[Paper Review] Exponential unitary divisors

László Tóth, Nicuşor Minculete|arXiv (Cornell University)|Oct 15, 2009

Analytic Number Theory Research12 references3 citations

TL;DR

This paper introduces and studies exponential unitary divisors—divisors formed by taking unitary divisors of the exponents in a number's prime factorization. It defines exponential unitary perfect numbers and proves there are no odd such numbers. The paper establishes asymptotic formulas for related arithmetic functions and shows their behavior closely mirrors that of exponential divisor functions, resolving a gap in the literature on convolutions of arithmetic functions.

ABSTRACT

We say that $d$ is an exponential unitary divisor of $n=p_1^{a_1}... p_r^{a_r}>1$ if $d=p_1^{b_1}... p_r^{b_r}$, where $b_i$ is a unitary divisor of $a_i$, i.e., $b_i\mid a_i$ and $(b_i,a_i/b_i)=1$ for every $i\in \{1,2,...,r\}$. We survey properties of related arithmetical functions and introduce the notion of exponential unitary perfect numbers.

Motivation & Objective

• To define and investigate exponential unitary divisors, combining exponential and unitary divisor concepts.
• To introduce and analyze exponential unitary perfect numbers, a new class of numbers analogous to unitary and exponential perfect numbers.
• To fill a gap in the literature by studying the exponential unitary convolution, previously unexplored in the context of arithmetic functions.
• To derive asymptotic formulas for the number, sum, and Euler-type functions associated with exponential unitary divisors.
• To resolve open problems regarding the existence of non-squarefree and 3-free exponential unitary perfect numbers.

Proposed method

• Define an exponential unitary divisor d of n = p₁^a₁⋯pᵣ^aᵣ as d = p₁^b₁⋯pᵣ^bᵣ where each bᵢ is a unitary divisor of aᵢ (i.e., bᵢ | aᵢ and gcd(bᵢ, aᵢ/bᵢ) = 1).
• Introduce the functions τ^(e*) (number of exponential unitary divisors), σ^(e*) (sum of exponential unitary divisors), μ^(e*) (exponential unitary Möbius function), and φ^(e*) (exponential unitary Euler function).
• Prove that τ^(e*) and σ^(e*) are multiplicative and derive explicit formulas: τ^(e*)(n) = ∏ᵢ 2^{ω(aᵢ)} and σ^(e*)(n) = ∏ᵢ ∑_{d|*aᵢ} pᵢ^d.
• Use a general result on multiplicative functions to establish the limsup asymptotic behavior of φ^(e*) (log F(n) log log n / log n → sup (log f(m))/m).
• Apply a general asymptotic formula for multiplicative functions bounded by 1 to derive ∑_{n≤x} τ^(e*)(n)/τ^(e)(n) = x ∏_p (1 + ∑_{a≥4} (2^{ω(a)}/τ(a) - 2^{ω(a-1)}/τ(a-1))/p^a) + O(x^{1/4} log x).
• Use Dirichlet convolution and properties of 4-full integers to estimate partial sums and control error terms in asymptotic expansions.

Experimental results

Research questions

• RQ1 Are there any exponential unitary perfect numbers that are not squarefree, and hence not exponential perfect numbers?
• RQ2 Do there exist exponential unitary perfect numbers not divisible by 3?
• RQ3 Is the number of exponential unitary perfect numbers finite or infinite?

Key findings

• There are no odd exponential unitary perfect numbers, as shown by contradiction using the evenness of the number of unitary divisors of exponents and bounds on the sum-of-divisors ratio.
• The asymptotic behavior of τ^(e*), σ^(e*), μ^(e*), and φ^(e*) closely mirrors that of their exponential divisor counterparts τ^(e), σ^(e), μ^(e), and φ^(e).
• An asymptotic formula is established: ∑_{n≤x} τ^(e*)(n)/τ^(e)(n) = x ∏_p (1 + ∑_{a≥4} (2^{ω(a)}/τ(a) - 2^{ω(a-1)}/τ(a-1))/p^a) + O(x^{1/4} log x).
• The quotient |μ^(e)(n)| / |μ^(e*)(n)| = |μ^(e)(n)| is the characteristic function of e-squarefree integers, and its asymptotic density has been studied.
• The smallest number that is exponential perfect but not exponential unitary perfect is 17,424 = 2⁴·3²·11².
• There are infinitely many exponential unitary perfect numbers, including all 36n for squarefree n coprime to 6.