48 research outputs found
Correlation of arithmetic functions over
For a fixed polynomial , we study the number of polynomials of
degree over such that and are both
irreducible, an -analogue of the twin primes problem. In the
large- limit, we obtain a lower-order term for this count if we consider
non-monic polynomials, which depends on in a manner which is
consistent with the Hardy-Littlewood Conjecture. We obtain a saving of if
we consider monic polynomials only and is a scalar. To do this, we use
symmetries of the problem to get for free a small amount of averaging in
. This allows us to obtain additional saving from equidistribution
results for -functions. We do all this in a combinatorial framework that
applies to more general arithmetic functions than the indicator function of
irreducibles, including the M\"{o}bius function and divisor functions.Comment: Incorporated referee comments. Accepted for publication in Math.
Annale
An improved lower bound for the union-closed set conjecture
Gilmer has recently shown that in any nonempty union-closed family of subsets of a finite set, there exists an element contained in at least a
proportion of the sets of . We improve the proportion from
to in this result. An improvement to
would be the Frankl union-closed set conjecture. We follow
Gilmer's method, replacing one key estimate by a sharp estimate. We then
suggest a new addition to this method and sketch a proof that it can obtain a
constant strictly greater than . We also disprove a
conjecture of Gilmer that would have implied the union-closed set conjecture.Comment: 10 page