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.

О поведении функций, родственных функции Чебышева

Abstract: Many problems of Number Theory are connected with investigation of Dirichlet series $$f(s)=\sum_{n=1}^{\infty} a_nn^{-s}$$ and the adding functions $$\Phi(x)=\sum_{n\leq x} a_n$$ of their coefficients. The most famous Dirichlet series is the Riemann zeta function $$\zeta(s),$$ defined for any $$s=\sigma+it$$ with $$\Re s=\sigma> 1$$ as $$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}.$$ The square of zeta function $$\zeta^{2}(s)=\sum_{n=1}^{\infty}\frac{\tau(n)}{n^s}, \,\, \Re s > 1,$$ is connected with the divisor function $$\tau (n)=\sum_ { d | n } 1,$$ giving the number of a positive integer divisors of positive integer number n. The adding function of the Dirichlet series $$\zeta^2(s)$$ is the function $$D (x)=\sum_ { n\leq x}\tau(n)$$; the questions of the asymptotic behavior of this function are known as Dirichlet divisor problem. Generally, $$ \zeta^{k}(s)=\sum_{n=1}^{\infty}\frac{\tau_k(n)}{n^s}, \,\, \Re s > 1, $$ where function $$\tau_k (n)=\sum_{n=n_1\cdot...\cdot n_k} 1$$ gives the number of representations of a positive integer number n as a product of k positive integer factors. The adding function of the Dirichlet series $$ \zeta^k (s)$$ is the function $$D_k (x)=\sum_ { n\leq x}\tau_k(n)$$; its research is known as the multidimensional Dirichlet divisor problem. The logarithmic derivative $$\frac{\zeta^{'}(s)}{\zeta(s)}$$ of zeta function can be represented as $$\frac{\zeta^{'}(s)}{\zeta(s)}=-\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s},$$ $$\Re s >1.$$ Here $$\Lambda(n)$$ is the Mangoldt function, defined as $$\Lambda(n)=\log p,$$ if $$n=p^{k}$$ for a prime number p and a positive integer number k, and as $$\Lambda(n)=0,$$ otherwise. So, the Chebyshev function $$\psi(x)=\sum_{n\leq x}\Lambda(n)$$ is the adding function of the coefficients of the Dirichlet series $$\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s},$$ corresponding to logarithmic derivative $$\frac{\zeta^{'}(s)}{\zeta(s)}$$ of zeta function. It is well-known in analytic Number Theory and is closely connected with many important number-theoretical problems, for example, with asymptotic law of distribution of prime numbers. In particular, the following representation of $$\psi(x)$$ is very useful in many applications: $$\psi(x)=x-\sum_{|\Im \rho|\leq T}\frac{x^{\rho}}{\rho}+O\left(\frac{x\ln^{2}x}{T}\right), $$ where x=n+0,5, $$n \in\mathbb{N},$$ $$2\leq T \leq x,$$ and $$\rho=\beta+i\gamma$$ are non-trivial zeros of zeta function, i.e., the zeros of $$\zeta(s),$$ belonging to the critical strip 0 2, $$T \geq 2,$$ and $$\rho=\beta+i\gamma$$ are non-trivial zeros of zeta function, i.e., the zeros of $$\zeta(s),$$ belonging to the critical strip 0 < Res < 1.

Embedding Divisor and Semi-Prime Testability in f-vectors of polytopes.

Abstract: We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The regime where we prove this computational difference (conditioned on standard conjectures on the density of primes and on $P eq NP$) is when the dimension $d$ tends to infinity and the number of facets is linear in $d$.