# Joshua Zelinsky

### Education

BA Yale 2007

PhD Boston University 2014

Research:

My research has been on three main topics: integer complexity, Artin representations and odd perfect numbers. More broadly, my research has focused on combinatorial aspects of analytic and algebraic number theory.

Publications:

The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number, (under review).

Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba's Function, (under review), with Tim McCormack.

Must a primitive non-deficient number have a component not much larger than its radical, (under review).

On 2-Near Perfect Numbers, (Integers), with Vedant Aryan, Dev Madhavani, Savan Parikh, and Ingrid Slattery.

A note on a result of Makowski, (under review), with Luis Gallardo.

Total Difference Labeling of Regular Infinite Graphs, (Involve, 2023), with Noam Benson-Tilsen, Even Brock, Brandon Faunce, Monish Kumar, and Noah Dokko Stein.

On the small prime factors of a non-deficient number, (Integers, 2022).

On the third largest prime divisor of an odd perfect numbers , (Integers, 2021), with Sean Bibby and Pieter Vyncke

On the total number of prime factors of an odd perfect number, (Integers, 2021).

Upper bounds on the second largest prime factor of an odd perfect number, (International Journal of Number Theory, 2019).

An improvement of an inequality of Ochem and Rao concerning odd perfect numbers, (Integers, 2018). Upper bounds for the number of primitive ray class characters with conductor below a given bound, (Acta Arithmetica, 2016)- Note that the published version has tighter results than the prepint version on the Arxiv.

Numbers with Integer Complexity Close to the Lower Bound, (Integers, 2012), with Harry Altman.

Tau Numbers: A Partial Proof of a Conjecture and Other Results (Journal of Integer Sequences, 2002).

Teaching:

I currently teach at the Hopkins School in New Haven, CT. This year I'm teaching pre-calculus. I'm also teaching the senior seminar which is focusing on number theory.

Other items:

Here is a very short (4 minutes) talk on some of my research on integer complexity. This video is intended for a general audience.

Here is a 20 minute, slightly more technical version of that material which was a pre-recorded talk for the Connecticut Summer School in Number Theory

Me being interviewed on Dan Rubin's math focused Youtube channel.