Correction to ‘Shifted convolution and the Titchmarsh divisor problem over Fq[t] [PDF]
, 2016 PublishedCorrection to original article: Phil. Trans. R. Soc. A 373, 20140308 (28 April 2015; Published online 23 March 2015) (doi:10.1098/rsta.2014.0308). Two of the equations in the original article contained a typographical error.
Andrade, JC, Bary-Soroker, L, Rudnick, Z
core +2 more sources
On the $k$-free values of the polynomial $xy^k+C$ [PDF]
, 2015 Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$.
Lapkova, Kostadinka
core +2 more sources
Uncertainty principles connected with the M\"{o}bius inversion formula [PDF]
, 2012 We say that two arithmetic functions f and g form a Mobius pair if f(n) = \sum_{d \mid n} g(d) for all natural numbers n. In that case, g can be expressed in terms of f by the familiar Mobius inversion formula of elementary number theory.
Pollack, Paul, Sanna, Carlo
core +2 more sources
Counting Fixed Points, Two-Cycles, and Collisions of the Discrete Exponential Function using p-adic Methods [PDF]
, 2011 Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g.
Bourbaki +9 more
core +1 more source
On the number of subgroups of a given exponent in a finite abelian group [PDF]
, 2017 This paper deals with the number of subgroups of a given exponent in a finite abelian group. Explicit formulas are obtained in the case of rank two and rank three abelian groups.
Tóth, László, Tărnăuceanu, Marius
core +2 more sources
The least common multiple of a sequence of products of linear polynomials
, 2011 Let $f(x)$ be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate: $\log {\rm lcm}(f(1), ..., f(n))\sim An$ as $n\rightarrow\infty $, where $A$ is a constant depending on $f$.Comment: To appear in ...
A. Selberg +12 more
core +1 more source
Asymptotic behavior of the least common multiple of consecutive arithmetic progression terms
, 2013 Let $l$ and $m$ be two integers with $l>m\ge 0$, and let $a$ and $b$ be integers with $a\ge 1$ and $a+b\ge 1$.
B. Farhi +10 more
core +1 more source
Inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs [PDF]
, 2017 Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of ...
Haukkanen, Pentti, Tóth, László
core +2 more sources
The shifted convolution of generalized divisor functions [PDF]
, 2016 We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously by Fouvry and ...
Topacogullari, Berke
core +2 more sources
Arithmetic functions at consecutive shifted primes [PDF]
, 2014 For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for $f(p_n-1) > f(p_{n+1 ...
Pollack, Paul, Thompson, Lola
core +2 more sources