相关论文: Multiplicative functions in arithmetic progression…
We introduce and study properties of certain new harmonic function spaces on products of upper half-spaces.Norm estimates for the so-called expanded Bergman projections are obtained.Sharp theorems on multipliers acting on certain Sobolev…
We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower…
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of…
Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…
Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes.
We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to $1$. Specifically, we consider $\alpha_N$-fold divisor functions, when $\alpha_N$ is a sequence of positive real numbers…
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…
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation".…
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers…
We investigate the construction of $\pm1$-valued completely multiplicative functions that take the value $+1$ at at most $k$ consecutive integers, which we call length-$k$ functions. We introduce a way to extend the length based on the idea…
A set of rational points on a curve is said to be in geometric progression if either the abscissae or the ordinates of the points are in geometric progression. Examples of three points in geometric progression on a circle are already known.…
We describe some of the machinery behind recent progress in establishing infinitely many arithmetic progressions of length $k$ in various sets of integers, in particular in arbitrary dense subsets of the integers, and in the primes.
In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.
In a previous paper we have introduced a class of multiplications of distributions in one dimension. Here we furnish different generalizations of the original definition and we discuss some applications of these procedures to the…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the…
We consider the sum of squares function in the ring $\mathbb{Z}_{n}$. We determine formulae in a number of cases when $n$ is a power of a prime.
We provide examples of multiplicative functions $f$ supported on the squarefree integers, such that on primes $f(p)=\pm1$ and such that $M_f(x):=\sum_{n\leq x} f(n)=o(\sqrt{x})$. Further, by assuming the Riemann hypothesis (RH) we can go…
We give characterizations of unital uniform topological algebras and saturated locally multiplicatively convex algebras by means of multiplicative linear functionals. Some automatic continuity theorems in advertibly complete uniform…