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Daniel A. Goldston

Publications and source records attributed to Daniel A. Goldston.

12 recordsLinked to original sources
arXiv2026

Zeta Zeros in a Narrow Vertical Box

In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note (arXiv:2511.20059 [math.NT]) we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery's proof that on assuming all the zeros are in a narrow vertical box between height $T$ and $2T$ of width $b/\log T$ and centered on the critical line, then, if $b=b(T)\to 0$ as $T\to \infty$, we have asymptotically at least 2/3 of the zeros are simple and on the critical line.

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arXiv2026

Zeta Zeros on the Critical Line

Montgomery in 1973 introduced the pair correlation method to study the vertical distribution of Riemann zeta-function zeros. This work assumed the Riemann Hypothesis (RH). One striking application was a short proof that at least 2/3 of zeta-zeros are simple zeros, the first result of its type. Over the last 50 years, most work on pair correlation of zeta-zeros has continued to assume RH. Here we show that if RH could be removed from Montgomery's simple zero proof, then this would also give a proof that 2/3 of the zeros are simple and on the critical line. This idea has been applied in several recent papers to obtain other results on the zeros.

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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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arXiv2023

On a smoothed average of the number of Goldbach representations

Assuming the Generalized Riemann Hypothesis for the zeros of the Dirichlet $L$-functions with characters modulo $q$, we obtain a smoothed version of the average number of Goldbach representations for numbers which are multiples of a positive integer $q$. Such an average was first considered by Granville \cite{Gran07,Gran08} but without any smoothing factor. In this short article, we also show how the smoothing can be removed.

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arXiv2023

On the Montgomery-Odlyzko method regarding gaps between zeros of the zeta-function

Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.

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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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arXiv2022

Small gaps and small spacings between zeta zeros

We show assuming RH that phenomena concerning pairs of zeros established $via$ pair correlations occur with positive density (with at most a slight adjustment of the constants). Also, while a double zero is commonly considered to be a close pair, we consider the difference between two $distinct$ zeros.

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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 Erdős 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 Erdős-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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arXiv2018

Distribution of Large Gaps Between Primes

We survey some past conditional results on the distribution of large differences between consecutive primes and examine how the Hardy-Littlewood prime k-tuples conjecture can be applied to this question.

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