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Landau’s problems on primes. (English) Zbl 1239.11101
This is a survey of results that have been obtained on Landau’s problems on primes, which are the four problems Landau listed at the 5th International Congress of Mathematicians in Cambridge in 1912. These problems are:
1) Are there infinitely many primes of the form \(n^2+1\)?
2) Is every even number greater than 2 equal to the sum of two primes?
3) Are there infinitely many twin primes?
4) Is there alway at least one prime between two consecutive squares?
None of these problems have been solved, but an enormous amount of work has been done on each, and this work has had a major influence on the whole field of analytic number theory. The survey provides a detailed history of what has been done on these problems, and includes 204 references. It should be pointed out that the paper also gives a nice introduction to the underlying ideas used in recent work on small gaps between primes which will be helpful for readers interested in this topic. Among the topics which the paper includes are upper bounds on large gaps between consecutive primes, the Cramér model for large gaps between primes, lower bounds for large gaps, small gaps including recent work of Goldston, Pintz, and Yıldırım, sieve results on almost primes and Chen’s theorem, results of Goldston, Graham, Pintz, and Yıldırım on almost primes and the Erdős problems concerning equal values of arithmetic functions on consecutive integers, the exceptional set in the Goldbach problem, the ternary Goldbach problem, gaps between Goldbach numbers, the Goldbach-Linnik problem, recent work of Pintz on the exceptional set, and almost primes values of \(n^2+1\) such as Iwaniec’s theorem.

MSC:
11N05 Distribution of primes
11P32 Goldbach-type theorems; other additive questions involving primes
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11N36 Applications of sieve methods
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References:
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