Generalized coefficients of the Dirichlet series

TL;DR

This paper introduces a method to convert divergent Dirichlet series into convergent ones using sigmoid-based coefficient transformation, enabling more accurate computation of the Riemann Zeta function and potentially other functions defined by Dirichlet series.

Contribution

It proposes a novel sigmoid-based approach to transform Dirichlet series coefficients, improving convergence and computational accuracy for the Riemann Zeta function.

Findings

Coefficients of finite Dirichlet series depend sigmoidally on their index.

The method achieves high-precision calculations of the Riemann Zeta function.

The approach may extend to other functions with Dirichlet series representations.

Abstract

The paper considers a method for converting a divergent Dirichlet series into a convergent Dirichlet series by directly converting the coefficients of the original series $1\rightarrow\delta_{n}(s)$ for the Riemann Zeta function. In the first part of the paper, we study the properties of the coefficients ${\delta}^*_n$ of a finite Dirichlet series for approximating the Riemann Zeta function on the interval $\Delta{H}$. In general, the coefficients ${\delta}^*_n$ of a finite Dirichlet series are complex numbers. The dependence of the coefficients ${\delta}^*_n$ of a finite Dirichlet series on the ordinal number of the coefficient $n$ is established, which can be set by a sigmoid, and for each $N$ there is a single sigmoid $\hat{\delta}_n$ and a single interval $\Delta{H}$ for which the condition is satisfied $$\Big|\sum\limits_{n}^{N}\{{\delta}^*_n- \hat\delta_n\}\Big| < \epsilon;$$ The second part of the paper presents the results of using the sigmoid to calculate the values of the generalized coefficients $\delta_{n}(s)$ of the Dirichlet series for the Riemann Zeta function. For the accuracy of the calculation, $\log_{10}(1/\epsilon)$ values of the Riemann Zeta Function by summing the resulting convergent Dirichlet series, the power of the imaginary part $t=Im(s)$ is established. Presumably, the sigmoid can be used to asymptotically calculate the values of the analytical continuation of any function defined by the Dirichlet series. Presumably, for any divergent series to which the generalized summation method is applicable, it is possible to find a direct transformation of the coefficients of the divergent series, so that the resulting series with the transformed coefficients will converge to the same function as the series of transformed partial sums.