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A Generalization of the Riemann Functional Equation with Three Variables
José Risomar Sousa
The discovery that the generalized Riemann functional equation from the literature can be extended to real , by exploiting symmetries of a new formula for the Hurwitz zeta…
Simplest Integrals for the Zeta Function and its Generalizations Valid in All
Jose Risomar Sousa
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, , the pol…
Lerch's and the Polylogarithm at the Negative Integers
Jose Risomar Sousa
At the negative integers, there is a simple relation between the Lerch function and the polylogarithm. Starting from that relation and a formula for the polylogarithm at the ne…
The Lerch Analytic Continuation
Jose Risomar Sousa
We demonstrate how to extend formulae for the Lerch transcendent function, , and the polylogarithm, , that only hold at the positive integers to…
A Reformulation of the Riemann Hypothesis
Jose Risomar Sousa
We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, , the zeta function's D…
Lerch's and the Polylogarithm at the Positive Integers
Jose Risomar Sousa
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…
The Hurwitz Zeta Function at the Positive Integers
Jose Risomar Sousa
A formula for the Hurwitz zeta function at the positive integers , , is created by solving the real and the imaginary parts separately and then combining them. A few dif…
Generalized Harmonic Progression Part II
Jose Risomar Sousa
In a previous paper, we saw how to create formulae for the sum of the terms of a harmonic progression of order , , with integer parameters, and . In thi…
Generalized Harmonic Progression
Jose Risomar Sousa
This paper presents formulae for the sum of the terms of a harmonic progression of order with integer parameters, , and for the partial sums of its two associ…
Generalized Harmonic Numbers
Jose Risomar Sousa
This paper presents new formulae for the harmonic numbers of order , , and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)…