Primes in short intervals

Contrary to what would be predicted on the basis of Cram\'er's model concerning the distribution of prime numbers, we develop evidence that the distribution of $\psi(x+H)- \psi(x)$, for $0\le x\le N$, is approximately normal with mean $\sim H$ and variance $\sim H\log N/H$, when $N^\delta \le H \le N^{1-\delta}$.Comment: 29 page

Mean values of multiplicative functions

Let f(n) be a totally multiplicative function such that | ƒ (n)|⪯ 1 for all n, and let F(s) = ∑ ∞ n=1 ƒ(n)n —∞ be the associated Dirichlet series. A variant of Halász"s method is developed, by means of which estimates for ∑ N n=1 ƒ(n)/n are obtained in terms of the size of | F(s) | for s near 1 with ℜs >1. The result obtained has a number of consequences, particularly concerning the zeros of the partial sum U N (s) =∑ N n=1 n-s s of the series for the Riemann zeta function.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43188/1/10998_2004_Article_400315.pd

Small gaps and small spacings between zeta zeros

We show assuming RH that phenomena concerning pairs of zeros established $via$ pair correlations occur with positive density (with at most a slight adjustment of the constants). Also, while a double zero is commonly considered to be a close pair, we consider the difference between two $distinct$ zeros.Comment: 17 page

Beurling primes with large oscillation

A Beurling generalized number system is constructed having integer counting function N B ( x ) = κ x + O ( x θ ) with κ >0 and 1/2 (4/ e )(1− θ ). The construction uses elements of classical analytic number theory and probability.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46253/1/208_2005_Article_638.pd

Real quadratic fields with large class number

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46217/1/208_2005_Article_BF01351721.pd

Products of polynomials in many variables

We study the product of two polynomials in many variables, in several norms, and show that under suitable assumptions this product can be bounded from below independently of the number of variables.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28840/1/0000675.pd

Population history and genome wide association studies of birth weight in a native high altitude Ladakhi population

Pathological low birth weight due to fetal growth restriction (FGR) is an important predictor of adverse obstetric and neonatal outcomes. It is more common amongst native lowlanders when gestating in the hypoxic environment of high altitude, whilst populations who have resided at high altitude for many generations are relatively protected. Genetic study of pregnant populations at high altitude permits exploration of the role of hypoxia in FGR pathogenesis, and perhaps of FGR pathogenesis more broadly. We studied the umbilical cord blood DNA of 316 neonates born to pregnant women managed at the Sonam Norboo Memorial Hospital, Ladakh (altitude 3540m) between February 2017 and January 2019. Principal component, admixture and genome wide association studies (GWAS) were applied to dense single nucleotide polymorphism (SNP) genetic data, to explore ancestry and genetic predictors of low birth weight. Our findings support Tibetan ancestry in the Ladakhi population, with subsequent admixture with neighboring Indo-Aryan populations. Fetal growth protection was evident in Ladakhi neonates. Although no variants achieved genome wide significance, we observed nominal association of seven variants across genes (ZBTB38, ZFP36L2, HMGA2, CDKAL1, PLCG1) previously associated with birthweight

Analysis of SARS-CoV-2 Emergent Variants Following AZD7442 (Tixagevimab/Cilgavimab) for Early Outpatient Treatment of COVID-19 (TACKLE Trial)

Introduction: AZD7442 (tixagevimab/cilgavimab) comprises neutralising monoclonal antibodies (mAbs) that bind to distinct non-overlapping epitopes on the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) spike protein. Viral evolution during mAb therapy can select for variants with reduced neutralisation susceptibility. We examined treatment-emergent SARS-CoV-2 variants during TACKLE (NCT04723394), a phase 3 study of AZD7442 for early outpatient treatment of coronavirus disease 2019 (COVID-19). // Methods: Non-hospitalised adults with mild-to-moderate COVID-19 were randomised and dosed ≤ 7 days from symptom onset with AZD7442 (n = 452) or placebo (n = 451). Next-generation sequencing of the spike gene was performed on SARS-CoV-2 reverse-transcription polymerase chain reaction-positive nasopharyngeal swabs at baseline and study days 3, 6, and 15 post dosing. SARS-CoV-2 lineages were assigned using spike nucleotide sequences. Amino acid substitutions were analysed at allele fractions (AF; % of sequence reads represented by substitution) ≥ 25% and 3% to 25%. In vitro susceptibility to tixagevimab, cilgavimab, and AZD7442 was evaluated for all identified treatment-emergent variants using a pseudotyped microneutralisation assay. // Results: Longitudinal spike sequences were available for 461 participants (AZD7442, n = 235; placebo, n = 226) and showed that treatment-emergent variants at any time were rare, with 5 (2.1%) AZD7442 participants presenting ≥ 1 substitution in tixagevimab/cilgavimab binding sites at AF ≥ 25%. At AF 3% to 25%, treatment-emergent variants were observed in 15 (6.4%) AZD7442 and 12 (5.3%) placebo participants. All treatment-emergent variants showed in vitro susceptibility to AZD7442. // Conclusion: These data indicate that AZD7442 creates a high genetic barrier for resistance and is a feasible option for COVID-19 treatment