Related papers: A note on Artin's constant
Work in progress concerning alternative formalizations of arithmetic.
Following an idea due to Euler, we evaluate the alternating sums of powers of consrcutive integers.
A new simple proof of Stirling's formula via the partial fraction expansion for the tangent function is presented.
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
In this note, we propose simple summations for primes, which involve two finite nested sums and Bernoulli numbers. The summations can also be expressed in terms of Bernoulli polynomials.
The authors review results implicit in their recent paper [2] on the product/quotient representation of rationals by rationals of the type $( an + b )/ ( An+ B )$ and give a detailed account of a particular related non-intuitive…
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…
I review some of the recent progress in two-dimensional string theory, which is formulated as a sum over surfaces embedded in one dimension.
In this article we furnish a new simple proof of a hard identity from the theory of cubature formulas via the method of coefficients.
By dividing hypergeometric series representations of the inverse sine by sin^-1 (x) and integrating, new double series representations of integers and constants arise. Binomial coefficients and the sine integral are thus combined in double…
A short proof of the linear nested Artin approximation property of the algebraic power series rings is given here.
We show that Somos' constant is universal in sense that is similar to the universality of the Khinchin constant. In addition we introduce generalized Somos' constants, which are universal in a similar sense.
As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessemberg sum is a…
One is expressed as the sum of the reciprocals of a certain set of integers. We give an elegant proof to the fact applying the polynomial theorem and basic calculus.
The Artin exponent induced from cyclic subgroups of finite groups was studied extensively by T.Y. Lam. A Burnside ring theoretic version of Lam's results for $p$-groups was given by the author in an earlier paper. Here we look at the Artin…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
We improve constants in the Rademacher-Menchov inequality.
Using probability theory we derive an expression for the sum of a series of definite integrals involving upper incomplete Gamma functions. In the proof, a normal variance mixture distribution with Beta mixing distributions plays a crucial…
We develop new closed form representations of sums of (n + {\alpha})th shifted harmonic numbers and reciprocal binomial coefficients in terms of {\alpha}th shifted harmonic numbers. Some interesting new consequences and illustrative…