# Selberg sieve variational problem

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- | | [http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255681 3. | + | | [http://terrytao.wordpress.com/2013/11/22/polymath8b-ii-optimising-the-variational-problem-and-the-sieve/#comment-255681 3.95608] |

| 4.1479398 | | 4.1479398 | ||

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All upper bounds come from (1). | All upper bounds come from (1). |

## Revision as of 20:32, 9 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.

## 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).