Related papers: Asymptotics on a class of Legendre formulas
We construct a smooth real-valued function P(n) in [0,1], defined via a triple integral with a periodic kernel, that approximates the characteristic function of prime numbers. The function is built to suppress when n is divisible by some m…
By establishing an improved level of distribution we study almost primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb Z$.
We prove and generalize a conjecture in arXiv:1610.0474(4) about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$, where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit…
Stirling numbers of the first kind are used in the derivation of several population genetics statistics, which in turn are useful for testing evolutionary hypotheses directly from DNA sequences. Here, we explore the cumulative distribution…
We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval.…
We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then…
We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…
The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but…
The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $\alpha$-mixing sequence of random variables. In particular the uniformly boundedness…
Some problems in the theory and applications of stochastic processes can be reduced to solving integral equations. While explicit solutions for these equations are often elusive, valuable insights can be gained through their asymptotic…
We study a sequence of differences related to the problem of finding the smallest factorial $n!$ greater than or equal to $a^n$, where $a > 1$, using the gamma function.
We prove weak approximation for isotrivial families of rationally connected varieties defined over the function field of a smooth projective complex curve.
We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…
Applying a theorem of Howard for a formula recently proved by Brassesco and M\'endez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion to the sequence of factorials. To our knowledge no explicit…
We investigate the rational approximation of fractional powers of unbounded positive operators attainable with a specific integral representation of the operator function. We provide accurate error bounds by exploiting classical results in…
Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions,…
Let f(t) be a rational function of degree at least 2 with rational coefficients. For a given rational number x_0, define x_{n+1}=f(x_n) for each nonnegative integer n. If this sequence is not eventually periodic, then the difference…
Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss…
Let $\mathfrak{p}_{\mathbb{P}_r}(n)$ denote the number of partitions of $n$ into $r$-full primes. We use the Hardy-Littlewood circle method to find the asymptotic of $\mathfrak{p}_{\mathbb{P}_r}(n)$ as $n \to \infty$. This extends previous…
The essential variables in a finite function $f$ are defined as variables which occur in $f$ and weigh with the values of that function. The number of essential variables is an important measure of complexity for discrete functions. When…