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: v3 · 2026-09-06 · History
Domain-specific #: 1685
Origin domain: mathematics
Subdomain: analytic number theory
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 \(\Re(s)>0\).^[1]

The recognition invariant is alternating reciprocal powers + convergence one half-plane farther left than the ordinary zeta series + the factor (1-2^{1-s}) + entire continuation + inherited and factor-generated zero structure.

Structural Signature¶

Complex argument \(s=\sigma+it\).
Alternating coefficients \(+1,-1,+1,-1,\ldots\).
Dirichlet terms (n^{-s}).
Conditional-convergence domain \(\Re(s)>0\).
Zeta factorization \(\eta=(1-2^{1-s})\zeta\).
Removable cancellation at (s=1).
Entire continuation.
Inherited zeta zeros plus zeros of the prefactor, with cancellation/multiplicity handled carefully.
Integral and Mellin representations involving (1/(e^x+1)).
Alternating-series acceleration for numerical evaluation.

What It Is Not¶

It is not the Riemann zeta function: it is entire rather than meromorphic and carries the additional prefactor zeros. It is not Dirichlet beta, which sums alternating odd reciprocal powers, nor Dirichlet lambda, which sums odd reciprocal powers. It is not a general Dirichlet L-function.

It is not the Dedekind eta function from modular-form theory, and not the Dirichlet kernel from Fourier analysis. Shared names do not imply shared identity.

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.^[2]

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.^[3]

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. Statements about eta zeros must distinguish inherited zeta zeros, prefactor zeros, and possible overlap.

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.

Examples¶

At one. The alternating harmonic series gives \(\eta(1)=1-1/2+1/3-\cdots=\log2\).

At two. \(\eta(2)=(1-2^{-1})\zeta(2)=\pi^2/12\).

At zero. Analytic continuation yields \(\eta(0)=1/2\), though the defining term sequence does not converge ordinarily there.

Prefactor zero. For nonzero integer (k), \(s=1+2\pi ik/\log2\) makes (1-2^{1-s}=0), producing an eta zero when zeta is finite there.

Structural Tensions¶

Series definition versus analytic continuation: one formula does not converge on the entire domain.
Pole cancellation versus numerical instability: algebraic regularity can hide loss of precision.
Inherited zeros versus added zeros: eta and zeta do not have identical zero sets.
Conditional convergence versus rearrangement: ordering cannot be treated casually outside absolute convergence.
Faster convergence versus complex-parameter tails: alternation helps but still needs error analysis.

Structural–Framed Character¶

Series, continuation, factorization, and zeros are structural. Representation and algorithm choice are computationally framed.

Structural Core vs. Domain Accent¶

The portable core is cancellation and convergence improvement through alternation. Complex Dirichlet series, zeta, analytic continuation, and zero structure are constitutive domain accent, so the identity is domain-specific.

Function (Mapping) is the proposed immediate parent. Gamma Function appears in integral/functional relations. Periodicity describes the prefactor-zero lattice. Dirichlet Kernel is a naming neighbor only.

The prospective queue contains one strict edge to prime:function_mapping. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Dirichlet Eta Function Domain-specific

Parents (1) — more general patterns this builds on

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

Function (Mapping) is the proposed immediate parent.

Hierarchy path (1) — routes to 1 parentless root

Dirichlet Eta Function → Function (Mapping)

Neighborhood in Abstraction Space¶

Dirichlet Eta Function sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Special Functions & Convergence Tests (6 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Riemann zeta function.
Dirichlet beta or lambda function.
General Dirichlet L-function.
Dedekind eta function.
Dirichlet kernel.
The alternating series only, without its analytic continuation.

References¶

[1] NIST Digital Library of Mathematical Functions, §25.2.3, “Other Infinite Series” for the Riemann zeta function, release 1.2.7, 2026. registry ↩

[2] Tom M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, Chapter 12. DOI 10.1007/978-1-4757-5579-4. registry ↩

[3] E. C. Titchmarsh, revised by D. R. Heath-Brown, The Theory of the Riemann Zeta-Function, 2^nd ed., Oxford University Press, 1986. registry ↩

[4] Peter Borwein, “An Efficient Algorithm for the Riemann Zeta Function,” Constructive Experimental and Nonlinear Analysis, CMS Conference Proceedings 27, 2000, 29–34. registry ↩