# Selberg sieve variational problem

### From Polymath1Wiki

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! colspan="2" | <math>M_{k,\varepsilon,1/2}</math> | ! colspan="2" | <math>M_{k,\varepsilon,1/2}</math> | ||

! colspan="2" | <math>M'_{k,\varepsilon,1/2}</math> (symmetric) | ! colspan="2" | <math>M'_{k,\varepsilon,1/2}</math> (symmetric) | ||

+ | ! colspan="2" | <math>M''_{k,\varepsilon,1/2}</math> (symmetric) | ||

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! | ! | ||

+ | ! Lower | ||

+ | ! Upper | ||

! Lower | ! Lower | ||

! Upper | ! Upper | ||

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| [http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268443 1.992] | | [http://terrytao.wordpress.com/2014/01/17/polymath8b-vi-a-low-dimensional-variational-problem/#comment-268443 1.992] | ||

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+ | | [http://terrytao.wordpress.com/2014/01/28/polymath8b-vii-using-the-generalised-elliott-halberstam-hypothesis-to-enlarge-the-sieve-support-yet-further/#comment-268732 2.0009] | ||

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## Revision as of 21:54, 29 January 2014

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

We will need some parameters *c*,*T*,τ > 0 and *a* > 1 to be chosen later (in practice we take c close to 1 / log*k*, T a small multiple of c, and τ a small multiple of c/k.

For any symmetric function F on the simplex , one has

and so by scaling, if F is a symmetric function on the dilated simplex , one has

after adjusting the definition of the functionals suitably for this rescaled simplex.

Now let us apply this inequality r in the interval [1,1 + τ] and to truncated tensor product functions

for some bounded measurable , not identically zero, with . We have the probabilistic interpretations

and

where , and are iid random variables in [0,T] with law , and we adopt the convention that vanishes when *b* < *a*. We thus have

- (*)

for any r.

We now introduce the random function *h* = *h*_{r} by

Observe that if *S*_{k − 1} < *r*, then

and hence by the Legendre identity

We also note that (using the iid nature of the *X*_{i} to symmetrise)

Inserting these bounds into (*) and rearranging, we conclude that

where is the defect from the upper bound. Splitting the integrand into regions where s or t is larger than or less than T, we obtain

where

and

We now focus on *Y*_{1}. It is only non-zero when . Bounding , we see that

where log_{ + }(*x*) is equal to log*x* when and zero otherwise. We can rewrite this as

We write and . Using the bound we have

and thus (bounding ).

- .

Symmetrising, we conclude that

where

For *Z*_{2}, which is a tiny term, we use the crude bound

For *Z*_{1}, we use the bound

valid for any *a* > 1, which can be verified because the LHS is concave for , while the RHS is convex and is tangent to the LHS as x=a. We then have

and thus

where

A good choice for *a* = *a*[*r*] here is (assuming ), in which case the formula simplifies to

Thus far, our arguments have been valid for arbitrary functions *g*. We now specialise to functions of the form

Note the identity

on [0,min(*r* − *S*_{k − 1},*T*)]. Thus

Bounding and using symmetry, we conclude

Since , we conclude that

where *Z*_{4} = *Z*_{4}[*r*] is the quantity

Putting all this together, we have

At this point we encounter a technical problem that *Z*_{4} diverges logarithmically (up to a cap of log*k*) as *S*_{k − 1} approaches r. To deal with this issue we average in r, and specifically over the interval [1,1 + τ]. One can calculate that

for all x if we have (because the two expressions touch at *x* = 1 + τ, with the RHS being convex with slope at least (1 + τ / 2) / τ there. and the LHS lying underneath this tangent line). Assuming this, we conclude that

provided that *k*μ < 1 − τ, and hence

Now we deal with the *Z*_{4} integral. We split into two contributions, depending on whether or not. If , then we may bound

when , so this portion of may be bounded by

Observe that

and so this portion of is bounded by

*W**X*

where

As for the portion when 1 + *u*τ − *S*_{k − 1} > 2*c*, we bound , and so this portion of may be bounded by

- =
*V**U*

where

We thus arrive at the final bound

provided that *k*μ < 1 − τ and the denominator is positive.

