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Modern times

The biggest technical change after 1950 has been the development of sieve methods, particularly in multiplicative problems. These are combinatorial in nature, and quite varied. The extremal branch of combinatorial theory has in return been greatly influenced by the value placed in analytic number theory on quantitative upper and lower bounds. Another recent development is probabilistic number theory, which uses methods from probability theory to estimate the distribution of number theoretic functions, such as how many prime divisors a number has.

Specifically, the breakthroughs by Yitang Zhang, James Maynard, Terence Tao and Ben Green have all used the Goldston – Pintz – Yıldırım method, which they originally used to prove that

p n + 1 − p n ≥ o (log p n). {\displaystyle p_{n+1}-p_{n}\geq o(\log p_{n}).}

Developments within analytic number theory are often refinements of earlier techniques, which reduce the error terms and widen their applicability. For example, the circle method of Hardy and Littlewood was conceived as applying to power series near the unit circle in the complex plane; it is now thought of in terms of finite exponential sums (that is, on the unit circle, but with the power series truncated). The needs of Diophantine approximation are for auxiliary functions that are not generating functions —their coefficients are constructed by use of a pigeonhole principle —and involve several complex variables. The fields of Diophantine approximation and transcendence theory have expanded, to the point that the techniques have been applied to the Mordell conjecture.