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Henryk Iwaniec

Publications and source records attributed to Henryk Iwaniec.

16 recordsLinked to original sources
arXiv2023

Sifting for small primes from an arithmetic progression

In this work and its sister paper [5] we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dipense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and reelied on in earlier proofs.

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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.

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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.

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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.

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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.

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arXiv2016

Twin Primes Via Exceptional Characters

We give an account of the arguments that lead from the assumption of the existence of exceptional characters to the asymptotics in related ranges for the counting function of twin primes.

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arXiv2011

Asymptotic Large Sieve

Motivated by applications to the study of L-functions, we develop an asymptotic version of the large sieve inequality for linear forms in primitive Dirichlet characters.

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arXiv2011

Critical zeros of Dirichlet $L$-functions

We use the Asymptotic Large Sieve and Levinson's method to obtain lower bounds for the proportion of simple zeros on the critical line of the twists by primitive Dirichlet characters of a fixed L-function of degree 1,2, or 3.

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arXiv2001

Spacing of zeros of Hecke L-functions and the class number problem

We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\sqrt{-q}), conditionally subject to the existence of many small (subnormal) gaps between zeros of the L-function associated with a character of the class group associated with this field. In particular, we prove that if the gap between consecutive zeros of the L-function is somewhat smaller than the average for sufficiently many pairs of zeros on the critical line, then h >> \sqrt q (log q)^{-A} for some constant A > 0. For the trivial character, the L-function is the Dedekind zeta-function of the number field and so contains the Riemann zeta-function as a factor. Thus, as a corollary to our main result, we prove that this lower bound for h follows from the hypothesis that there are sufficiently many pairs of adjacent zeros of the Riemann zeta-function on the critical line whose spacing is slightly smaller than 1/2 of the average spacing.

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arXiv1999

Low lying zeros of families of L-functions

In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.

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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.

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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.

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