In this post, Kevin O’Bryant has considered “least nonzero digit”-type examples . Encouraged by his remark that there are only very few possibilities, i.e. feasible sets S, for each modulus m I did a complete enumeration up to modulus 200. The goal was (and still is) to find a slowly growing example possibly beating the current record holder (please read my last post for more information).
Much to my surprise there are certain “magic” numbers with comparatively many feasible sets S. What follows is a table (modulus m: number of sets S) for those m with more than 4 sets:
All other m < 200 have 4 or less feasible sets.
For example, there is no set S for m=105. This can be seen as follows: Logarithmic discrepancy of implies for . Since complete multiplicativity implies . Now and again since is completely multiplicative produces a contradiction.
What makes these number “special”? Why is there this huge jump from 4 to “many”? Is this just one more “random” number theoretic artifact or can we understand this by considering how the m-examples are constructed from the d|m-examples?
I do not know the answers, but let me close with something I do know. In my last post I gave a proof that any that grows slower than must have . Subsequent computer searches have increased that bound to 1000.