Andrew Ledoan

JUMPING CHAMPIONS AND GAPS BETWEEN CONSECUTIVE PRIMES

The most common difference that occurs among the consecutive primes less than or equal to

Limit Points of the Sequence of Normalized Differences Between Consecutive Prime Numbers

x

Distribution of Large Gaps Between Primes

is called a jumping champion. Occasionally there are ties. Therefore there can be more than one jumping champion for a given

Zeros of Partial Sums of the Square of the Riemann Zeta-Function

x

Chapter 3: On the Differences Between Consecutive Prime Numbers, I

. 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

Square-Full Divisors of Square-Full Integers

p

Asymptotic behavior of the irrational factor

divides all sufficiently large jumping champions. In this paper we extend a method of P. Erd\H{o}s 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.

THE JUMPING CHAMPION CONJECTURE

Let p n denote the nth prime number and let \(d_{n} = p_{n+1} - p_{n}\) denote the nth difference in the sequence of prime numbers. Erdős and Ricci independently proved that the set of limit points of \(\frac{d_{n}} {\log p_{n}}\), the normalized differences between consecutive prime numbers, forms a set of positive Lebesgue measure. Hildebrand and Maier answered a question of Erdős and proved that the Lebesgue measure of the set of limit points of \(\frac{d_{n}} {\log p_{n}}\) in the interval [0, T] is ≫ T as \(T \rightarrow \infty\). Currently, the only specific limit points known are 0 and \(\infty\). In this note, we use the method of Erdős to obtain specific intervals within which a positive Lebesgue measure of limit points exist. For example, the intervals \(\left [\frac{1} {8},2\right ]\) and \(\left [ \frac{1} {40},1\right ]\) both have a positive Lebesgue measure of limit points.

A Universality Property of Gaussian Analytic Functions

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

Discrepancy of Fractions with Divisibility Constraints

Let s be the complex variable σ + it, let d(n) denote the number of divisors of n, and let X be a real number greater than or equal to 2. In this paper, we establish the zero-free regions for the partial sums of the square of the Riemann zeta-function, defined by \(\zeta _{X}^{2}(s) =\sum _{ n=1}^{X}d(n)n^{-s}\) for σ > 1, and estimate the number of zeros up to a given height T.