Paul Pollack

Following Johnson, Queen, and Sevilla, we call an order $$\mathcal {O}$$ O in a number field K if, for all $$\alpha ,\beta \in \mathcal {O}$$ α , β ∈ O with $$\beta \ne 0$$ β ≠ 0 for which $$\alpha \mathcal {O}+\beta \mathcal {O}$$ α O + β O is a principal ideal of $$\mathcal {O}$$ O , there are $$\gamma , \rho \in \mathcal {O}$$ γ , ρ ∈ O with $$\alpha =\beta \gamma + \rho $$ α = β γ + ρ and $$|N_{K/\mathbb {Q}}(\rho )| < |N_{K/\mathbb {Q}}(\beta )|$$ | N K / Q ( ρ ) | < | N K / Q ( β ) | . We prove that only finitely many quadratic orders are generalized Euclidean. Additionally, assuming GRH, we show that for each fixed $$d > 1,$$ d > 1 , there are only finitely many integers m for which $$T^d-m$$ T d - m is irreducible over $$\mathbb {Q}$$ Q and $$\mathbb {Z}[m^{1/d}] = \mathbb {Z}[T]/(T^d-m)$$ Z [ m 1 / d ] = Z [ T ] / ( T d - m ) is generalized Euclidean. When d is even, we can remove the assumption of GRH.

For each positive integer $n$, we denote by $ω^*(n)$ the number of shifted-prime divisors $p-1$ of $n$, i.e., \[ω^*(n):=\sum_{p-1\mid n}1.\] First introduced by Prachar in 1955, this function has interesting applications in primality testing and bears a strong connection with counting Carmichael numbers. Prachar showed that for a certain constant $c_0 > 0$, \[ω^*(n)>\exp\left(c_0\frac{\log n}{(\log\log n)^2}\right)\] for infinitely many $n$. This result was later improved by Adleman, Pomerance and Rumely, who established an inequality of the same shape with $(\log\log n)^2$ replaced by $\log\log n$. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, Prachar also proved the stronger inequality \[ω^*(n)>\exp\left(\left(\frac{1}{2}\log2+o(1)\right)\frac{\log n}{\log\log n}\right)\] for infinitely many $n$. By refining the arguments of Prachar and of Adleman, Pomerance and Rumely, we improve on their results by establishing \begin{align*} ω^*(n)&>\exp\left(0.6736\log 2\cdot\frac{\log n}{\log\log n}\right) \quad\text{(unconditionally)},\\ ω^*(n)&>\exp\left(\left(\log\left(\frac{1+\sqrt{5}}{2}\right)+o(1)\right)\frac{\log n}{\log\log n}\right) \quad\text{(under GRH)}, \end{align*} for infinitely many $n$.

Schinzel and Wójcik have shown that for every $$\alpha ,\beta \in \mathbb {Q}^{\times }\hspace{0.55542pt}{\setminus }\hspace{1.111pt}\{\pm 1\}$$ α , β ∈ Q × \ { ± 1 } , there are infinitely many primes p where $$v_p(\alpha )=v_p(\beta )=0$$ v p ( α ) = v p ( β ) = 0 and where $$\alpha $$ α and $$\beta $$ β generate the same multiplicative group mod p. We prove a weaker result in the same direction for algebraic numbers $$\alpha , \beta $$ α , β . Let $$\alpha , \beta \in \overline{\mathbb {Q}} ^{\times }$$ α , β ∈ Q ¯ × , and suppose $$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1$$ | N Q ( α , β ) / Q ( α ) | ≠ 1 and $$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1$$ | N Q ( α , β ) / Q ( β ) | ≠ 1 . Then for some positive integer $$C = C(\alpha ,\beta )$$ C = C ( α , β ) , there are infinitely many prime ideals P of "Equation missing" where $$v_P(\alpha )=v_P(\beta )=0$$ v P ( α ) = v P ( β ) = 0 and where the group $$\langle \beta \bmod {P}\rangle $$ ⟨ β mod P ⟩ is a subgroup of $$\langle \alpha \bmod {P}\rangle $$ ⟨ α mod P ⟩ with $$[\langle \alpha \bmod {P}\rangle \,{:}\, \langle \beta \bmod {P}\rangle ]$$ [ ⟨ α mod P ⟩ : ⟨ β mod P ⟩ ] dividing C. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shifted S-units.

We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log{f})^{c_1 \log\log\log{f}}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log{f})^{c_2\log\log\log{f}}$ for infinitely many $f$. Here $c_1, c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^{\times}$.

We provide a uniform bound on the partial sums of multiplicative functions under very general hypotheses. As an application, we give a nearly optimal estimate for the count of $n \le x$ for which the Alladi–Erdős function $A(n) = \sum_{p^k \parallel n} k p$ takes values in a given residue class modulo q, where q varies uniformly up to a fixed power of $\log x$ . We establish a similar result for the equidistribution of the Euler totient function $\phi(n)$ among the coprime residues to the ‘correct’ moduli q that vary uniformly in a similar range and also quantify the failure of equidistribution of the values of $\phi(n)$ among the coprime residue classes to the ‘incorrect’ moduli.

Schinzel and Wójcik have shown that if $\alpha ,\beta \in \mathbb Q^{\times} \setminus \{\pm 1\}$, then there are infinitely many primes $p$ where $v_p(\alpha )=v_p(\beta )=0$ and where $\alpha ,\beta $ share the same multiplicative order modulo $p$. We p