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

A bound for the least prime ideal in the Chebotarev Density Theorem

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46605/1/222_2005_Article_BF01390234.pd

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