Why is the Simple Zeros Conjecture said to be stronger than the Riemann Hypothesis? Let the "Simple Zeros Conjecture (SZC)" be the statement that all zeros of the Riemann zeta function are simple.
I have often heard of the statement that the SZC is stronger than the Riemann Hypothesis (RH). However, I have never seen or heard of any justification of this claim, and a quick internet search doesn't seem to reveal any result like SZC $\implies$ RH.
Therefore, can someone explain why the SZC is said to be stronger than the RH ?
 A: SZC is thought to be stronger than RH not because any proof exists that SZC implies RH
but because all existing hypotheses implying SZC are stronger than RH.
The most important of these involve the Mertens function $M(x)=\sum_{n\leq x}\mu(n)$
and include the generalised Mertens Hypothesis or GMH ($M(x) = O(x^{\frac{1}{2}}$))
and the slightly less drastic hypothesis that $\int_{1}^{X}(\frac{M(x)}{x})^2dx = O(\log(X)$
which GMH obviously implies. As far as is known however, neither of these follows from RH.
A: As Peter Humphries points out, the precise claim is that "RH + Simple Zeroes" is stronger than "RH". Of course, this is formally trivial.
So what's really meant is that "RH + Simple Zeroes" is a natural strengthening of RH
The reason for this is a generalization of the following simple fact:
Let $f$ be a monic polynomial in one variable of degree $n$ with real coefficients. Then $f$ has all roots real if and only if the coefficients of $f$ lie in a certain closed subset of $\mathbb R^n$, and $f$ has all roots real and simple if and only if the coefficients of $f$ lie in the interior of that closed subset.
So real roots + simple is just slightly stronger than real roots alone in a very natural way.
