arXiv subjects
Publications and source records attributed to Cooper O'Kuhn.
We prove an averaged version of a claim suspected to be true by Alladi, Erdös, and Vaaler. Qualitatively, the result states that a divisor sum of a multiplicative function, which obeys certain size constraints, derives most of its value from its "small" divisors.
Let $K$ be a finite Galois extension of $\mathbb{Q}$. We count primes in short intervals represented by the norm of a prime ideal of $K$ satisfying a small sector condition determined by Hecke characters. We also show that such primes are well-distributed in arithmetic progressions in the sense of Bombieri-Vinogradov. This extends previous work of Duke and Coleman.
In response to the Numberphile video regarding the Mondrian Puzzle https://www.youtube.com/watch?v=49KvZrioFB0, we provide a lower bound on how many integers less than a given threshold $x$ satisfy $M(n) \neq 0$ where $M(n)$ is the quantity in which the Mondrian Puzzle is interested, i.e. the minimal difference in area between the largest and smallest rectangle in a set of incongruent, integer-sided rectangles which tile an $n$ by $n$ square.
We obtain partial progress towards answering the question of whether the quantity defined in the Mondrian Puzzle can ever equal 0. More specifically, we obtain a nontrivial lower bound for the cardinality of the set $\{n\leq x: M(n)\neq0 \}$ where $M(n)$ is the quantity appearing in the Mondrian Puzzle and $x$ is the usual quantity that one thinks of as tending to infinity. More surprisingly, we do so by use of number theoretic techniques in juxtaposition to the innately geometric nature of the problem.