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).
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An important breakthrough was the application of analytic tools to the problem by Hardy and Littlewood.
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In the early 20th century G. H. Hardy and Littlewood proved many results about the zeta function in an attempt to prove the Riemann Hypothesis.
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The founders of the theory were Paul Erdős, Aurel Wintner and Mark Kac during the 1930s, one of the periods of investigation in analytic number theory.
Timothy Browning is a mathematician working in number theory, examining the interface of analytic number theory and Diophantine geometry.
Theodor Estermann (1902–1991), mathematician, working in analytic number theory