# Selberg sieve variational problem

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(→More general variational problems) |
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:<math>{\mathcal R}'_k = \{ (t_1,\ldots,t_k) \in [0,1]^k: t_1+\ldots+t_k \leq 1 + \min(t_1,\ldots,t_k) \}</math> | :<math>{\mathcal R}'_k = \{ (t_1,\ldots,t_k) \in [0,1]^k: t_1+\ldots+t_k \leq 1 + \min(t_1,\ldots,t_k) \}</math> | ||

- | provided that one works with a generalisation of <math>EH[\theta]</math> which controls more general Dirichlet convolutions than the von Mangoldt function (a precise assertion in this regard may be found in BFI). In fact one should be able to work in | + | provided that one works with a generalisation of <math>EH[\theta]</math> which controls more general Dirichlet convolutions than the von Mangoldt function (a precise assertion in this regard may be found in BFI). In fact one should be able to work in any larger region <math>R</math> for which |

- | :<math> | + | :<math>R + R \subset \{ (t_1,\ldots,t_k) \in [0,2/\theta]^k: t_1+\ldots+t_k \leq 2 + \max(t_1,\ldots,t_k) \} \cup \frac{2}{\theta} \cdot {\mathcal R}_k</math> |

provided that all the marginal distributions of F are supported on <math>{\mathcal R}_{k-1}</math>, thus (assuming F is symmetric) | provided that all the marginal distributions of F are supported on <math>{\mathcal R}_{k-1}</math>, thus (assuming F is symmetric) | ||

:<math>\int_0^\infty F(t_1,\ldots,t_{k-1},t_k)\ dt_k = 0 </math> when <math>t_1+\ldots+t_{k-1} > 1.</math> | :<math>\int_0^\infty F(t_1,\ldots,t_{k-1},t_k)\ dt_k = 0 </math> when <math>t_1+\ldots+t_{k-1} > 1.</math> | ||

+ | |||

+ | For instance, one can take <math>R = \frac{1}{\theta} \cdot {\mathcal R}_k</math>, or one can take <math>R = \{ (t_1,\ldots,t_k) \in [0,1/\theta]^k: t_1 +\ldots +t_{k-1} \leq 1 </math> (although the latter option breaks the symmetry for F). Perhaps other choices are also possible. |

## Revision as of 06:07, 13 December 2013

Let *M*_{k} be the quantity

where *F* ranges over square-integrable functions on the simplex

with being the quadratic forms

and

It is known that *D**H**L*[*k*,*m* + 1] holds whenever *E**H*[θ] holds and . Thus for instance, *M*_{k} > 2 implies *D**H**L*[*k*,2] on the Elliott-Halberstam conjecture, and *M*_{k} > 4 implies *D**H**L*[*k*,2] unconditionally.

## Contents |

## Upper bounds

We have the upper bound

- (1)

that is proven as follows.

The key estimate is

- . (2)

Assuming this estimate, we may integrate in to conclude that

which symmetrises to

giving the desired upper bound (1).

It remains to prove (2). By Cauchy-Schwarz, it suffices to show that

But writing , the left-hand side evaluates to

as required.

## Lower bounds

...

## World records

k
| Lower bound | Upper bound |
---|---|---|

4 | 1.845 | 1.848 |

5 | 2.001162 | 2.011797 |

10 | 2.53 | 2.55842 |

20 | 3.05 | 3.1534 |

30 | 3.34 | 3.51848 |

40 | 3.52 | 3.793466 |

50 | 3.66 | 3.99186 |

59 | 3.95608 | 4.1479398 |

All upper bounds come from (1).

## More general variational problems

It appears that for the purposes of establish DHL type theorems, one can increase the range of F in which one is taking suprema over (and extending the range of integration in the definition of accordingly). Firstly, one can enlarge the simplex to the larger region

provided that one works with a generalisation of *E**H*[θ] which controls more general Dirichlet convolutions than the von Mangoldt function (a precise assertion in this regard may be found in BFI). In fact one should be able to work in any larger region *R* for which

provided that all the marginal distributions of F are supported on , thus (assuming F is symmetric)

- when

For instance, one can take , or one can take (although the latter option breaks the symmetry for F). Perhaps other choices are also possible.