Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the correlation between M{o}bius and polynomial phases in short arithmetic progressions
70
0
0.0
(
0
)
Authors
Fei Wei
Ask ChatGPT about the research
No Arabic abstract
We obtain an estimate for the average value of the product of the Mobius function and any polynomial phase over short intervals and arithmetic progressions simultaneously. As a consequence, we prove that the product of M{o}bius and any polynomial phase is disjoint from arithmetic functions realized in certain rigid dynamical systems, such as any finite products of translations of Mobius squared.
rate research
Read More
We establish cancellation in short sums of certain special trace functions over $mathbb{F}_q[u]$ below the P{o}lya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $mathbb{F}_q[u]$-analog of Chowlas conjecture on cancellation in M{o}bius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $mathbb{F}_q[u]$.
A vast class of exponential functions is showed to be deterministic. This class includes functions whose exponents are polynomial-like or piece-wise close to polynomials after differentiation. Many of these functions are indeed disjoint from the Mobius function. As a consequence, we show that Sarnaks Disjointness Conjecture for the Mobius function (from deterministic sequences) is equivalent to the disjointness in average over short intervals
We study a Dirichlet series in two variables which counts primitive three-term arithmetic progressions of squares. We show that this multiple Dirichlet series has meromorphic continuation to $mathbb{C}^2$ and use Tauberian methods to obtain counts for arithmetic progressions of squares and rational points on $x^2+y^2=2$.
Given an abelian group $G$, it is natural to ask whether there exists a permutation $pi$ of $G$ that destroys all nontrivial 3-term arithmetic progressions (APs), in the sense that $pi(b) - pi(a) eq pi(c) - pi(b)$ for every ordered triple $(a,b,c) in G^3$ satisfying $b-a = c-b eq 0$. This question was resolved for infinite groups $G$ by Hegarty, who showed that there exists an AP-destroying permutation of $G$ if and only if $G/Omega_2(G)$ has the same cardinality as $G$, where $Omega_2(G)$ denotes the subgroup of all elements in $G$ whose order divides $2$. In the case when $G$ is finite, however, only partial results have been obtained thus far. Hegarty has conjectured that an AP-destroying permutation of $G$ exists if $G = mathbb{Z}/nmathbb{Z}$ for all $n eq 2,3,5,7$, and together with Martinsson, he has proven the conjecture for all $n > 1.4 times 10^{14}$. In this paper, we show that if $p$ is a prime and $k$ is a positive integer, then there is an AP-destroying permutation of the elementary $p$-group $(mathbb{Z}/pmathbb{Z})^k$ if and only if $p$ is odd and $(p,k) otin {(3,1),(5,1), (7,1)}$.
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
