I am interested to know other examples vacuously true statements that are non-trivial. My starting example is Turan's result in regards to the Riemann hypothesis, which states


Suppose that for each $N \in \mathbb{N}\_{>0}$ the function $\displaystyle \sum\_{n=1}^N n^{-s}$ has no zeroes for $\mathfrak{R}(s) > 1$. Then the function
$$\displaystyle T(x) = \sum\_{1\leq n \leq x} \frac{\lambda(n)}{n}$$
is non-negative for $x \geq 0$. In particular, this would imply the Riemann hypothesis.


Here $\lambda(n) = \lambda(p\_1^{a\_1} \cdots p\_r^{a\_r}) = (-1)^r$ is the Liouville function.


The interesting thing about this statement is that both the hypothesis and the consequence can be proven false independently. In particular, Montgomery showed in 1983 that for all sufficiently large $N$ the above sums have zeroes with real parts larger than one, and Haselgrove showed in 1958 that $T(x)$ is negative for infinitely many values of $x$. Peter Borwein et al. found the smallest such $x$ in 2008.


I find this result fascinating because it relates to a well-known conjecture, and both the hypothesis and consequence were plausible. Are there any other mathematical facts of this nature, perhaps in other areas?



Famous vacuously true statements
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