6 papers
A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound
Tristan Freiberg
We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime can cover an interval of l…
Long runs of integers with small prime factors and the divisor function of
Tristan Freiberg
Let be the divisor function, and let be the least positive integer for which . Erdős, Graham, Ivić and Pomerance proved that, for infinitely…
Sparser variance for primes in arithmetic progression
Roger Baker, Tristan Freiberg
We obtain an analog of the Montgomery-Hooley asymptotic formula for the variance of the number of primes in arithmetic progressions. In the present paper the moduli are restricted…
Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers
Tristan Freiberg, Pär Kurlberg, Lior Rosenzweig
We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hard…
Limit points and long gaps between primes
Roger Baker, Tristan Freiberg
Let , where denotes the th smallest prime, and let (the "Erd{\H o}s--Rankin" function). We consider the…
Short intervals with a given number of primes
Tristan Freiberg
A well-known conjecture asserts that, for any given positive real number and nonnegative integer , the proportion of positive integers for which the interval $(n,n…