arXiv subjects

Jordan Schettler

Publications and source records attributed to Jordan Schettler.

11 recordsLinked to original sources

arXiv2025

Pair Correlation Conjecture for the zeros of the Riemann zeta-function II: The Alternative Hypothesis

In an earlier paper, we proved that Montgomery's Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function can be used to prove without the assumption of the Riemann Hypothesis (RH) that asymptotically 100% of the zeros are both simple and on the critical line. This is based on a method of Gallagher and Mueller from 1978. We formulate an appropriate form of the Alternative Hypothesis (AH), which determines a different PCC, and, using the same method as above, prove that asymptotically, 100% of the zeros are both simple and on the critical line. As in our previous paper, we do not assume RH.

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arXiv2025

Pair Correlation Conjecture for the Zeros of the Riemann Zeta-function I: Simple and Critical Zeros

Montgomery in 1973 introduced the Pair Correlation Conjecture (PCC) for zeros of the Riemann zeta-function. He also conjectured that asymptotically 100% of the zeros are simple. His reasoning to support these two conjectures used the Riemann Hypothesis (RH). Building on Montgomery's approach, Gallagher and Mueller proved in 1978 that PCC under RH implies that 100% of the zeros are simple. Actually, the method of Gallagher and Mueller does not depend on RH, and thus Montgomery's second simplicity conjecture follows unconditionally from his PCC conjecture. We clarify this result by explicitly not assuming RH and considering PCC as a conjecture only concerning the vertical distribution of zeros. We then show that, for the first time, PCC can also be used to obtain information on the horizontal distribution of zeros. Using Gallagher and Mueller's method and a new idea concerning "horizontal multiplicity", we use PCC to prove that asymptotically 100% of the zeros are not only simple but also on the critical line.

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arXiv2024

On Product Formulas of Guillera and Sondow

In this note, we evaluate a multivariable family of infinite products which generalize Guillera's infinite product for $e$, and Ser's formula (rediscovered by Sondow) for $e^\gamma$. We describe formulas for the products in terms of special values of the Hurwitz zeta function $\zeta(s,u)$ and its $s$ derivative. Additionally, we derive integral and double integral representations for the logarithms of these infinite products.

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arXiv2023

The Average Number of Goldbach Representations and Zero-Free Regions of the Riemann Zeta-Function

In this paper, we prove an unconditional form of Fujii's formula for the average number of Goldbach representations and show that the error in this formula is determined by a general zero-free region of the Riemann zeta-function, and vice versa. In particular, we describe the error in the unconditional formula in terms of the remainder in the Prime Number Theorem which connects the error to zero-free regions of the Riemann zeta-function.

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arXiv2022

Explicit Calculations for Sono's Multidimensional Sieve of $E_2$-Numbers

We derive explicit formulas for integrals of certain symmetric polynomials used in Keiju Sono's multidimensional sieve of $E_2$-numbers, i.e., integers which are products of two distinct primes. We use these computations to produce the currently best-known bounds for gaps between multiple $E_2$-numbers. For example, we show there are infinitely many occurrences of four $E_2$-numbers within a gap size of 94 unconditionally and within a gap size of 32 assuming the Elliott-Halberstam conjecture for primes and sifted $E_2$-numbers.

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arXiv2021

Small Gaps Between Three Almost Primes and Almost Prime Powers

A positive integer is called an $E_j$-number if it is the product of $j$ distinct primes. We prove that there are infinitely many triples of $E_2$-numbers within a gap size of $32$ and infinitely many triples of $E_3$-numbers within a gap size of $15$. Assuming the Elliot-Halberstam conjecture for primes and $E_2$-numbers, we can improve these gaps to $12$ and $5$, respectively. We can obtain even smaller gaps for almost primes, almost prime powers, or integers having the same exponent pattern in the their prime factorizations. In particular, if $d(x)$ denotes the number of divisors of $x$, we prove that there are integers $a,b$ with $1\leq a < b \leq 9$ such that $d(x)=d(x+a)=d(x+b) = 192$ for infinitely many $x$. Assuming Elliot-Halberstam, we prove that there are integers $a,b$ with $1\leq a < b \leq 4$ such that $d(x)=d(x+a)=d(x+b) = 24$ for infinitely many $x$.

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arXiv2020

Small gaps between almost primes, the parity problem, and some conjectures of Erd\H{o}s on consecutive integers II

This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$, so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e., $x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was done for shifts $n$ which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift $n$.

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arXiv2014

Rational Hyperbolic Triangles and a Quartic Model of Elliptic Curves

The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle.

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arXiv2014

Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

Let $\ell>3$ be a prime such that $\ell \equiv 3 \pmod{4}$ and $\mathbb{Q}(\sqrt{\ell})$ has class number 1. Then Hirzebruch and Zagier noticed that the class number of $\mathbb{Q}(\sqrt{-\ell})$ can be expressed as $h(-\ell) = (1/3)(b_1+b_2 + \cdots + b_m) - m$ where the $b_i$ are partial quotients in the `minus' continued fraction expansion $\sqrt{\ell} = [[b_0; \overline{b_1, b_2, \ldots, b_m}]]$. For an odd prime $p \neq \ell$, we prove an analogous formula using these $b_i$ which computes the sum of Iwasawa lambda invariants $\lambda_p(-\ell)+\lambda_p(-4)$ of $\mathbb{Q}(\sqrt{-\ell})$ and $\mathbb{Q}(\sqrt{-1})$. In the case that $p$ is inert in $\mathbb{Q}(\sqrt{-\ell})$, the formula pleasantly simplifies under some additional technical assumptions.

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arXiv2012

Generalizations of Iwasawa's 'Riemann-Hurwitz' Formula for Cyclic p-Extensions of Number Fields

We produce generalizations of Iwasawa's `Riemann-Hurwitz' formula for number fields. These generalizations apply to cyclic extensions of number fields of degree p^n for any positive integer n. We first deduce some congruences and inequalities and then use these formulas to establish a vanishing criterion for Iwasawa \lambda-invariants which generalizes a result of Takashi Fukuda et. al. for totally real number fields.

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arXiv2012

An Alternative Proof and Generalization of Ferrero's Computations of Iwasawa \lambda-Invariants

We prove a slight generalization of Iwasawa's `Riemann-Hurwitz' formula for number fields and use it to generalize Ferrero's and Kida's well-known computations of Iwasawa \lambda-invariants for the cyclotomic Z_2-extensions of imaginary quadratic number fields. In particular, we show that if p is a Fermat prime, then similar computations of Iwasawa \lambda-invariants hold for certain imaginary quadratic extensions of the unique subfield k of Q(\zeta_{p^2}) such that [k:Q] = p. In fact, we actually prove more by explicitly computing cohomology groups of principal ideals. The computation of lambda invariants obtained is a special case of a much more general result concerning relative lambda invariants for cyclotomic Z_2-extensions of CM number fields due to Yuji Kida. However, the approach used here significantly differs from that of Kida, and the intermediate computations of cohomology groups found here do not hold in Kida's more general setting.

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