ID: math/0412220

An invitation to additive prime number theory

December 10, 2004

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Additive Combinatorics: A Menu of Research Problems

May 21, 2017

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Bela Bajnok
Number Theory
Combinatorics

This text contains over three hundred specific open questions on various topics in additive combinatorics, each placed in context by reviewing all relevant results. While the primary purpose is to provide an ample supply of problems for student research, it is hopefully also useful for a wider audience. It is the author's intention to keep the material current, thus all feedback and updates are greatly appreciated.

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On a new proof of the Prime Number Theorem

March 30, 2011

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Dhurjati Prasad Datta
General Mathematics

A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more simplified manner. Basic properties of some key concepts such as infinitesimals, the associated nonarchimedean absolute values, invariance of measure and cardinality of a compact subset of the real line under an IFS are discussed more thorou...

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Inverse questions for the large sieve

November 24, 2013

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Ben J. Green, Adam J. Harper
Number Theory

Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $O(X^{1/2})$, and the quadratic examples show that this is sharp. The simplest form of the inverse large sieve problem asks whether they are the only examples. W...

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Combinatorial and Additive Number Theory Problem Sessions: '09--'23

June 13, 2014

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Steven J. Miller
Number Theory
Combinatorics

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to pro...

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Remarks on additive representations of natural numbers

September 7, 2023

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Runbo Li
Number Theory

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $a p+b P_{2}$ and give a new lower bound for it, where $a p$ and $b P_{2}$ are both square-free, $p$ is a prime, and $P_{2}$ has at most two prime factors. We also consider some special cases where $p$ is small, $p$ and $P_2$ are within short intervals, $p$ and $P_2$ are within arithmetical progressions and a Goldbach-type upper bound result. Our new results generalize and improv...

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Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim

May 27, 2006

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K. Soundararajan
Number Theory

This is an expository article on the recent marvellous theorem of Goldston, Pintz, and Yildirim on small gaps between prime numbers.

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Generalising the Hardy-Littlewood Method for Primes

January 10, 2006

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Ben Green
Number Theory

The Hardy-Littlewood method is a well-known technique in analytic number theory. Among its spectacular applications are Vinogradov's 1937 result that every sufficiently large odd number is a sum of three primes, and a related result of Chowla and Van der Corput giving an asymptotic for the number of 3-term progressions of primes, all less than N. This article surveys recent developments of the author and T. Tao, in which the Hardy-Littlewood method has been generalised to obt...

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Rota meets Ramanujan: Probabilistic interpretation of Ramanujan - Fourier series

September 30, 2002

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H. Gopalkrishna Gadiyar, R. Padma
Mathematical Physics
Number Theory

In this paper the ideas of Rota and Ramanujan are shown to be central to understanding problems in additive number theory. The circle and sieve methods are two different facets of the same theme of interplay between probability and Fourier series used to great advantage by Wiener in engineering.

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Harmonic Analysis on the Positive Rationals II: Multiplicative Functions and Maass Forms

May 28, 2014

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P. D. T. A. Elliott, Jonathan Kish
Number Theory

This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.

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The problem of the least prime number in an arithmetic progression and its applications to Goldbach's conjecture

December 1, 2009

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Shaohua Zhang
General Mathematics

The problem of the least prime number in an arithmetic progression is one of the most important topics in Number Theory. In [11], we are the first to study the relations between this problem and Goldbach's conjecture. In this paper, we further consider its applications to Goldbach's conjecture and refine the result in [11]. Moreover, we also try to generalize the problem of the least prime number in an arithmetic progression and give an analogy of Goldbach's conjecture.

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