arXiv subjects
Publications and source records attributed to D. A. Goldston.
We prove that the error in the prime number theorem can be quantitatively improved beyond the Riemann Hypothesis bound by using versions of Montgomery's conjecture for the pair correlation of zeros of the Riemann zeta-function which are uniform in long ranges and with suitable error terms.
Fujii obtained a formula for the average number of Goldbach representations with lower order terms expressed as a sum over the zeros of the Riemann zeta-function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result, and obtain applications conditional on various conjectures on zeros of the Riemann zeta-function.
We generalize the work of Fei, Bhowmik and Halupczok, and Jia relating the Goldbach conjecture to real zeros of Dirichlet $L$-functions.
We show that the Riesz mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the Riemann zeta-function. Unconditionally this error term can be shown to oscillate, while conditionally it can be shown to oscillate between sharp bounds.
We show that the error term in the asymptotic formula for the Ces{à}ro mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures cannot be too small and oscillates.
Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average spacing. To obtain this result, we generalize a set of weights that were developed by Xiaosheng Wu, who used them to find a positive proportion of large and small gaps between zeros of the Riemann zeta-function.
In this paper, we obtain bounds on the $L^1$ norm of the sum $\sum_{n\le x}τ(n) e(αn)$ where $τ(n)$ is the divisor function.
A Prime Difference Champion (PDC) for primes up to $x$ is defined to be any element of the set of one or more differences that occur most frequently among all positive differences between primes $\le x$. Assuming an appropriate form of the Hardy-Littlewood Prime Pair Conjecture we can prove that for sufficiently large $x$ the PDCs run through the primorials. Numerical results also provide evidence for this conjecture as well as other interesting phenomena associated with prime differences. Unconditionally we prove that the PDCs go to infinity and further have asymptotically the same number of prime factors when counted logarithmically as the primorials.
Assuming the Riemann Hypothesis, we obtain asymptotic formulas for the average number representations of an even integer as the sum of two primes. We use the method of Bhowmik and Schlage-Puchta and refine their results slightly to obtain a more recent result of Languasco and Zaccagnini, and a new result on a smoother average.
We obtain an asymptotic formula for a weighted sum of the square of the tail of the singular series for the Goldbach and prime-pair problems.
We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.
An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. For the $n$th prime $p_{n}$, the $n$th primorial $p_{n}^{\sharp}$ is defined as the product of the first $n$ primes. In 1999, Odlyzko, Rubinstein and Wolf provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $p_{1}^{\sharp}, p_{2}^{\sharp}, p_{3}^{\sharp}, p_{4}^{\sharp}, p_{5}^{\sharp}, ...$, that is, $2, 6, 30, 210, 2310, ....$ In this paper, we prove that an appropriate form of the Hardy-Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$.
We show that a positive proportion of all gaps between consecutive primes are small gaps. We provide several quantitative results, some unconditional and some conditional, in this flavour.
We prove that a positive proportion of the gaps between consecutive primes are short gaps of length less than any fixed fraction of the average spacing between primes.
The most common difference that occurs among the consecutive primes less than or equal to $x$ is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given $x$. In 1999 A. Odlyzko, M. Rubinstein, and M. Wolf provided heuristic and empirical evidence in support of the conjecture that the numbers greater than 1 that are jumping champions are 4 and the primorials 2, 6, 30, 210, 2310,... As a step towards proving this conjecture they introduced a second weaker conjecture that any fixed prime $p$ divides all sufficiently large jumping champions. In this paper we extend a method of P. Erdős and E. G. Straus from 1980 to prove that the second conjecture follows directly from the prime pair conjecture of G. H. Hardy and J. E. Littlewood.
In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of exactly two primes. We use that result to prove that there are inifinitely many integers $x$ that simultaneously satisfy $$ω(x)=ω(x+1)=4, Ω(x)=Ω(x+1)=5, \text{and} d(x)=d(x+1)=24.$$ Here, $ω(x), Ω(x), d(x)$ represent the number of prime divisors of $x$, the number of prime power divisors of $x$, and the number of divisors of $x$, respectively. We also prove similar theorems where $x+1$ is replaced by $x+b$ for an arbitrary positive integer $b$. Our results sharpen earlier work of Heath-Brown, Pinner, and Schlage-Puchta.
We prove that there are infinitely often pairs of primes much closer than the average spacing between primes - almost within the square root of the average spacing. We actually prove a more general result concerning the set of values taken on by the differences between primes.
This paper is based on a talk given to motivated high school (and younger) students at a BAMA (Bay Area Math Adventure) event. Some of the methods used to study primes and twin primes are introduced.