Polymath5 – A somewhat surprising observation

In this post, Kevin O’Bryant has considered “least nonzero digit”-type examples \lambda_{m, S}. 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 \mu_{3} 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:

(39: 24)
(87: 560)
(95: 3.360)
(111: 5.280)
(119: 42.772)
(135: 5.940)
(145: 11.440)
(159: 64.064)
(183: 13.728)

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 \lambda_{105} implies \sum_{i=1}^{d-1} \lambda_{105}(i)=0 for d|105. Since 3,5,7|105 complete multiplicativity implies \lambda_{105}(2)=\lambda_{105}(3)=\lambda_{105}(5)=-1. Now 45^2\equiv 30 \mod 105 and again since \lambda_{105} is completely multiplicative 1=\lambda_{105}(30)=\lambda_{105}(2)\lambda_{105}(3)\lambda_{105}(5)=-1 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 \lambda_{m,S} that grows slower than \mu_3 must have m>729. Subsequent computer searches have increased that bound to 1000.

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