arXiv subjects

Jose Risomar Sousa

Publications and source records attributed to Jose Risomar Sousa.

11 recordsLinked to original sources

arXiv2026

Generalized Harmonic Progression Part II

In a previous paper, we saw how to create formulae for the sum of the terms of a harmonic progression of order $k$, $\mathrm{HP}_k(n)$, with integer parameters, $a$ and $b$. In this new paper we make those formulae even more general by lifting the restriction that the parameters be integers. The new formula holds always, except when $i\,b/a \in \mathbb{Z}$. Here a slightly modified version of the reasoning used before is introduced, in a standalone exposition that does not require prior knowledge of the precursor paper. A notable advantage of this new approach is that it can be used to find summation formulae for the reciprocal of polynomials, that is, $\sum_{j}1/p(j)$.

math.NT↗

arXiv2026

Simplest Integrals for the Zeta Function and its Generalizations Valid in All $\mathbb{C}$

A formula for the Riemann zeta function is obtained as an extension of the Faulhaber formula and, from there, formulae for its generalizations (the Hurwitz zeta, $ζ(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^z)$, and the Lerch transcendent, $Φ(e^z,-k,b)$), are derived that coincide with their Abel-Plana expressions. It allows one to find, for example, the Taylor series expansion of $H_{-k}(n)$ about $n=0$ (when $k$ is a positive integer, a finite Taylor series is obtained, which is nothing but the Faulhaber formula). The used method requires evaluating the limit of $Φ\left(e^{2πi \,x},-2k+1,n+1\right)+πi \,x\,Φ\left(e^{2πi \,x},-2k,n+1\right)/k$ when $x$ goes to zero, which is an interesting problem in itself.

math.NT↗

arXiv2026

Lerch's $Φ$ and the Polylogarithm at the Positive Integers

We review the closed forms of the partial Fourier sums associated with $\HP_k(n)$ from a previous paper and create an asymptotic expression for $\HP(n)$ as a way to obtain formulae for the full Fourier series (if $|b|<1$, one obtains a surprising pattern, $\HP(n) \sim H(n)-\sum_{k\ge 2}(-1)^kζ(k)b^{k-1}$). Finally, the derived Fourier series formulae are used to obtain a formula for the Lerch transcendent function, $Φ(e^z,k,b)$, and by extension the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, at the positive integers $k$.

math.NT↗

arXiv2026

The Hurwitz Zeta Function at the Positive Integers

A formula for the Hurwitz zeta function at the positive integers $k$, $ζ(k,b)$, is created by solving the real and the 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 $\Re{(k)}>1$. The advantage of formulae that only hold at the positive integers is the fact that they are simpler and easier to work with. An analytic continuation of the generating function of $ζ(k,b)$ is also obtained as $\sum_{k\ge 2}x^k(ζ(k,b)-1/b^k)$, where the term $1/b^k$ was subtracted for convenience.

math.NT↗

arXiv2026

Generalized Harmonic Progression

This paper presents formulae for the sum of the terms of a harmonic progression of order $k$ with integer parameters, $\mathrm{HP}_k(n)$, and for the partial sums of its two associated Fourier series, $C^z_{k}(a,b,n)$ and $S^z_{k}(a,b,n)$. $\mathrm{HP}_k(n)$ is built from the ground up, with a power series for $1/(aj+b)^k$ that is summed over $j$ using Faulhaber's formula. These new formulae are a generalization of the formulae created in a previous paper and were achieved using a slightly modified version of the reasoning employed before.

math.NT↗

arXiv2026

Generalized Harmonic Numbers

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 a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $ζ(2k+1)$ and a new generating function for $ζ(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

math.NT↗

arXiv2025

A Reformulation of the Riemann Hypothesis

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to the right half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $φ(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.

math.NT↗

arXiv2024

Lerch's $Φ$ and the Polylogarithm at the Negative Integers

At the negative integers, there is a simple relation between the Lerch $Φ$ function and the polylogarithm. Starting from that relation and a formula for the polylogarithm at the negative integers known from the literature, we can deduce a simple closed formula for the Lerch $Φ$ function at the negative integers, where the Stirling numbers of the second kind are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch $Φ$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we extend formulae for the Hurwitz zeta function only valid at the positive integers to the complex half-plane using this novelty.

math.NT↗

arXiv2024

The Lerch $Φ$ Analytic Continuation

We demonstrate how to extend formulae for the Lerch transcendent function, $Φ(e^z,k,b)$, and the polylogarithm, $\mathrm{Li}_{k}(e^{z})$, that only hold at the positive integers to the right half of the complex $k$-plane, that is, $\Re{(k)}>0$. The same is done for the partial sums of each of these functions.

math.NT↗

arXiv2022

The Faulhaber Formula Analytic Continuation

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 negative integers, $ζ(-k,b)$, to the whole complex plane, following a similar procedure.

math.CV↗

arXiv2021

An Exact Formula for the Prime Counting Function

This paper discusses a few main topics in Number Theory, such as the Möbius function and its generalization, leading up to the derivation of neat power series for the prime counting function, $π(x)$, and the prime-power counting function, $J(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which implies the Riemann hypothesis, and enabled the creation of a formula for $π(x)$ in the first place), and the realization that sums of divisors and the Möbius function are particular cases of a more general concept. From this result, one concludes that it's not necessary to resort to the zeros of the analytic continuation of the zeta function to obtain $π(x)$.

math.GM↗