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 asymptotic bounds for the Dirichlet kernel and minor arc bounds for exponential sum with γ-tau-k, where γ is the number of vertices in the circle.
Abstract: We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc bounds for the exponential sum with $\tau_k$, and asymptotics for high moments of the Dirichlet kernel.
1 citations
TL;DR: In this article, the Chung-Feller theorem was used to give an answer to the analogous question for another divisor of the central binomial coefficient, and the main result follows from a more general observation regarding binomial coefficients with n and k relatively prime.
Abstract: It is well known that for all $$n\ge 1$$
the number $$n+1$$
is a divisor of the central binomial coefficient $${2n\atopwithdelims ()n}$$
. Since the nth central binomial coefficient equals the number of lattice paths from (0, 0) to (n, n) by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of $$n+1$$
paths or $$n+1$$
equinumerous sets of paths. The Chung–Feller theorem gives an elegant answer to this question. We pose and deliver an answer to the analogous question for $$2n-1$$
, another divisor of $${2n\atopwithdelims ()n}$$
. We then show our main result follows from a more general observation regarding binomial coefficients $${n\atopwithdelims ()k}$$
with n and k relatively prime. A discussion of the case where n and k are not relatively prime is also given, highlighting the limitations of our methods. Finally, we come full circle and give a novel interpretation of the Catalan numbers.
1 citations
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TL;DR: In this article, the admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct primes.
Abstract: We calculate admissible values of r such that a square-free polynomial with integer coefficients, no fixed prime divisor and irreducible factors of degree at most 3 takes infinitely many values that are a product of at most r distinct primes.
1 citations
TL;DR: In this article, a relative homotopy invariant K-theory of a closed immersion of schemes was constructed for a smooth scheme and a divisor, and a canonical homomorphism from the Chow groups with modulus $CH^i(X|D)$ to the relative motivic cohomology groups appeared in the above spectral sequence.
Abstract: We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles.
Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme $X$ and a divisor $D \subset X$, we construct a canonical homomorphism from the Chow groups with modulus $\CH^i(X|D)$ to the relative motivic cohomology groups $H^{2i}(X|D, \Z(i))$ appearing in the above spectral sequence. This map is shown to be an isomorphism when $X$ is affine and $i = \dim(X)$.
1 citations