Showing papers by "Henryk Iwaniec published in 2014"
07 Oct 2014
TL;DR: In this paper, the Dirichlet polynomials were used to approximate the functional equation of the product formula over the zeros, and the asymptotic formula for the zero-free region and the PNT approximated functional equation.
Abstract: Classical topics Panorama of arithmetic functions Sums of basic arithmetic functions Tchebyshev's prime seeds Elementary prime number theorem The Riemann memoir The analytic continuation The functional equation The product formula over the zeros The asymptotic formula for N(T) The asymptotic formula for ?(x) The zero-free region and the PNT Approximate functional equations The Dirichlet polynomials Zeros off the critical line Zeros on the critical line The critical zeros after Levinson Introduction Detecting critical zeros Conrey's construction The argument variations Attaching a mollifier The Littlewood lemma The principal inequality Positive proportion of the critical zeros The first moment of Dirichlet polynomials The second moment of Dirichlet polynomials The diagonal terms The off-diagonal terms Conclusion Computations and the optimal mollifier Smooth bump functions The gamma function Bibliography Index
49 citations
TL;DR: This paper was written in July and August of 2013, apart from one technical correction, and it appeared in the current volume of "Opera de Cribro" as mentioned in this paper, commemorating the one hundred and twenty-fifth anninversary of the birth of Y. Zhang.
Abstract: This paper was written, apart from one technical correction, in July and August of 2013. The, then very recent, breakthrough of Y. Zhang (18) had revived in us an intention to produce a second edition of our book "Opera de Cribro", one which would include an account of Zhang's result, stressing the sieve aspects of the method. A complete and connected version of the proof, in our style but not intended for journal publication, seemed a natural first step in this project. Following the further spectacular advance given by J. Maynard (arXiv:1311. 4600, Nov 20, 2013), we have had to re-think our position. Maynard's method, at least in its current form, proceeds from GPY in quite a different direction than does Zhang's, and achieves numerically superior results. Consequently, although Zhang's contribution to the distribution of primes in arithmetic pro- gressions certainly retains its importance, the fact remains that much of the material in this paper would no longer appear in a new edition of our book. Because this paper contains some innovations that we do not wish to become lost, we have decided to make the work publicly available in its original for- mat. We are extremely pleased to be able to include it in the current volume, commemorating the one hundred and twenty-fifth anninversary of the birth of
4 citations