L. Washington

activity

20082022

most citedOn Taking Square Roots without Quadratic Nonresidues over Finite Fields

13 citations · 27 across the 15 of their papers we have counts for

collaborators

16 papers

math.NT2022★ 1 cited

Sums of Powers of Primes II

Lawrence C. Washington

For a real number $k$ , define $π_{k} (x) = \sum_{p \leq x} p^{k}$ . When $k > 0$ , we prove that $π_{k} (x) - π (x^{k + 1}) = Ω_{\pm} (\frac{x ^{\frac{1}{2} + k}}{lo g x} lo g lo g lo g x)$ as…

math.NT2022

Heuristics for anti-cyclotomic $Z_{p}$ -extensions

Debanjana Kundu, Lawrence C. Washington

This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: the…

math.NT2022

Relative Ideal Classes of Arbitrary Order

David L. Pincus, Lawrence C. Washington

We adapt a known technique for searching for ideal classes of arbitrary order and then apply it to three families of number fields. We show that a family of cyclic sextic number fi…

math.NT2021

An Elliptic Curve Analogue of Pillai's Lower Bound on Primitive Roots

Steven Jin, Lawrence C. Washington

Let $E / Q$ be an elliptic curve. For a prime $p$ of good reduction, let $r (E, p)$ be the smallest non-negative integer that gives the $x$ -coordinate of a point of maximal or…

math.NT2021★ 4 cited

Dirichlet Series Expansions of p-adic L-Functions

Heiko Knospe, Lawrence C. Washington

We study $p$ -adic $L$ -functions $L_{p} (s, χ)$ for Dirichlet characters $χ$ . We show that $L_{p} (s, χ)$ has a Dirichlet series expansion for each regularization parameter $c$ that is prim…

math.NT2020

Class groups of real cyclotomic fields

Mohit Mishra, Rene Schoof, Lawrence C. Washington

We prove that every finite abelian group G occurs as a subgroup of the class group of infinitely many real cyclotomic fields.

math.NT2020

Class group of real cyclotomic fields

Mohit Mishra

In this article, we prove that every finite abelian group $G$ of odd order occurs as a subgroup of the class group of infinitely many real cyclotomic fields.

math.NT2020★ 3 cited

Analogues of the Robin-Lagarias Criteria for the Riemann Hypothesis

Lawrence C. Washington, Ambrose Yang

Robin's criterion states that the Riemann hypothesis is equivalent to $σ (n) < e^{γ} n lo g lo g n$ for all integers $n \geq 5041$ , where $σ (n)$ is the sum of divisors of $n$ and $γ$ is…

math.NT2018★ 2 cited

Explicit computations in Iwasawa theory

Reinier Broker, David Hubbard, Lawrence C. Washington

We give two algorithms to compute layers of the anticyclotomic $Z_{3}$ -extension of an imaginary quadratic field. The first is based on complex multiplication techniques for no…

math.NT2017

A family of cyclic quartic fields with explicit fundamental units

Steve Balady, Lawrence C. Washington

We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.

math.NT2017★ 1 cited

The Coefficient-Choosing Game

William Gasarch, Lawrence C. Washington, Sam Zbarsky

Let $D$ be an integral domain. Two players, Nora and Wanda, alternately choose coefficients from $D$ for a polynomial of degree $d$ . When they are done, if the polynomial has a roo…

math.NT2017★ 3 cited

Iwasawa Invariants of Some Non-Cyclotomic $Z_{p}$ -extensions

David Hubbard, Lawrence C. Washington

Iwasawa showed that there are non-cyclotomic $Z_{p}$ -extensions with positive $μ$ -invariant. We show that these $μ$ -invariants can be evaluated explicitly in many situations…

math.HO2015

$\sum_{p \leq n} 1/ p = ln (ln n) + O (1)$ : An Exposition

William Gasarch, Larry Washington

It is well known that $\sum_{p \leq n} 1/ p = ln (ln (n)) + O (1)$ where $p$ goes over the primes. We give several known proofs of this. We first present a a proof that $\ge \ln(\ln(n))…

math.NT2008★ 13 cited

On Taking Square Roots without Quadratic Nonresidues over Finite Fields

Tsz-Wo Sze

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is det…

math.NT2008

Kummer generators and lambda invariants