Jose Risomar Sousa

Jose Risomar Sousa
University of São Paulo | USP

Master of Science
Researching new formulas in math like there's no tomorrow. Wisdom is to be interested only in solvable problems.

About

16
Publications
33,767
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Introduction
I'm an enthusiast of math and physics. I read a lot about physics, I've learned a lot about it even though it's not my trade field. In math I like to research new formulas. I found a few and am struggling to find a few more. It's very important to me that my papers are pleasant and interesting to read. I don't want to just throw formulas at the reader without any backdrop or story telling. I try to avoid cold and hard papers, like most in the field.
Education
January 1992 - December 1997
University of São Paulo
Field of study
  • Math and Statistics

Publications

Publications (16)
Preprint
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The Riemann hypothesis is true. In this paper I present a solution for it in a very short and condensed way, making use of one of its equivalent problems. But as Carl Sagan once famously said, extraordinary claims require extraordinary evidence. The evidence here is the newly discovered inversion formula for Dirichlet series. PS-This is not really...
Article
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At the negative integers, there is a simple relation between the Lerch Φ function andthe polylogarithms. The literature has a formula for the polylogarithm at the negativeintegers, which utilizes the Stirling numbers of the second kind. Starting from thatformula, we deduce a simple closed formula for the Lerch Φ function at the negativeintegers, wh...
Article
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I present a method to solve the general cubic polynomial equation based on six years of research that started back in 1985 when, in the fifth grade, I first learned of Bhaskara's formula for the quadratic equation. I was fascinated by Bhaskara's formula and naively thought I could easily replicate his method for the third degree equation, but only...
Preprint
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In this paper we derive the possibly simplest integral representations for the Riemann zeta function and its generalizations (the Lerch function, Φ(e m , −k, b), the Hurwitz zeta, ζ(−k, b), and the polylogarithm, Li −k (e m)), valid in the whole complex plane relative to all parameters, except for singularities. We also present the relations betwee...
Article
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We review the closed-forms of the partial Fourier sums associated with HP k (n) and create an asymptotic expression for HP (n) as a way to obtain formulae for the full Fourier series (if b is such that |b| < 1, we get a surprising pattern, HP (n) ∼ H(n) − k≥2 (−1) k ζ(k)b k−1). Finally, we use the found Fourier series formulae to obtain the values...
Preprint
At the negative integers, there is a simple relation between the Lerch $\Phi$ function and the polylogarithms. The literature has a formula for the polylogarithm at the negative integers, which utilizes the Stirling numbers of the second kind. Starting from that formula, we can deduce a simple closed formula for the Lerch $\Phi$ function at the neg...
Article
Full-text available
A new relation between the Lerch's transcendet, Φ, and the Hurwitz zeta, ζ(k, b), at the positive integers is introduced. It is derived simply by inverting the relation presented in the precursor paper with one of two approaches (its generating function or the binomial theorem). This enables one to go from Lerch as a function of Hurwitz zetas (of d...
Preprint
We extend the Faulhaber formula to the whole complex plane, obtaining an expression that fully resembles the Euler-Maclaurin summation formula, only it's exact. Thereafter, an expression for the generalized harmonic progressions valid in the whole complex plane is also derived. Lastly, we extend a formula for the Hurwitz zeta function valid at the...
Article
Full-text available
We demonstrate how to obtain the analytic continuation of some formulae that are only valid at the positive integers. This results in alternative expressions for the Lerch Φ and the polylogarithm functions, Φ(e m , k, b) and Li k (e m), respectively, and for their partial sums, valid in the complex half-plane. Similarly, from a formula for the Hurw...
Article
Full-text available
We extend the Faulhaber formula to the whole complex plane, obtaining an expression that fully resembles the Euler-Maclaurin summation formula, only it's exact. Thereafter, an expression for the generalized harmonic progressions valid in the whole complex plane is also derived. Lastly, we extend a formula for the Hurwitz zeta function valid at the...
Preprint
Full-text available
An elementary proof of the Riemann hypothesis was provided in a prior paper. This new paper aims to provide some novelties, unrelated to that solution. Using the analytic continuation we created for the polylogarithm, Li k (e m), we extend the zeta function from (k) > 1 to the half-complex plane, (k) > 0, by means of the Dirichlet eta function. Mor...
Preprint
This is not the latest version, it was automatically generated. Go to my page for the latest version. PLEASE DON'T RECOMMEND THIS!
Article
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This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $\pi(x)$, and the prime-power counting function, $J(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series...
Article
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We address the problem of finding out the values of the Hurwitz zeta function at the positive integers k, ζ(k, b), by working out their real and imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known from the literature, but they are very general and usually hold for (k) > 1. The advanta...
Article
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This paper presents formulae for the sum of the terms of a harmonic progression of order k with integer parameters, HP k (n), and for the partial sums of its two associated Fourier series, C m k (a, b, n) and S m k (a, b, n). These new formulae are a generalization of the formulae created in a previous paper and were achieved using a slightly modif...
Article
Full-text available
This paper presents new formulae for the harmonic numbers of order k, H k (n), and for the partial sums of two Fourier series associated with them, denoted here by C m k (n) and S m k (n). I believe this new formula for H k (n) is an improvement over the digamma function, ψ, because it's simpler and it stems from Faulhaber's formula, which provides...

Questions

Questions (29)

Projects

Projects (2)
Archived project
Welcome to all the distinguished professors, doctors and researchers in this project, which aims to exchange experiences, research and achievements on RG, and Every researcher must helps the others by reading their work and give them some advises.
Project
Sharing of achievements and accomplishments by all members. Support everyone by recommending, commenting, and following their research and achievements. We should promote each other's work as researchers and keep motivating everyone for further research work. (If you want to add as collaborator in this project request any collaborator by adding comments on their updates, they'll surely add you)