arXiv subjects

Geoffrey D. Smith

Publications and source records attributed to Geoffrey D. Smith.

2 recordsLinked to original sources

arXiv2014

An explicit approach to the Ahlgren-Ono conjecture

Let $p(n)$ be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers $N$ for which $p(N)$ is not congruent to $0\pmod{3}$. Radu proved this conjecture in 2010 using work of Deligne and Rapoport. In this note, we give a simpler proof of Ahlgren and Ono's conjecture in the special case where the modulus of the arithmetic progression is a power of $3$ by applying a method of Boylan and Ono and using work of Bellaïche and Khare generalizing Serre's results on the local nilpotency of the Hecke algebra.

math.NT↗

arXiv2014

Bounded gaps between primes in special sequences

We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor nα\rfloor\}$, where $α$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some properties described by Leitmann, we show that for all $m$ there are infinitely many bounded intervals containing $m$ primes and at least one integer of the form $\lfloor f(q)\rfloor$ with $q$ a positive integer.

math.NT↗