After the first world war, Cramer began studying the distribution of prime numbers, guided by Riesz and Mittag-Leffler. His works then, and later in the mid-thirties, have had a profound influence on the way mathematicians think about the distribution of prime numbers. In this article, we shall focus on how Cramer's ideas have directed and motivated research ever since.

http://empslocal.ex.ac.uk/people/staff/mrwatkin//zeta/cramer.pdf

Harald Cramér and the distribution of prime numbers

Au congres international de Cambridge en 1912, Laudau dressa la liste de quatre problemes de base sur les nombres premiers. Ces problemes furent caracterises dans son discours comme "inaccessibles en l'etat actuel de la science". Ces problemes sont les suivants: (1) Existe-t-il une infinite de nombres premiers de la forme n 2 + 1 ? (2) La conjecture (binaire) de Goldbach, que chaque nombre pair superieur a 2 est somme de deux nombres premiers. (3) La conjecture des nombres premiers jumeaux. (4) Existe-t-il toujours un nombre premier entre deux carres consecutifs ? Tous ces problemes sont encore ouverts. Le travail presente ici est un expose des resultats partiels aux problemes (2)-(4), avec une attention particuliere concernant les resultats recents de D. Goldston, C. Yildirim et de l'auteur sur les petits ecarts entre nombres premiers.

/pdf/landau-s-problems-on-primes-3vk35tu8kd.pdf

Landau’s problems on primes

We investigate some extremal problems in Fourier analysis and their connection to a problem in prime number theory. In particular, we improve the current bounds for the largest possible gap between consecutive primes assuming the Riemann hypothesis.

Fourier optimization and prime gaps

We prove some results concerning the distribution of primes assuming the Riemann hypothesis. First, we prove the explicit result that there exists a prime in the interval $(x-\frac{4}{\pi} \sqrt{x}...

On the Riemann hypothesis and the difference between primes

The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research. 2000 MSC: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.

An invitation to additive prime number theory

Some Theorems Concerning Prime Numbers

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Harald Cramér

Author Tools

Chat about Author

Papers

Some Theorems Concerning Prime Numbers