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The Riemann hypothesis (or, how to earn $1 million)

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One of the oldest theorems in mathematics is that there are infinitely many primes. Euclid (the geometer!) proved this around 300 BCE, and for centuries afterwards the story ended there. There are infinitely many primes, what else is there to ask? However, some infinite sets are more `common' than others. For example, even numbers are pretty common, but powers of 2 are pretty rare: between 1 1 and 1000 , 1000, there are 500 even numbers, but only ten powers of 2. Thus, an interesting follow up question to Euclid's proof is to ask: can we quantify how common prime numbers are? One way of expressing the idea of that there are more even numbers than powers of 2 is to note that 1 2 + 1 4 + 1 6 + 1 8 + 1 10 + ⋯ = ∞ , 2 1 ​ + 4 1 ​ + 6 1 ​ + 8 1 ​ + 10 1 ​ +⋯=∞, but 1 2 + 1 2 2 + 1 2 3 + 1 2 4 + 1 2 5 + ⋯ = 1. 2 1 ​ + 2 2 1 ​ + 2 3 1 ​ + 2 4 1 ​ + 2 5 1 ​ +⋯=1. In other words, the powers of 2 are so far apart that when we replace 2 𝑛 2 n with 1 2 𝑛 2 n 1 ​ (turn

The Riemann hypothesis (or, how to earn $1 million) ← back The Riemann hypothesis (or, how to earn $1 million) February 13, 2026 One of the oldest theorems in mathematics is that there are infinitely many primes. Euclid (the geometer!) proved this around 300 BCE, and for centuries afterwards the story ended there. There are infinitely many primes, what else is there to ask? However, some infinite sets are more `common' than others. For example, even numbers are pretty common, but powers of 2 are pretty rare: between \(1\) and \(1000,\) there are 500 even numbers, but only ten powers of 2. Thus

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