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I heard that many theorems in modern mathematics depend on the unproven Riemann Hypothesis. What are some examples of such theorems, and what will be the impact on these results if the Riemann Hypothesis is proven to be true?

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  • $\begingroup$ Not exactly theorem, but certain probabilistic primality tests such as Miller-Rabin can be turned into deterministic primality tests with polynomial complexity assuming the Generalized Riemann Hypothesis. $\endgroup$ – Tob Ernack Mar 15 '18 at 0:10
  • $\begingroup$ @TobErnack What would happen to the MR primality test if RH was false? $\endgroup$ – user83024 Mar 15 '18 at 0:13
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    $\begingroup$ Perhaps take a look at this Wikipedia entry for several consequences related to error bounds $\endgroup$ – sharding4 Mar 15 '18 at 0:23
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The first response here gives some great answers. Most of which directly relate to the distribution of primes.

Also, there are some interesting consequences of the Generalized Riemann Hypothesis.

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