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Documents  11N05 | enregistrements trouvés : 4

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We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively. We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...

11N05 ; 11B05

Post-edited  Large gaps between primes in subsets
Maynard, James (Auteur de la Conférence) | CIRM (Editeur )

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao. All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long ...

11N05 ; 11N35 ; 11N36

Multi angle  Primes, exponential sums, and L-functions
Banks, William (Auteur de la Conférence) | CIRM (Editeur )

This talk will survey some recent directions in the study of prime numbers that rely on bounds of exponential sums and advances in sieve theory. I will also describe some new results on the Riemann zeta function and Dirichlet functions, and pose some open problems.

11L20 ; 11N05 ; 11L07 ; 11N36 ; 11S40

Multi angle  Primes with missing digits
Maynard, James (Auteur de la Conférence) | CIRM (Editeur )

We will talk about recent work showing there are infinitely many primes with no $7$ in their decimal expansion. (And similarly with $7$ replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically vey difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.
We will talk about recent work showing there are infinitely many primes with no $7$ in their decimal expansion. (And similarly with $7$ replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically vey difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, com...

11N05 ; 11A63

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