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4

Let's do a back of the envelope computation for a slightly simpler problem (when voting for one's own talk is allowed). It doesn't seem like it matters too much, but formally it is a bit different. There will be no formal proof, just an "educated guess" anyway. We have $n$ numbered urns (talks) and $n$ balls (votes) and the question is what ...

6

I threw together a quick Java program to calculate this for $N$ up to 10, and it actually seems to be more likely as $N$ grows, with a limit of slightly below $\frac12$: N = 2 ties: 1, no ties: 0, ratio: 1.0 N = 3 ties: 2, no ties: 6, ratio: 0.25 N = 4 ties: 21, no ties: 60, ratio: 0.25925925925925924 N = 5 ties: 344, no ties: 680, ratio: 0.3359375 N = 6 ...

6

James Maynard has a survey paper Digits of primes. In Primes with restricted digits he shows that for any digit $d\in\{0,1,\dotsc,9\}$ there are infinitely many primes that do not have $d$ in their decimal expansion. This might be the deepest known result in Digit Theory.

1

It may or may not be results of the type you are looking for, but one have deduced asymptotic formulae for the average value of the digit sum $S_b(n)$ considered a function of $n$ and considered a function of $b$. (See this post.) Specifically, one has $$\sum \limits_{n=1}^{N}S_b(n)\sim\frac{(b-1)\log(N)}{2N \log b} \quad \text{as} \; n\to\infty.$$ For the ...

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