arXiv subjects
Publications and source records attributed to Amir Parvardi.
The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory (through mid-2024) together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.
For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by the product of some subset of $\{1,\dots,y\}$. It is easy to see that $y(p)$ is at least as large as the smallest quadratic nonresidue $\pmod p$; we prove that $y(p) \ll_\varepsilon p^{1/(4 \sqrt e)+\varepsilon}$, thus strengthening Burgess's classical result. This result is of intermediate strength between two other results, namely Burthe's proof that the multiplicative group $\pmod p$ is generated by the integers up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$, and Munsch and Shparlinski's result that every reduced residue class $\pmod p$ is represented by the product of some subset of the primes up to $O_\varepsilon(p^{1/(4 \sqrt e)+\varepsilon}$. Unlike the latter result, our proof is elementary and similar in structure to Burgess's proof for the least quadratic nonresidue.