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Some series representing the zeta function for $\Re s>1$

Jean-François Burnol

TL;DR

This work develops geometrically convergent series for the Riemann zeta function $\zeta(s)$ in $\Re s>1$ by using a digit-restriction framework on words in a base $b$ (specialized to $b=2$). It builds a moment/recurrent structure with a moment generating function $E_{A,s}(t)$ and moments $u_{A,s}(m)$ to produce restricted Dirichlet series $K_{b,A,s}$, which admit fast geometric tails for $\Re s>\log_b N$. Specializing to the unrestricted digits yields explicit Bernoulli-number expressions for the coefficients, together with a practical, recurrence-based computation of $\zeta(s)$ via a finite-sum base-$b$ block plus an alternating tail. The approach offers an easy-to-implement, numerically efficient alternative for obtaining dozens to hundreds of digits in moderate precision, while highlighting trade-offs for very high precision due to linear-term recurrence costs and gamma-growth with large $|\Im s|$. Overall, the paper extends digit-restriction techniques to explicit, geometric-convergence zeta representations with clear numerical implications and potential applications in high-precision computations at moderate $s$.

Abstract

We present series converging to the Riemann zeta function in its half-plane of convergence, and possessing remainders whose sizes decrease geometrically. They are easy to implement numerically, using only polynomial and power functions, and are efficient for obtaining dozens or hundreds of digits (when the imaginary part is not too large). They may prove less suited to very high precision (tens of thousands of digits), due to a linear cost for each new term. One can express the coefficients as linear combinations of Bernoulli numbers, but this is not advantageous numerically. The method is a development of tools introduced by the author for the evaluation of harmonic series with restricted digits in a given radix.

Some series representing the zeta function for $\Re s>1$

TL;DR

This work develops geometrically convergent series for the Riemann zeta function in by using a digit-restriction framework on words in a base (specialized to ). It builds a moment/recurrent structure with a moment generating function and moments to produce restricted Dirichlet series , which admit fast geometric tails for . Specializing to the unrestricted digits yields explicit Bernoulli-number expressions for the coefficients, together with a practical, recurrence-based computation of via a finite-sum base- block plus an alternating tail. The approach offers an easy-to-implement, numerically efficient alternative for obtaining dozens to hundreds of digits in moderate precision, while highlighting trade-offs for very high precision due to linear-term recurrence costs and gamma-growth with large . Overall, the paper extends digit-restriction techniques to explicit, geometric-convergence zeta representations with clear numerical implications and potential applications in high-precision computations at moderate .

Abstract

We present series converging to the Riemann zeta function in its half-plane of convergence, and possessing remainders whose sizes decrease geometrically. They are easy to implement numerically, using only polynomial and power functions, and are efficient for obtaining dozens or hundreds of digits (when the imaginary part is not too large). They may prove less suited to very high precision (tens of thousands of digits), due to a linear cost for each new term. One can express the coefficients as linear combinations of Bernoulli numbers, but this is not advantageous numerically. The method is a development of tools introduced by the author for the evaluation of harmonic series with restricted digits in a given radix.
Paper Structure (4 sections, 11 theorems, 64 equations)

This paper contains 4 sections, 11 theorems, 64 equations.

Table of Contents

  1. Main result
  2. Measures on words, and moments
  3. Moment generating function
  4. The case of the zeta function

Key Result

Lemma 1

For any non-negative bounded function $\phi$ on $\mathcal{W}$, such that there is at least one admissible non-empty word $w$ with $\phi(w)>0$, the function $g(\sigma)= \int_{\mathcal{W}} \phi(w)d\mu_{A,\sigma}(w)$ is positive, strictly decreasing, and log-convex on $\Ioo{\log_b(N)}{\infty}$.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • ...and 17 more