[6] In 1959, Gallagher received a PhD from Princeton University with a doctoral dissertation entitled Metric Diophantine Approximation in One and Several Dimensions completed under the supervision of Donald C.
[1] He spent one year living in the Latin Quarter of Paris before becoming an assistant professor at Columbia University in 1962.
[10][11][1] Gallagher received the Columbia University Presidential Teaching Award in 2005[7] and became director of undergraduate studies in the department of mathematics in 2013.
[3] In the 1960s and 1970s, Gallagher proved several results in large sieve methods in analytic number theory and simplified key ingredients used in the proof of the Bombieri–Vinogradov theorem.
[12][13] He also applied the large sieve to study the asymptotics of Galois groups of monic integral polynomials of bounded height, improving on results by van der Waerden.