Related papers: Dirichlet product of derivative arithmetic with an…
Under two assumptions, we determine the distribution of the difference between two functions each counting the numbers < x that are in a given arithmetic progression modulo q and the product of two primes. The two assumptions are (i) the…
We derive a product formula for the multiple stochastic integrals with respect to Levy process. The idea is to use exponential vectors and the polarization technique which greatly simplify the argument.
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
A formula expressing the fermionic determinant as an infinite product of smaller determinants is derived and discussed. These smaller determinants are of a fixed size, independent of the size of the lattice and are indexed by loops of…
We show that the module of integral points on a Drinfeld module satisfies a an analogue of Dirichlet's unit theorem, despite its failure to be finitely generated. As a consequence, we obtain a construction of a canonical finitely generated…
An integer--valued function is an entire function which maps the nonnegative integers $\mathbb N$ to the integers. An example is $2^z$. A Hurwitz function is an entire function having all derivatives taking integer values at $0$. An example…
We define the e\~ne product for the multiplicative group of polynomials and formal power series with coefficients on a commutative ring and unitary constant coefficient. This defines a commutative ring structure where multiplication is the…
There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a…
Given two infinite sequences with known binomial transforms, we compute the binomial transform of the product sequence. Various identities are obtained and numerous examples are given involving sequences of special numbers: Harmonic…
A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…
We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable non-degeneracy condition. Using Geometry of Numbers, we unconditionally show their…
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating…
By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).
The classical Khintchine and Jarn\'ik theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real…
The discrete Fourier transform of the greatest common divisor is a multiplicative function, if taken with respect to the same order of the primitive root of unity, which is a well known fact. As such, the transform can be expressed in the…
The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…
We study a multiplicative function analogue of Linnik's problem on the least prime in an arithmetic progression. Let $h\colon \mathbb{N}\to\mathbb{R}\setminus\{0\}$ be a multiplicative function, and let $a \pmod q$ be a reduced residue…
The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar function in matrix arguments may require the computation of…