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Amirali Fatehizadeh

Publications and source records attributed to Amirali Fatehizadeh.

5 recordsLinked to original sources
arXiv2026

Prime and Nonprime Totatives: A Sharp Construction and Exact Thresholds

For \(n\ge2\), let \(A(n)=π(n)-ω(n)\) and \(B(n)=ϕ(n)-π(n)+ω(n)\) denote the numbers of prime and nonprime totatives of \(n\), respectively, with \(1\) included in \(B(n)\). We construct an explicit injectively parametrized family \(\mathcal C_n\) of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove \(\lvert\mathcal C_n\rvert/A(n)\ge (e^{-γ}-o(1))\log A(n)/\log\log A(n)\), realizing the sharp classical leading constant \(e^{-γ}\). For the same family, with the same defining parameters, we obtain an effective refinement with error of order \((\log\log A(n))^{-1/4}\) and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that \(e^{-γ}\) is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let \(N_k\) and \(M_k\) be the least integers such that \(ϕ(n)>kπ(n)\) for every \(n\ge N_k\) and \(B(n)>kA(n)\) for every \(n\ge M_k\), respectively. We prove \(M_k\le N_{k+1}\) for every \(k\ge1\), determine \(N_1,\ldots,N_6\) and \(M_1,\ldots,M_5\) exactly, and obtain \(M_k=N_{k+1}\) for \(1\le k\le5\).

math.NT↗
arXiv2026

On the Divisibility Relation $σ(n)\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture

For each fixed positive integer $h$, we study the divisibility relation $σ(n)\midσ(n+h)$. We isolate an explicit regular family arising from integral quotients of shifted abundancy indices and show that the complementary set satisfies a subexponential saving; in particular, the number of solutions up to $x$ is $O_h(x/(\log x)^2)$. We also study the proportionality equation $σ(n+h)=λσ(n)$. For every fixed nonzero integer $h$, uniformly for all real $λ>0$, the number of solutions up to $x$ is $O(x/\sqrt{\log\log\log x})$, with an absolute implied constant once $x$ exceeds an $h$-dependent threshold. Finally, we give an explicit family which, under Schinzel's Hypothesis $H$, produces infinitely many solutions of $σ(n+1)=2σ(n)$; the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to $x$. We conjecture that $σ(n+h)=kσ(n)$ has infinitely many positive integer solutions for every fixed $h,k\ge1$.

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arXiv2026

A generalization of the Erdős-Sierpiński conjecture

In this paper, we investigate the combinatorial structure and asymptotic distribution of the solution set of the equation $σ(n+1) = kσ(n)$ for a given integer $k>1$. From a combinatorial perspective, the solutions to this equation are closely related to the concept of $k$-layered numbers, which are a generalization of Zumkeller numbers. In the analytic section, which constitutes the core of this research, we employ the framework of probabilistic number theory and an extension of the classical Kubilius model to study the oscillatory and local behavior of the sum-of-divisors function. Utilizing the truncation technique for arithmetic functions and applying the Chinese Remainder Theorem, the problem is reduced to a synthetic measure space equipped with independent random variables. Subsequently, by applying the optimized version of the Kolmogorov-Rogozin anti-concentration inequality (Petrov's theorem) to the difference of additive variables and finely tuning the error parameters, we prove that the natural density of this set is zero. The main quantitative outcome of this approach is the derivation of the explicit upper bound $A_k(x) \ll_k \frac{x}{\sqrt{\log \log \log x}}$ for the counting function of the solutions. Finally, alongside the zero asymptotic density, relying on the framework of polynomials and Schinzel's H Hypothesis, we establish the conditional infinitude of the solution set for the case $k=2$ and formulate the existential results.

math.NT↗
arXiv2026

On the Natural Density of Monic Integer Polynomials with Roots in a Fixed Number Field

In this article, we investigate the statistical distribution and asymptotic behavior of the family of monic integer polynomials of degree $n$ having at least one root in a fixed number field $K$. Although the framework of thin sets implies that the natural density of this family in the parameter space of bounded height is zero, explicitly quantifying this vanishing rate is a central challenge in arithmetic statistics. Employing a hybrid approach that integrates the Mahler measure, Dirichlet's unit theorem, and residue analysis of the Dedekind zeta function, we demonstrate that the rate of convergence of this density to zero is strictly dependent on the degree $n$. Specifically, we prove that the degrees of the factors induce a phase transition in the asymptotic behavior; for polynomials of degree $n = 2$, the decay rate is bounded by $O(H^{-1} \log H)$, whereas for higher degrees, the asymptotic behavior is dominated by the contribution of rational roots, yielding a bound of $O(H^{-1})$. Beyond deriving these asymptotic estimates, we apply principles from the geometry of numbers to establish explicit combinatorial bounds for counting both the reducible and irreducible components of these polynomials. These explicit bounds provide practical tools for computational evaluations within this domain.

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