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Let $\mathbb{F}_q[t]$ be the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements, and let $p$ be the characteristic of $\mathbb{F}_q$. We denote $\widetilde{G}_q(k)$ to be the least integer $t_0$ with the property that…
Defining a family of recurrences, we generalize Comtet's formula for the generating function of the enumeration of indecomposable permutations. Consequently, we generalize Panaitopol's asymptotic expansion for the prime counting function,…
Let A be a set of integers. For every integer n, let r_{A,2}(n) denote the number of representations of n in the form n = a_1 + a_2, where a_1 and a_2 are in A and a_1 \leq a_2. The function r_{A,2}: Z \to N_0 \cup {\infty} is the…
We improve the classical discrete Hardy inequality for $ 1<p<\infty $ for functions on the natural numbers. For integer values of $ p $ the Hardy weight is an absolutely monotonic function.
We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression.…
In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the…
One of the questions of distribution of prime numbers is considered in the article. It is shown what error is obtained from the assumption that the asymptotic density of a sequence of primes is a probability. Various forms of an analogue of…
We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…
We investigate the structure of branching asymptotics appearing in solutions to elliptic edge problems. The exponents in powers of the half-axis variable, logarithmic terms, and coefficients depend on the variables on the edge and may be…
The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given…
Extending classical results on polytopal approximation of convex bodies, we derive asymptotic formulas for the weighted approximation of smooth convex functions by piecewise affine convex functions as the number of their facets tends to…
The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and…
The so-called polynomial equations play an important role both in algebra and in the theory of functional equations. If the unknown functions in the equation are additive, relatively many results are known. However, even in this case, there…
For $n \geq 3$, an asymptotic formula is derived for the number of representations of a sufficiently large natural number $N$ as a sum of $r = 2^n + 1$ summands, each of which is an $n$-th power of natural numbers $x_i$, $i = \overline{1,…
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.
We give results on the asymptotic in Waring's problem over function fields that are stronger than the results obtained over the integers using the main conjecture in Vinogradov's mean value theorem. Similar estimates apply to Manin's…
The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
We give asymptotic expansions for the moments of the $M_2$-rank generating function and for the $M_2$-rank generating function at roots of unity. For this we apply the Hardy-Ramanujan circle method extended to mock modular forms. Our…
We study various asymptotic approximations of Good's special functions arising in atomic physics. These special functions are situated beyond Anger's functions to which they are closely related. Our major tool is the method of the…