Topic
Divisor
About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.
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TL;DR: In this paper, the authors obtained a full asymptotic expansion for the shifted convolution sum for the generalized Titchmarsh divisor problem under the assumption that the multiplicative function is periodic over the primes.
Abstract: Given a multiplicative function~$f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|
6 citations
TL;DR: In this article, the steady Swift-Hohenberg partial differential equation (SSEPDE) was considered and the existence of small divisor solutions for small parameter values was proved.
Abstract: We consider the steady Swift - Hohenberg partial differential equation. It is a one-parameter family of PDE on the plane, modeling for example Rayleigh - B\'enard convection. For values of the parameter near its critical value, we look for small solutions, quasiperiodic in all directions of the plane and which are invariant under rotations of angle \pi/q, q\geq 4. We solve an unusual small divisor problem, and prove the existence of solutions for small parameter values.
6 citations
TL;DR: In this paper, it was shown that for all even integers n > 2 and a positive proportion of the odd integers n, namely those having a dominating prime power factor, there exists a multiplicative divisor set D such that ICG ( n, D ) is Ramanujan.
6 citations
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TL;DR: In this article, an upper bound for the expected error term in the asymptotic formula was obtained for the case of δ(n) = 3, where δ is the error term of the divisor function generated by δ.
Abstract: We obtain a new upper bound for $\sum_{h\le H}\Delta_k(N,h)$ for $1\le H\le N$, $k\in \N$, $k\ge3$, where $\Delta_k(N,h)$ is the (expected) error term in the asymptotic formula for $\sum_{N < n\le2N}d_k(n)d_k(n+h)$, and $d_k(n)$ is the divisor function generated by $\zeta(s)^k$. When $k=3$ the result improves, for $H\ge N^{1/2}$, the bound given in the recent work \cite{[1]} of Baier, Browning, Marasingha and Zhao, who dealt with the case $k=3$.
6 citations
01 Nov 2016
TL;DR: In this paper, it was shown that the proportion of polynomials with a divisor of every degree below a given value of n is given by O(n −1 + O (n −2 ) for a finite field with n elements, where n is the number of elements in the field.
Abstract: We show that the proportion of polynomials of degree $n$ over the finite field with $q$ elements, which have a divisor of every degree below $n$, is given by $c_q n^{-1} + O(n^{-2})$. More generally, we give an asymptotic formula for the proportion of polynomials, whose set of degrees of divisors has no gaps of size greater than $m$. To that end, we first derive an improved estimate for the proportion of polynomials of degree $n$, all of whose non-constant divisors have degree greater than $m$. In the limit as $q \to \infty$, these results coincide with corresponding estimates related to the cycle structure of permutations.
6 citations