Related papers: On the irrationality measure function in average
We express continuous $\times p,\times q$-invariant measures on the unit circle via some simple forms. On one hand, a continuous $\times p,\times q$-invariant measure is the weak-$*$ limit of average of Dirac measures along an irrational…
In this paper we study the mean values of some multiplicative functions connected with the divisor function on the short interval of summation. The asymptocic values for such mean values are proved.
Given a monomial ideal $I$, we study two functions that quantify ways to measure the difference between symbolic powers and usual powers of $I$. In many cases we determine the asymptotic growth rate of these two functions. We also perform…
The main purpose of this article is to get a handle on determining how far a non-rational singularity is from being rational, or in other words, introduce a measure of the failure of a singularity being rational.
This paper is concerned with the asymptotic behavior of sums of terms which are a test function f evaluated at successive increments of a discretely sampled semimartingale. Typically the test function is a power function (when the power is…
Rational inner functions are a generalization of finite Blaschke products to several variables. In this article we survey a variety of results about rational inner functions related to interpolation, sums of squares formulas, and boundary…
We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of…
A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…
This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums…
A survey of work on motivic integration.
A general divergence measure for monotonic functions is introduced. Its connections with the f-divergence for convex functions are explored. The main properties are pointed out.
If the Euclidean norm is strongly concentrated with respect to a measure, the average distribution of an average marginal of this measure has Gaussian asymptotics that captures tail behaviour. If the marginals of the measure have…
Using a recently derived integral in terms of elementary functions, we derive new asymptotic expansions of the normal inverse Gaussian cumulative distribution function. One of the asymptotic representations is in terms of the normal…
There have been a number of developments in measuring inconsistency in logic-based representations of knowledge. In contrast, the development of inconsistency measures for computational models of argument has been limited. To address this…
We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.
In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to 0. The function of the $i$th increment may depend on the current time, and also…
We examine the counting function for rational points on conics, and show how the point where the asymptotic behaviour begins depends on the size of the smallest zero.
We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the…
In this note we show how the irrationality measure of $\zeta(s) = \pi^2/6$ can be used to obtain explicit lower bounds for $\pi(x)$. We analyze the key ingredients of the proof of the finiteness of the irrationality measure, and show how to…
In this paper, we estimate the integral T(x) mentioned in the title, where {t} denotes the fractional part of the real number t, and x is any positive real number.