- Euclid proved there are infinitely many primes around 300 BCE, but later mathematicians sought to quantify how common primes are compared to other infinite sets like even numbers or powers of 2.
- Euler investigated whether the sum of the reciprocals of primes (1/2 + 1/3 + 1/5 + ...) is infinite or finite, linking it to the Riemann zeta function via the Euler product formula, which connects number theory and calculus.
- By taking logarithms and using series expansions, Euler showed that the sum of reciprocal primes diverges to infinity, indicating primes are not too sparse, though the sum grows extremely slowly, like log(log(n)).
- Gauss conjectured that the prime-counting function π(N) is approximately N/log(N) or the logarithmic integral Li(N), based on empirical data, providing a probabilistic estimate for primes.
- Riemann extended Gauss's work by deriving an explicit formula for π(x) involving zeros of the zeta function: π(x) ≈ Li(x) - Σ Li(x^ρ), where ρ are non-trivial zeros, with the main term Li(x) dominating.
- The Riemann hypothesis conjectures that all non-trivial zeros of the zeta function have real part 1/2, which would optimize Gauss's approximation and has profound implications for prime distribution but remains unproven.