arXiv subjects

Pietro Sgobba

Publications and source records attributed to Pietro Sgobba.

6 recordsLinked to original sources

arXiv2025

Unconditional results for Artin-type problems over number fields

Let $K$ be a number field and let $G$ be a finitely generated subgroup of $K^\times$. For all but finitely many primes $\mathfrak p$ of $K$, the reduction $(G \bmod \mathfrak p)$ generates a well-defined subgroup of the multiplicative group of the residue field at $\mathfrak p$, and we may consider its index. We study the primes of $K$ for which this index lies in a given set of positive integers $S$. In particular, we prove that under certain convergence conditions on series associated to $S$ this problem can be addressed without assuming the Generalized Riemann Hypothesis (GRH), and we provide asymptotic formulas for the corresponding prime-counting functions. Problems of this type are related to Artin's primitive root conjecture, which has been proven under the assumption of GRH (Hooley, 1967).

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arXiv2024

The distribution of the multiplicative index of algebraic numbers over residue classes

Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative group of the residue field at $\mathfrak p$. Under the Generalized Riemann Hypothesis we determine the natural density of primes of $K$ for which this index is in a prescribed set $S$ and has prescribed Frobenius in a finite Galois extension $F$ of $K$. We study in detail the natural density in case $S$ is an arithmetic progression, in particular its positivity.

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arXiv2023

Divisibility conditions on the order of the reductions of algebraic numbers

Let $K$ be a number field, and let $G$ be a finitely generated subgroup of $K^\times$. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes $\mathfrak p$ of $K$ such that the order of $(G\bmod\mathfrak p)$ is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes $\mathfrak p$ for which the order is $k$-free, and those for which the order has a prescribed $\ell$-adic valuation for finitely many primes $\ell$. An additional condition on the Frobenius conjugacy class of $\mathfrak p$ may be considered. In order to establish these results, we prove an unconditional version of the Chebotarev density theorem for Kummer extensions of number fields.

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arXiv2022

Unified treatment of Artin-type problems II

This work concerns Artin's Conjecture on primitive roots and related problems for number fields. Let $K$ be a number field and let $W_1$ to $W_n$ be finitely generated subgroups of $K^\times$ of positive rank. We consider the index map, which maps a prime $\mathfrak p$ of $K$ to the $n$-tuple of the indices of $(W_i \bmod \mathfrak p)$. Conditionally under GRH, any preimage under the index map admits a density, and the aim of this work is describing it. For example, we express the density as a limit in various ways. We study in particular the preimages of sets of $n$-tuples that are defined by prescribing valuations for their entries. Under some mild assumptions we can express the density as a multiple of a (suitably defined) Artin-type constant.

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arXiv2022

Prime divisors of $\ell$-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level $\ell$

Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it divides at least one of the $\ell$-Genocchi numbers $G_2,G_4,\ldots, G_{p-3}$, and $\ell$-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of $\ell$-Genocchi irregular primes in a prescribed arithmetic progression in case $\ell$ is odd. The case $\ell=2$ was already dealt with by Hu, Kim, Moree and Sha (2019). Using similar methods we study the prime factors of $(1-\ell^n)B_{2n}/2n$ and $(1+\ell^n)B_{2n}/2n$. This allows us to estimate the number of primes $p\leq x$ for which there exist modulo $p$ Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level $\ell$.

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arXiv2020

On the distribution of the order and index for the reductions of algebraic numbers

Let $α_1,\ldots,α_r$ be algebraic numbers in a number field $K$ generating a subgroup of rank $r$ in $K^\times$. We investigate under GRH the number of primes $\mathfrak p$ of $K$ such that each of the orders of $(α_i\bmod\mathfrak p)$ lies in a given arithmetic progression associated to $α_i$. We also study the primes $\mathfrak p$ for which the index of $(α_i\bmod\mathfrak p)$ is a fixed integer or lies in a given set of integers for each $i$. An additional condition on the Frobenius conjugacy class of $\mathfrak p$ may be considered. Such results are generalizations of a theorem of Ziegler from 2006, which concerns the case $r=1$ of this problem.

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