Note on a question of Wilf

Abstract

Let $S$ be a numerical semigroup with Frobenius number $f$, genus $g$ and embedding dimension $e$. % In 1978 Wilf asked the question, whether $\frac{f+1-g}{f+1}\ge \frac{1}{e}$. As is well known, this holds in the cases $e=2$ and $e=3$. For $e\ge 4$, we derive from results of Zhai [5] the following (substantially weaker) lower bound \[\frac{f+1-g}{f+1}>\left(\frac{2N+1}{(2N+2)(e-2)}\right)^e\text{ with }\lfloor N\rfloor=104978\,.\] To the best of our knowledge this is the first explicit lower bound for $\frac{f+1-g}{f+1}$ in terms of the embedding dimension.

Note that our new paper arXiv:2107.06752 [math.NT] contains a stronger estimate for d(S). Typing error corrected, 4 pages