Recently I read about some attempts of finding the exact order of Mertens function. It is conjectured that $$0<\limsup_{x\to\infty}\frac{M(x)}{\sqrt x(\log\log\log x)^{5/4}}<\infty\ (\text{Gonek's conjecture}),$$ and this conjecture is supported by probabilistic evidence that $M(e^y)e^{-y/2}$ has a limiting distribution. It would be surprising either if this conjecture is proven (which leads to Riemann Hypothesis) or if it is proven false despite of the strong heuristic probabilistic evidence.

Question: is there any famous conjecture which has probabilistic evidence to suggest it is true initially, but later it was proven false?

By probabilistic evidence I mean evidence such as $S=\sum_{n=1}^\infty\frac{\sin n^2x}n$ converges with probability one, suggesting that series $\sum_{n=1}^\infty\frac{\sin n^2}n$ converges. Note that it does not mean for all $x$, $S$ converges, but such an $x$ making $S$ diverges is hard to construct so even if there is an $x$ satisfying that condition, it does not count as an example.

Another possible example will be the normality of $\sqrt2$ in a chosen base. Borel proved$^{[1]}$ that almost all (w.r.t. Lebesgue measure) real numbers are normal. This is still open, and I wonder whether there is an irrational number which is proven not to be normal, but is not constructed digit-by-digit such as $0.123456789101112131415161718\cdots$. Here the probabilistic evidence is the result Borel proved.

[1]Borel, É. "Les probabilités dénombrables et leurs applications arithmétiques." Rend. Circ. Mat. Palermo 27, 247-271, 1909.

  • $\begingroup$ I think that probabilistic heuristics in a discrete setting like number theory have more power than the examples you give. In continuous settings, generally many of the subsets you are interested have measure zero (e.g. rationals, algebraic numbers, etc. in $\mathbb{R}$) so the fact that something holds almost-everywhere doesn't have much force in my mind. $\endgroup$ – Jair Taylor Apr 29 at 17:18
  • $\begingroup$ I agree that I should not look for those zero-measured sets. Actually I'm just trying to convince myself that "almost surely" does not guarantee the freedom of use of the result by using a counterexample. $\endgroup$ – Kemono Chen Apr 29 at 17:33
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    $\begingroup$ I'm not sure the heuristic evidence for this conjecture is so strong. This paper (on p. $480$) says that the conjecture doesn't fit well with their data, and suggests that the exponent should be $\frac12$ instead of $\frac54$. $\endgroup$ – joriki Apr 29 at 18:04

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