arXiv subjects

Paul Pollack

Publications and source records attributed to Paul Pollack.

60 records · Page 4Linked to original sources

arXiv2012

On the degrees of divisors of T^n-1

Fix a field $F$. In this paper, we study the sets $\D_F(n) \subset [0,n]$ defined by [\D_F(n):= {0 \leq m \leq n: T^n-1\text{has a divisor of degree $m$ in} F[T]}.] When $\D_F(n)$ consists of all integers $m$ with $0 \leq m \leq n$, so that $T^n-1$ has a divisor of every degree, we call $n$ an $F$-practical number. The terminology here is suggested by an analogy with the practical numbers of Srinivasan, which are numbers $n$ for which every integer $0 \leq m \leq σ(n)$ can be written as a sum of distinct divisors of $n$. Our first theorem states that, for any number field $F$ and any $x \geq 2$, [#{\text{$F$-practical $n\leq x$}} \asymp_{F} \frac{x}{\log{x}};] this extends work of the second author, who obtained this estimate when $F=\Q$. Suppose now that $x \geq 3$, and let $m$ be a natural number in $[3,x]$. We ask: For how many $n \leq x$ does $m$ belong to $\D_F(n)$? We prove upper bounds in this problem for both $F=\Q$ and $F=\F_p$ (with $p$ prime), the latter conditional on the Generalized Riemann Hypothesis. In both cases, we find that the number of such $n \leq x$ is $\ll_{F} x/(\log{m})^{2/35}$, uniformly in $m$.

math.NT↗

arXiv2012

Practical pretenders

Following Srinivasan, an integer n\geq 1 is called practical if every natural number in [1,n] can be written as a sum of distinct divisors of n. This motivates us to define f(n) as the largest integer with the property that all of 1, 2, 3,..., f(n) can be written as a sum of distinct divisors of n. (Thus, n is practical precisely when f(n)\geq n.) We think of f(n) as measuring the "practicality" of n; large values of f correspond to numbers n which we term practical pretenders. Our first theorem describes the distribution of these impostors: Uniformly for 4 \leq y \leq x, #{n\leq x: f(n)\geq y} \asymp \frac{x}{\log{y}}. This generalizes Saias's result that the count of practical numbers in [1,x] is \asymp \frac{x}{\log{x}}. Next, we investigate the maximal order of f when restricted to non-practical inputs. Strengthening a theorem of Hausman and Shapiro, we show that every n > 3 for which f(n) \geq \sqrt{e^γ n\log\log{n}} is a practical number. Finally, we study the range of f. Call a number m belonging to the range of f an additive endpoint. We show that for each fixed A >0 and ε> 0, the number of additive endpoints in [1,x] is eventually smaller than x/(\log{x})^A but larger than x^{1-ε}.

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arXiv2011

The average least character nonresidue and further variations on a theme of Erdos

For each nonprincipal Dirichlet character $χ$, let $n_χ$ be the least $n$ with $χ(n) \notin \{0,1\}$. We show that as the average of $n_χ$ over all nonprincipal characters $χ$ modulo $q$ is $\ell(q) + o(1)$, where $\ell(q)$ denotes the least prime not dividing $q$. Moreover, if one averages over all nonprincipal characters of modulus at most $x$, the average approaches a particular limiting value 2.5350541804. We also prove a result of this type for cubic number fields: If one averages over all cubic fields $K$, ordered by the absolute value of their discriminant, then the mean value of the least rational prime that does not split completely in $K$ is another particular constant 2.1211027269.

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arXiv2010

On common values of $ϕ(n)$ and $σ(m)$, II

Let phi(n) be Euler's totient function and let sigma(n) be the sum of the positive divisors of n. We show that most phi-values (integers in the range of phi) are not sigma-values and vice versa.

math.NT↗

arXiv2010

On common values of phi(n) and sigma(n), I

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x(log x)^{-1+o(1)} numbers not exceeding x common to the ranges of Euler's function phi(n) and the sum-of-divisors function sigma(m).

math.NT↗

arXiv2009

On quantitative analogues of the Goldbach and twin prime conjectures over F_q[t]

We study the number of ways to decompose a monic polynomial in F_q[t] of degree n as a sum of two monic irreducible polynomials in F_q[t]. Our principal result is an asymptotic formula for the number of such representations in the case when q is large compared to n. In its range of validity, this formula agrees with what is suggested by heuristic arguments from the rational setting. We also present similar results towards an analogue of the twin prime conjecture.

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