Twists of Hooley's Δ\Delta-function over number fields

We prove tight estimates for averages of the twisted Hooley $\Delta$-function over arbitrary number fields

Counting rational points on quartic del Pezzo surfaces with a rational conic

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over Q that contains a conic defined over Q

Gaps between prime divisors and analogues in Diophantine geometry

Erd\H{o}s considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes $p$ for which there is no $\mathbb{Q}_p$-point on a random variety are Poisson distributed.Comment: Added analogues in Diophantine geometr

The density of fibres with a rational point for a fibration over hypersurfaces of low degree

We prove asymptotics for the proportion of fibres with a rational point in aconic bundle fibration. The basis of the fibration is a general hypersurface oflow degree.<br

Vinogradov's three primes theorem with primes having given primitive roots

The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation

Isolation and Screening of Microorganisms for Their Ability to Reduce the Amount of Cholesterol in a Model System

ABSTRACT The objective of this study was to isolate and screen the ability of microorganisms to reduce the amount of cholesterol in a model system supplemented with cholesterol. The cholesterol-degrading bacteria were isolated from samples of various fats and authentic strains were also used. Analyses were performed to determine the rate of cholesterol degradation in model system, effect of the medium pH on cholesterol reduction, and identification of cholesterol-degrading bacteria. Rhodococcus equi ATCC 21107, Rhodococcus equi ATCC 33706, Leuconostoc cremoris, Serratia maicescens ATCC 13880 and several bacterial isolates from fats were capable of degrading cholesterol in a model system supplemented with pure soluble cholesterol. Morphological and biochemical characteristics revealed that some of the bacterial isolates tentatively classified as belonged to the genus Pseudomona

Cyclotomic polynomials with prescribed height and prime number theory

Given any positive integer $n,$ let $A(n)$ denote the height of the $n^{\text{th}}$ cyclotomic polynomial, that is its maximum coefficient in absolute value. It is well known that $A(n)$ is unbounded. We conjecture that every natural number can arise as value of $A(n)$ and prove this assuming that for every pair of consecutive primes $p$ and $p'$ with $p\ge 127$ we have $p'-p<\sqrt{p}+1.$ We also conjecture that every natural number occurs as maximum coefficient of some cyclotomic polynomial and show that this is true if Andrica's conjecture that always $\sqrt{p'}-\sqrt{p}<1$ holds. This is the first time, as far as the authors know, a connection between prime gaps and cyclotomic polynomials is uncovered. Using a result of Heath-Brown on prime gaps we show unconditionally that every natural number $m\le x$ occurs as $A(n)$ value with at most $O_{\epsilon}(x^{3/5+\epsilon})$ exceptions. On the Lindel\"of Hypothesis we show there are at most $O_{\epsilon}(x^{1/2+\epsilon})$ exceptions and study them further by using deep work of Bombieri--Friedlander--Iwaniec on the distribution of primes in arithmetic progressions beyond the square-root barrier

Schinzel Hypothesis on average and rational points

We resolve Schinzel’s Hypothesis (H) for 100% of polynomials of arbitrary degrees. We deduce that a positive proportion of diagonal conic bundles over Q with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations

Averages of arithmetic functions over principal ideals

For a general class of non-negative functions defined on integral ideals of number fields, upper bounds are established for their average over the values of certain principal ideals that are associated to irreducible binary forms with integer coefficients