Related papers: A Riemann-Farey Computation
Particular solutions of the Benney equations are constructed. Their properties are discussed.
New lower bounds involving sum, difference, product, and ratio sets for $A\subset \C$ are given.
The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
In this note we give a closed formula for Faltings' delta-invariant of a hyperelliptic Riemann surface.
A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.
An integral representation for matrix Airy function is presented
We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable…
We offer some partition functions related to ternary quadratic forms, and note on their upper bounds and related properties. We offer these results as an application of a simple method related to conjugate Bailey pairs presented in a prior…
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
In this paper, we will present a new iterative construction for the auxiliary equation of Waring's problem, which seems a little simpler than the one of so called "smooth numbers" in papers [4] and [8], and give same upper bounds of G(k) as…
The known lower bound for the the classical Ramsey number $R(5,6)$ is improved from $58$ to $59$. The method used to construct the graph is a simple variant of computational methods that have been previously used to construct Ramsey graphs.…
Let $\boldsymbol{\alpha}\in \mathbb{R}^N$ and $Q\geq 1$. We consider the sum $\sum_{\boldsymbol{q}\in [-Q,Q]^N\cap\mathbb{Z}^N\backslash\{\boldsymbol{0}\}}\|\boldsymbol{\alpha}\cdot\boldsymbol{q}\|^{-1}$. Sharp upper bounds are known when…
We establish sharp upper bounds for shifted moments of quadratic Dirichlet $L$-function under the generalized Riemann hypothesis. Our result is then used to prove bounds for moments of quadratic Dirichlet character sums.
Analytical expressions are derived for the position of irreducible fractions in the Farey sequence $F_N$ of order $N$ for a particular choice of $N$. The asymptotic behaviour is derived obtaining a lower error bound than in previous results…
Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…
Let phi(n) denote the Euler totient function. We study the analytic part associated with the summatory function of sigma_1(n) and obtain explicit bounds under the Riemann Hypothesis. In particular, we establish an upper bound of order…
The work is devoted to the construction of a new interval arithmetic which would combine algorithmic efficiency and high quality estimation of the ranges of expressions.
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…
We give a short proof of the convergence to the boundary of Riemann maps on varying domains. Our proof provides a uniform approach to several ad-hoc constructions that have recently appeared in the literature.