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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 ...


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 ...


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.


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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