Selberg sieve variational problem

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(World records)
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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 Mk be the quantity

\displaystyle M_k := \sup_F \frac{\sum_{m=1}^k J_k^{(m)}(F)}{I_k(F)}

where F ranges over square-integrable functions on the simplex

\displaystyle {\mathcal R}_k := \{ (t_1,\ldots,t_k) \in [0,+\infty)^k: t_1+\ldots+t_k \leq 1 \}

with I_k, J_k^{(m)} being the quadratic forms

\displaystyle I_k(F) := \int_{{\mathcal R}_k} F(t_1,\ldots,t_k)^2\ dt_1 \ldots dt_k

and

\displaystyle J_k^{(m)}(F) := \int_{{\mathcal R}_{k-1}} (\int_0^{1-\sum_{i \neq m} t_i} F(t_1,\ldots,t_k)\ dt_m)^2 dt_1 \ldots dt_{m-1} dt_{m+1} \ldots dt_k.

It is known that DHL[k,m + 1] holds whenever EH[θ] holds and M_k > \frac{2m}{\theta}. Thus for instance, Mk > 2 implies DHL[k,2] on the Elliott-Halberstam conjecture, and Mk > 4 implies DHL[k,2] unconditionally.

Upper bounds

We have the upper bound

\displaystyle M_k \leq \frac{k}{k-1} \log k (1)

that is proven as follows.

The key estimate is

 \displaystyle \int_0^{1-t_2-\ldots-t_k} F(t_1,\ldots,t_k)\ dt_1)^2 \leq \frac{\log k}{k-1} \int_0^{1-t_2-\ldots-t_k} F(t_1,\ldots,t_k)^2 (1 - t_1-\ldots-t_k+ kt_1)\ dt_1.. (2)

Assuming this estimate, we may integrate in t_2,\ldots,t_k to conclude that

\displaystyle J_k^{(1)}(F) \leq \frac{\log k}{k-1} \int F^2 (1-t_1-\ldots-t_k+kt_1)\ dt_1 \ldots dt_k

which symmetrises to

\sum_{m=1}^k J_k^{(m)}(F) \leq k \frac{\log k}{k-1} \int F^2\ dt_1 \ldots dt_k

giving the desired upper bound (1).

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

\displaystyle \int_0^{1-t_2-\ldots-t_k} \frac{dt_1}{1 - t_1-\ldots-t_k+ kt_1} \leq \frac{\log k}{k-1}.

But writing s = t_2+\ldots+t_k, the left-hand side evaluates to

\frac{1}{k-1} (\log k(1-s) - \log (1-s) ) = \frac{\log k}{k-1}

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).

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