Daniel Goldston

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.

Goldston was named to the 2021 class of fellows of the American Mathematical Society "for contributions to analytic number theory".[4]

• Landau's problems
• Yitang Zhang
• James Maynard (mathematician)

• Dan Goldston's Homepage