Alternating sum of the Báez-Duarte coefficients

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Number

$S$

0.7825279853253842345766884742837840768058009796525991599926472087826073767199003522927165598264149835

Definition

The constant $S=\lim_{K\to\infty}\sum_{k=0}^{K}(-1)^k c_k$ is defined by ordinary partial sums, where $c_k=\sum_{j=0}^{k}(-1)^j\binom{k}{j}/\zeta(2j+2)$ [1], equations (4) and (49).

Formulas

(1)

$\displaystyle S=\sum_{m=1}^{\infty}\frac{1}{2^m\zeta(2m)}$ [1], equation (49).

(2)

$\displaystyle S=\sum_{n=1}^{\infty}\frac{\mu(n)}{2n^2-1}$, where $\mu$ is the usual Möbius function with $\mu(1)=1$. This follows from the reciprocal-zeta Dirichlet series and geometric summation [1], equations (12) and (47).

Comments

(3)

The Báez-Duarte coefficients $c_k$ use the ordinary Riemann zeta function $\zeta$. The alternating sum starts at $k=0$ and retains the signs of $c_k$ before multiplication by $(-1)^k$.

(4)

The alternating sum converges in the ordinary partial-sum sense, without assuming the Riemann hypothesis. The arithmetic series in (2) is absolutely convergent. The attached numerical evidence gives a dominated-convergence proof of their equality.

(5)

The decimal expansion of $S$ is recorded in [2].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def alternating_sum(digits=100):
    F = RealBallField(numberdb.bits(digits, losing=80))
    N = 4*digits
    partial = sum((F(QQ(1)/2**m)/F(2*m).zeta()
                   for m in range(1,N+1)), F(0))
    return partial + F(0, QQ(1)/2**N)

alternating_sum()

References

[1]

J. Cisło and M. Wolf, On the Riesz and Báez-Duarte criteria for the Riemann Hypothesis (2008), §§2 and 4, equations (4), (12), (14), (47), and (49). (arXiv)

Links

[2]

OEIS A345412: decimal expansion of the alternating sum

[3]

DLMF 25.6: integer arguments of the Riemann zeta function

Similar tables

Báez-Duarte coefficients — The coefficients $c_k$ are the terms whose alternating sum defines $S$.

Values of the Riesz function — The Riesz function is the continuous transform paired with the coefficients $c_k$ in the comparison of the Riesz and Báez-Duarte criteria [1], §2.

Values of the Riemann zeta function at rational numbers — The reciprocals of the positive even zeta values are summed with the weights $2^{-m}$ in (1).

Bernoulli numbers — Euler’s even-zeta identity expresses the reciprocal even-zeta values used by the independent calculation in terms of exact Bernoulli numbers and powers of $\pi$.

Data properties

Entries are of type: real number

Table is complete: yes (it holds the single constant defined by the ordinary alternating sum; there are no free arguments or sampled generating-function inputs)

How well the digits are known: proven

How they were obtained:

The positive reciprocal-zeta series is evaluated with Arb real balls at 413 working bits through $m=400$. Since $0<1/\zeta(2m)<1$, the omitted tail is smaller than $2^{-400}$; this bound is included in the result. The enclosure retains 398 relative accuracy bits and lies strictly inside the interval represented by the written 100 significant digits.

An independent 700-bit calculation uses the Möbius series, sums its first 16 terms, and expands the remaining geometric factors through order 60 with an explicit remainder bound. It recomputes reciprocal even-zeta values from exact Bernoulli numbers and a Machin-series enclosure of $\pi$ [3], equation 25.6.2. Its enclosure retains 550 relative accuracy bits and lies strictly inside the same written interval. The independent calculation shares Arb arithmetic infrastructure but uses neither the main series implementation nor its zeta evaluator.

A 130-digit repeat, the 87 digits displayed in [2], a direct 10000-term Möbius sum with its $1/10000$ tail bound, and finite alternating-sum identities at five truncations provide further checks. Known-answer controls and deliberate-error controls precede the comparisons. After database readback, the stored interval contains both certified enclosures and passes generator verification; all 257 stored intervals in Báez-Duarte coefficients agree with freshly computed coefficients. The attached checks and numerical evidence reproduce these results.