arXiv subjects

Milad Fakhari

Publications and source records attributed to Milad Fakhari.

2 recordsLinked to original sources

arXiv2026

Counting metacyclic fields

By a metacyclic field $K$ we mean the Galois closure of a pure field $\mathbb{Q}(\sqrt[\ell]{D})$, $D\in\mathbb{Z}$, of odd prime degree $\ell$. Let $\mathcal{M}_\ell$ denote the collection of isomorphism classes of metacyclic fields of degree $\ell(\ell-1)$. Write $N_\ell(X)=\#\{K\in\mathcal{M}_\ell: |\Delta_K|\leq X\}$ for the associated counting function, where $\Delta_K$ denotes the discriminant of $K$. We show $$N_\ell(X)\sim A_\ell X^{\frac{1}{(\ell-1)^2}}(\log X)^{\ell-2}\,,$$ for an explicit constant $A_\ell$. We express $A_\ell$ as a rational number times a product over primes of a degree $\ell$ polynomial in $1/p$.

math.NT↗

arXiv2024

Constants for Artin like problems in Kummer and division fields

We apply the character sums method of Lenstra, Moree, and Stevenhagen, to explicitly compute the constants in the Titchmarsh divisor problem for Kummer fields and for division fields of Serre curves. We derive our results as special cases of a general result on the product expressions for the sums in the form $$\sum_{n=1}^{\infty}\frac{g(n)}{\#G(n)}$$ in which $g(n)$ is a multiplicative arithmetic function and $\{G(n)\}$ is a certain family of Galois groups. Our results extend the application of the character sums method to the evaluation of constants, such as the Titchmarsh divisor constants, that are not density constants.

math.NT↗