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May
21 |
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Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
After a first look at the paper, I actually have the impression that Zhang does not use automorphic results or techniques, but only some of the dispersion method / combinatorial ideas which occur as ingredients (together with automorphic results) in the works of Bombieri-Fouvry-Friedlander-Iwaniec... |

May
21 |
comment |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
To be more precise, it is another estimate of Friedlander-Iwaniec (for the exponent of distribution of the $d_3$ divisor function) which depends on the estimate of Bombieri and Birch. This bound is indeed for a three-variable character sum, and requires square-root cancellation, so that there is certainly no known elementary proof (although it will probably be scrutinized rather carefully...) |

May
21 |
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How much of character theory can be done without Schur's lemma or the Artin-Wedderburn theorem?
Schur's Lemma is really old, I think. Burnside used it in 1905 almost with no justification, for instance... A completely off-hand suggestion: Deligne has defined a category that is supposed to behave like the representations of the symmetric group $S_t$ where $t$ is not a positive integer. Maybe those could be useful? |

May
20 |
comment |
Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture
Note: the first result of "Bombieri-Friedlander-Iwaniec style", going beyond the Riemann Hypothesis but not as far as the exponent $4/7$, is due to Fouvry-Iwaniec ("Primes in arithmetic progressions", Acta Arith. 42 (1983), 197-218; the exponent there is $9/17$). All these results depend crucially on the spectral theory of automorphic forms to estimates sum of Kloosterman sums, and especially on the results of Deshouillers and Iwaniec in this direction. |

May
19 |
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Good effective versions of theorems of Artin and Brauer
This is to some extent a question about lattices: there is the lattice of virtual representations of $G$, the basis of irreducible characters, the generating set of induced characters from cyclic subgroups, and the question is to minimize a certain complexity for expressing the basis vectors in terms of the other generating set. Maybe, once interpreted this way, there might be some information in the literature on lattices? |