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Ethan Simpson Lee

Publications and source records attributed to Ethan Simpson Lee.

14 recordsLinked to original sources
arXiv2026

Minimal zero-free regions for results on primes between consecutive perfect $k$th powers

We compute minimal zero-free regions for the Riemann zeta-function of the Littlewood form which ensure there is always a prime between consecutive perfect $k$th powers. Our computations cover powers $k\geq 65$ and quantify how far we are away from proving certain milestones toward an infamous open problem (Legendre's conjecture). In addition, we prove there is always a prime between consecutive perfect $86$th powers, and identify an integer sequence (that is a subset of the positive integers) for which there is always a prime between consecutive $70$th powers.

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arXiv2025

Sharper bounds for the error in the prime number theorem assuming the Riemann Hypothesis

In this paper, we establish new bounds for classical prime-counting functions. All of our bounds are explicit and assume the Riemann Hypothesis. First, we prove that $|ψ(x) - x|$ and $|\vartheta(x) - x|$ are bounded from above by $$\frac{\sqrt{x}\log{x}(\log{x} - \log\log{x})}{8π}$$ for all $x\geq 101$ and $x \geq 2\,657$ respectively, where $ψ(x)$ and $\vartheta(x)$ are the Chebyshev $ψ$ and $\vartheta$ functions. Using the extra precision offered by these results, we also prove new explicit descriptions for the error in each of Mertens' theorems which improve earlier bounds by Schoenfeld.

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arXiv2024

On the error in the prime number theorem in short intervals

Assuming the Riemann Hypothesis, we derive explicit bounds for the error terms in short interval analogues of the prime number theorem using a new smoothing argument. Our results improve upon earlier bounds in both constant terms and applicable ranges. In addition, we apply our bounds to establish sharper conditional bounds for a broad class of weighted sums over primes in a short interval.

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arXiv2022

Explicit upper bounds for the number of primes simultaneously representable by any set of irreducible polynomials

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of $n\leq x$ such that a non-empty set of irreducible polynomials $F_i(n)$ with integer coefficients are simultaneously prime; this set can contain as many polynomials as desired. To demonstrate, we present computations for some irreducible polynomials and obtain an explicit upper bound for the number of Sophie Germain primes up to $x$, which have practical applications in cryptography.

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arXiv2022

New explicit bounds for Mertens function and the reciprocal of the Riemann zeta-function

IWe prove new unconditional and explicit upper bounds for the reciprocal of the Riemann zeta function in the critical strip, of the orders $(\log t)^{11/12}$, $(\log t)^{11/12}(\log\log t)^{-3/4}$, and $(\log t)^{2/3}(\log\log t)^{1/4}$; these hold in prescribed classical, Littlewood, and Korobov--Vinogradov zero-free regions, respectively. From these bounds, we also derive new unconditional and explicit upper bounds for the Mertens function $M(x)$, of the orders $x (\log x)\exp\!\left(-\eta_1 \sqrt{\log x}\right)$, $x (\log x) \exp\!\left(-\eta_2 \sqrt{\log x \log\log x}\right)$, and $x (\log x)\exp\!\left(-\eta_3 (\log x)^{3/5}(\log\log x)^{-1/5}\right)$, for suitable constants $\eta_1,\eta_2,\eta_3>0$. A key feature of our approach is the use of computational smoothing, which ensures the resulting numerical bounds are strong.

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arXiv2021

Unconditional Explicit Mertens' Theorems for Number Fields and Dedekind Zeta Residue Bounds

We obtain unconditional, effective number-field analogues of the three Mertens' theorems, all with explicit constants and valid for $x\geq 2$. Our error terms are explicitly bounded in terms of the degree and discriminant of the number field. To this end, we provide unconditional bounds, with explicit constants, for the residue of the corresponding Dedekind zeta function at $s=1$.

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arXiv2020

An effective analytic formula for the number of distinct irreducible factors of a polynomial

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb{Z}[x]$. We use an explicit version of Mertens' theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb{Z}[x]$, our result yields a finite list of primes that certifies the number of distinct irreducible factors of $f$.

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