Daniel Goldston

Daniel Alan Goldston (born January 4, 1954 in Oakland, California) is an American mathematician who specializes in number theory. He is currently a professor of mathematics at San Jose State University.

Daniel Goldston
Born	(1954-01-04) January 4, 1954 Oakland, California, US
Nationality	American
Alma mater	UC Berkeley
Known for	GPY theorem in number theory
Awards	Cole Prize (2014)
Scientific career
Fields	Mathematics
Institutions	San Jose State University
Thesis	Large differences between consecutive prime numbers (1981)
Doctoral advisor	Russell Lehman
Influenced	Yitang Zhang

Goldston is best known for the following result that he, János Pintz, and Cem Yıldırım proved in 2005:[1]

${\displaystyle \liminf _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}=0}$

where ${\displaystyle p_{n}}$ denotes the n^th prime number. In other words, for every ${\displaystyle c>0\ }$, there exist infinitely many pairs of consecutive primes ${\displaystyle p_{n}\ }$ and ${\displaystyle p_{n+1}\ }$ which are closer to each other than the average distance between consecutive primes by a factor of ${\displaystyle c\ }$, i.e., ${\displaystyle p_{n+1}-p_{n}<c\log p_{n}\ }$.

This result was originally reported in 2003 by Goldston and Yıldırım but was later retracted.[2][3] Then Pintz joined the team and they completed the proof in 2005.

In fact, if they assume the Elliott–Halberstam conjecture, then they can also show that primes within 16 of each other occur infinitely often, which is related to the twin prime conjecture.