log r)^{29/9}} left( 1 + frac{loglog r}{log r - loglog r} + frac{mathcal{O}(1)}{log r}right) right],$$ where $pi(t)$ is the number of primes $pleq t$. We also show that $C_r$ does not take all integer values, specifically that it cannot be equal to $4$." @default.
- W2767928218 created "2017-11-17" @default.
- W2767928218 creator A5087889635 @default.
- W2767928218 date "2017-11-08" @default.
- W2767928218 modified "2023-09-27" @default.
- W2767928218 title "On the Number of Connected Components of Ranges of Divisor Functions" @default.
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