A teenage Gauss scribbled the rule for primes in his logarithm table
Around 1792, aged 15 or 16, Carl Friedrich Gauss jotted a short note in his book of logarithms guessing how many primes lie below any large number. He never published it and only mentioned it in a letter almost sixty years later. Proving his hunch took until 1896 and a detour through complex numbers.
Analytic number theory attacks questions about whole numbers with the tools of calculus and analysis. Its founding moment is usually dated to 1837, when Peter Gustav Lejeune Dirichlet used new functions, now called L-functions, to show that any arithmetic sequence whose starting value and step share no common factor contains infinitely many primes. The field splits roughly into two camps: one studies how primes are scattered, the other how integers break into sums.
The prime number theorem is the centrepiece. Counting primes up to 10 gives four, namely 2, 3, 5 and 7, and the theorem says that for huge numbers N the count is close to N divided by the natural logarithm of N. Adrien-Marie Legendre proposed a version around 1797, refined it in 1808, and Dirichlet offered a better approximation in 1838. Pafnuty Chebyshev, in papers of 1848 and 1850, showed the ratio stays pinned between two constants near 1, enough to prove there is always a prime between any n of at least 2 and double n.
Bernhard Riemann's single paper on number theory, from 1859, linked the distribution of primes to the zeros of the zeta function and threw off the Riemann hypothesis along the way. Building on it, Jacques Hadamard and Charles de la Vallée-Poussin independently proved the theorem in the same year, 1896.
The additive side has its own triumphs. Lagrange showed in 1770 that every positive integer is a sum of at most four squares, and Hilbert proved in 1909 that similar bounds exist for any power, though his method gave no actual numbers; Hardy and Littlewood later brought analysis to bear. Some questions still resist. Nobody has proved there are infinitely many twin primes differing by 2, but it is now known unconditionally that infinitely many prime pairs sit at most 246 apart, a result built on a method used by Yitang Zhang, James Maynard and Terence Tao.
Source: Analytic number theory