### Asymptotic analysis

We work in the asymptotic regime . Setting

for some absolute constants and β,γ > 0, one calculates

and so the constraint 1 − *k*μ > τ becomes

With for , one has

and one also calculates

*Z*_{2},*Z*_{3}=*o*(1)

and hence

assuming that α + logβ − 1 > γ and

and where .

In particular, by setting α,β,γ as absolute constants obeying these constraints, we have , and so

*M*_{k}= log*k*+*O*(1).

If we set *c* = τ: = 1 / log*k*, with *T* a small multiple of c, then for a large absolute constant A, and σ^{2} is a small multiple of . This makes the denominator comparable to 1; one can check that all the terms in the numerator are O(1), finally giving the bound

- Δ
_{k}=*O*(1)

and thus we have the lower bound

- .

## 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 (as shown here) one can 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). See this blog post for more discussion.

If the marginal distributions of F are supported in instead of , one still has a usable lower bound in which is replaced by the slightly smaller quantity ; see this blog post for more discussion.

## World records

k
| M_{k}
| M'_{k}
| M''_{k} (prism)
| M''_{k} (symmetric)
| M''_{k} (non-convex)
| ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1D | Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | |

2 | 1.383 | 1.38593... | 1.38593... | 2 | 2 | 2 | 2 | 2 | 2 | ||

3 | 1.635 | 1.646 | 1.648 | 1.8429 | 2.080 | 1.914 | 2.38593... | 1.902626 | 3 | 1.847 | 3 |

4 | 1.820 | 1.845 | 1.848 | 1.937 | 2.198 | 1.951 | 2.648 | 1.937 | 4 | ||

5 | 1.965 | 2.001162 | 2.011797 | 2.059 | 2.311 | 2.848 | |||||

10 | 2.409 | 2.53 | 2.55842 | 2.7466 | |||||||

20 | 2.810 | 3.05 | 3.1534 | 3.2716 | |||||||

30 | 3.015 | 3.34 | 3.51848 | 3.6079 | |||||||

40 | 3.145 | 3.52 | 3.783467 | 3.8564 | |||||||

50 | 3.236 | 3.66 | 3.99186 | 4.0540 | |||||||

54 | 3.266 | 4.001115 | 4.0642 | 4.1230 | |||||||

55 | 3.273 | 4.0051689 | 4.0816 | 4.1396 | |||||||

59 | 3.299 | 4.06 | 4.1478399 | 4.2030 |

k
| (symmetric) | (prism) | (symmetric) | (non-convex) | (symmetric) | (symmetric) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | Lower | Upper | |

2 | 1.38593... | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1.690608 | 2 | |||||

3 | 1.8615 | 3 | 1.956713 | 3 | 1.936708 | 3 | 1.962998 | 3 | 1.9400 | 3 | 1.87459 | 1.992 | 2.0009 | |||

4 | 2.0023 | 4 | 2.0023 | 4 | 2.0023 | 4 | 2.0023 | 4 | 2.01869 |

For k>2, all upper bounds on *M*_{k} come from (1). Upper bounds on *M*'_{k} come from the inequality that follows from an averaging argument, and upper bounds on *M*''_{k} (on EH, using the prism as the domain) come from the inequality by comparing *M*''_{k} with a variational problem on the prism (details here). The 1D bound is the optimal value for *M*_{k} when the underlying function *F* is restricted to be of the "one-dimensional" form .

For k=2, *M*'_{2} = *M*''_{2} = 2 can be computed exactly by taking F to be the indicator function of the unit square (for the lower bound), and by using Cauchy-Schwarz (for the upper bound). can be computed exactly as the solution to the equation .

The quantity is defined as the supremum of where F is now supported on , and is defined as but now restricted to , and is a parameter to be optimised in. The quantity is defined similarly, but with F now supported on . Finally, is also defined similarly, but with F supported on a region R as above, with all marginals supported on .

For k=3, we also have the non-convex candidate .

The crude upper bound of *k* for any of the *M*_{k} type quantities comes from the parity problem obstruction that each separate event "*n* + *h*_{i} prime" can occur with probability at most 1/2.