Dirichlet Eta Function¶

Extend the alternating reciprocal-power series into an entire complex function tied to zeta by η(s)=(1−2^(1−s))ζ(s), gaining convergence on Re(s)>0 and a characteristic extra zero lattice.

Version: v4 · 2026-09-19 · History
Domain-specific #: 1685
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Analytic Number Theory → Mathematics
Aliases: Alternating zeta function, Dirichlet's eta function

Core Idea¶

The Dirichlet Eta Function is initially defined by the alternating Dirichlet series

\[ \eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s} =1-2^{-s}+3^{-s}-4^{-s}+\cdots, \]

which converges for complex (s) with \(\Re(s)>0\). On the half-plane \(\Re(s)>1\), separating odd and even terms of the absolutely convergent zeta series gives

\[ \eta(s)=(1-2^{1-s})\zeta(s). \]

This relation continues \(\eta\) to an entire function: the zero of (1-2^{1-s}) at (s=1) cancels the simple pole of \(\zeta\). NIST DLMF records the alternating representation of zeta on \(\Re(s)>0\).

How would you explain it like I'm…

The Back-and-Forth Sum

Take 1, then take away a half, add a third, take away a quarter, and keep going — adding and subtracting smaller and smaller pieces. The Dirichlet eta function is this kind of back-and-forth sum, with a knob that controls how fast the pieces shrink.

Plus-Minus Fraction Sum

The Dirichlet eta function is built by adding and subtracting fractions in turn: 1, minus 1 over 2 to a power, plus 1 over 3 to that power, minus 1 over 4 to that power, and so on. The power, called s, is the input to the function. Because the signs switch back and forth, the sum settles to a definite answer for more inputs than the plain all-plus version, which is the famous zeta function. There is a simple formula connecting the two: eta equals zeta times a correction factor. Using that link, mathematicians can make sense of the eta function for every input.

Alternating Zeta Series

The Dirichlet eta function is defined by the alternating series η(s) = 1 − 1/2ˢ + 1/3ˢ − 1/4ˢ + ⋯, where s can be a complex number. It converges whenever the real part of s is greater than 0, which is farther than the ordinary zeta series ζ(s) = 1 + 1/2ˢ + 1/3ˢ + ⋯, which needs real part greater than 1. Where both converge nicely, you can split the terms into odd and even ones and show that η(s) = (1 − 2^{1−s}) ζ(s). This formula extends eta to every complex number as a smooth (entire) function: at s = 1 the zeta function blows up, but the factor 1 − 2^{1−s} equals zero there and cancels it. So eta inherits zeta's zeros and also gets extra zeros from that factor.

The Dirichlet eta function is η(s) = Σ_{n≥1} (−1)^{n−1} n⁻ˢ = 1 − 2⁻ˢ + 3⁻ˢ − ⋯, which converges for Re(s) > 0, one half-plane farther left than the zeta series. On Re(s) > 1, where the zeta series converges absolutely, separating odd and even terms gives η(s) = (1 − 2^{1−s}) ζ(s). This identity continues η to an entire function, because the zero of 1 − 2^{1−s} at s = 1 cancels the simple pole of ζ there. Conversely, it gives an alternating representation of ζ on Re(s) > 0 away from the zeros of the factor. The zeros of η are those inherited from ζ together with those generated by the factor 1 − 2^{1−s}, which vanishes at s = 1 + 2πik/log 2 for nonzero integers k (its zero at s = 1 is used up cancelling the pole of ζ).

Scope of Application¶

Eta provides a direct convergent representation connected to zeta throughout \(\Re(s)>0\), including much of the critical strip where the ordinary zeta Dirichlet series diverges. It supplies special values such as \(\eta(1)=\log 2\) and \(\eta(0)=1/2\) by continuation, and connects zeta zeros to a function without a pole.

Euler transformation and related acceleration methods make its alternating series useful for computation. Mellin-transform representations connect it to gamma factors and Fermi–Dirac-style kernels.

Clarity¶

The defining series converges only for \(\Re(s)>0\); the entire function outside that half-plane is its unique analytic continuation, not convergence of the original series. At (s=1), writing \(0\cdot\infty\) informally hides a removable limit: \(\eta(1)=\log2\).

Zeros from (1-2^{1-s}=0) lie on \(\Re(s)=1\) at periodic imaginary spacing, but the point (s=1) is not a zero because of pole cancellation.

Manages Complexity¶

Alternation shifts the direct convergence boundary from \(\Re(s)>1\) to \(\Re(s)>0\), replacing a pole-bearing series with an entire related function. The factorization separates zeta structure from a simple periodic factor and supports stable series acceleration.

That compression has a cost: division by (1-2^{1-s}) becomes ill-conditioned near its zeros, so eta is not uniformly a numerically superior route to zeta.

Abstract Reasoning¶

Fix a domain and choose series, integral, functional relation, or continuation representation valid there.
For \(\Re(s)>1\), derive the factor by separating even and odd zeta terms.
Extend identities by analytic continuation only after proving domain overlap.
Treat (s=1) as a removable cancellation and evaluate the limit.
Classify zeros by source and check overlaps/multiplicity.
Accelerate alternating sums with explicit error control.
Avoid dividing by a nearly zero prefactor in numerical zeta evaluation.

Knowledge Transfer¶

The alternating-series/factor-cancellation structure transfers across analytic number theory, special-function computation, Mellin transforms, and statistical-physics integrals. The strict parent is Function (Mapping): eta is a single-valued complex analytic mapping with unusually rich extra structure.

Relationships to Other Abstractions¶

Current abstraction Dirichlet Eta Function Domain-specific

Parents (1) — outgoing typed relations

Dirichlet Eta Function is a kind of Function (Mapping) Prime

Function (Mapping) is the broader abstraction this entry instantiates.

Hierarchy path (1) — routes to 1 parentless root

Dirichlet Eta Function → is a kind of Function (Mapping)

Neighborhood in Abstraction Space¶

Dirichlet Eta Function sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Special Functions & Series Convergence (7 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25