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John Friedlander

Publications and source records attributed to John Friedlander.

10 recordsLinked to original sources
arXiv2023

When Euler met Brun

We survey and study some aspects of the distribution of primes in very short intervals.

math.NT↗
arXiv2022

Selberg's sieve of irregular density

We study certain aspects of the Selberg sieve, in particular when sifting by rather thin sets of primes. We derive new results for the lower bound sieve suited especially for this setup and we apply them in particular to give a new sieve-propelled proof of Linnik's theorem on the least prime in an arithmetic progression in the case of the presence of exceptional zeros.

math.NT↗
arXiv2021

Note on a note of Goldston and Suriajaya

We show that the assumption of a weak form of the Hardy-Littlewood conjecture on the Goldbach problem suffices to disprove the possible existence of exceptional zeros of Dirichlet L-functions. This strengthens a result of the authors named in the title.

math.NT↗
arXiv2019

Sums over Vanishing Determinants

We study sums of arithmetic functions, defined on Gaussian integers and taken over those pairs of integers whose coordinates give rise to a singular system.

math.NT↗
arXiv2018

Coordinate Distribution of Gaussian Primes

We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.

math.NT↗
arXiv2007

Residue Classes Having Tardy Totients

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $ϕ(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as $k\to\infty$. Here, $ϕ(n)$ is the Euler function. This answers a question raised in \cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $λ(n)$ and the least such $n$ satisfies $n\ll m^{13}$.

math.NT↗
arXiv1998

The polynomial X^2+Y^4 captures its primes

This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} Λ(a^2 + b^4) = 4π^{-1}κx^{3/4} (1 + O(\log\log x / \log x)) where a, b run over positive integers and κ= \int^1_0 (1 - t^4)^{1/2} dt = Γ(1/4)^2 /6\sqrt{2π}. Here of course, Λdenotes the von Mangoldt function and Γthe Euler gamma function.

math.NT↗
arXiv1998

Asymptotic sieve for primes

This article develops a new sieve method which by adding an additional axiom to the classical formulation breaks the well-known parity problem and allows one to detect primes in thin, interesting integer sequences. In the accompanying paper [math.NT/9811184] the practicality of the axiom is demonstrated by verifying it (and the other axioms) to produce primes in the sequence a^2+b^4 with the relevant asymptotics.

math.NT↗