Robert J. Lemke Oliver

Unexpected biases in the distribution of consecutive primes

Significance Prime numbers play a central role in analytic number theory, and are well known to be very well distributed among the reduced residue classes (mod q). Surprisingly, the same does not appear to be true for sequences of consecutive primes, with different patterns occurring with wildly different frequencies. We formulate a precise conjecture, based on the Hardy−Littlewood conjectures, which explains this phenomenon. In particular, we predict that all patterns do occur their fair share of the time in the limit, but that there are secondary terms only very slowly tending to zero that create the observed biases. Although the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy−Littlewood conjectures. The conjectures are then compared with numerical data, and the observed fit is very good.

Bounded gaps between primes in number fields and function fields

The Hardy--Littlewood prime

Class groups of cyclic p -groups

k

Proof of the Alder-Andrews conjecture

-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field

THE DISTRIBUTION OF 2-SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION

\mathbb{F}_q(t)

Gauss sums over finite fields and roots of unity

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Pretentiously detecting power cancellation

As usual, if π is a finite group and ℳ ⊆ ℚπ a maximal order containing , we set This is well known to be a finite group, and independent of the choice of ℳ.

The Generalized Nagell–Ljunggren Problem: Powers with Repetitive Representations

Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, Alder investigated q d (n) and Q d (n), the number of partitions of n into d-distinct parts and into parts which are ±1(modd+3), respectively. He conjectured that q d (n) ≥ Q d (n). Andrews and Yee proved the conjecture for d = 2 s - 1 and also for d > 32. We complete the proof of Andrewss refinement of Alders conjecture by determining effective asymptotic estimates for these partition functions (correcting and refining earlier work of Meinardus), thereby reducing the conjecture to a finite computation.

Best Possible Densities of Dickson m -Tuples, as a Consequence of Zhang–Maynard–Tao

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion.

Smooth compact Lie group actions on disks

Let χ be a non-trivial character of F × q , and let g(x) be its associated Gauss sum. It is well known that g(χ) = e(χ)√q, where |e(χ)| = 1. Using the p-adic gamma function, we give a new proof of a result of Evans which gives necessary and sufficient conditions for e(χ) to be a root of unity.