Dirichlet Eta Function Calculator and Zeta Relation

Dirichlet eta function eta(s) calculator: alternating series, Riemann zeta relation, special values via Bernoulli numbers and critical strip analysis tool.

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Dirichlet eta function

Dirichlet eta function, introduced by Peter Gustav Lejeune Dirichlet in 19th century, is a fundamental special function in analytic number theory. Defined for complex numbers with Re(s) > 0 via alternating series:

eta(s) = Sigma_{k=1}^{infinity} (-1)^(k+1) / k^s = 1 - 1/2^s + 1/3^s - 1/4^s + ...

Also called 'alternating zeta function'. Closely linked to Riemann zeta but with important analytic differences.

Relation to Riemann zeta

Fundamental relation between eta and zeta:

eta(s) = (1 - 2^(1-s)) * zeta(s)

Proof: zeta(s) - 2 * 2^(-s) * zeta(s) = zeta(s) * (1 - 2^(1-s)) = eta(s). Subtracted factor 2^(-s) eliminates all even terms 1/2^s + 1/4^s + ... from zeta series, leaving only odd terms with alternating signs.

Important consequence: eta is well-defined for every complex s, while zeta has simple pole at s=1 with residue 1. Eta at s=1 equals lim_{s->1} (1-2^(1-s)) * zeta(s) = ln(2) via Laurent expansion of zeta.

Special values

  • eta(0) = 1/2 (analytic continuation)
  • eta(1) = ln(2) = 0.6931471805... (famous alternating harmonic series)
  • eta(2) = pi^2 / 12 = 0.8224670334...
  • eta(3) = (3/4) * zeta(3) = 0.9015426773... (Apery's constant)
  • eta(4) = 7 * pi^4 / 720 = 0.9470328295...
  • eta(6) = 31 * pi^6 / 30240 = 0.9855510912...
  • eta(-1) = 1/4 (analytic continuation)
  • eta(-2) = 0 (trivial zero)

General formula for eta(2n) with n >= 1:

eta(2n) = (2^(2n-1) - 1) / 2^(2n-1) * |B_{2n}| * pi^(2n) / (2n)!

Series convergence

  • For Re(s) > 1: absolute convergence.
  • For 0 < Re(s) <= 1: conditional convergence (Leibniz). Example: Re(s) = 1 gives eta(1) = ln(2).
  • For Re(s) <= 0: series diverges classically. Eta defined via analytic continuation or convergence acceleration (Euler transform).

Unlike zeta requiring non-trivial analytic continuation, eta has 'natural' convergence on more than half-plane.

Five calculator modes

  1. Real eta(s): computation via Euler-Borwein acceleration, 12+ digit precision.
  2. Zeta relation: eta(s) zeta(s) via (1 - 2^(1-s)).
  3. Partial alternating sum: convergence and error vs term count.
  4. Critical line analysis: behavior at s=1/2 and critical strip 0 < Re(s) < 1.
  5. Special values: notable values table in Bernoulli and pi.

Riemann hypothesis

Riemann hypothesis states all non-trivial zeros of zeta lie on line Re(s) = 1/2. Since eta(s) = (1 - 2^(1-s)) * zeta(s), zeros of eta are: (a) zeros of zeta (RH), (b) zeros of (1 - 2^(1-s)), i.e., s = 1 + 2k pi i / ln 2 for integer k. Eta zero analysis equivalent to RH.

Euler-Borwein acceleration

For high-precision computation using few terms:

  • Euler transform on alternating series: few terms (~50) give 15+ digits precision.
  • Borwein algorithm (1991): exact closed form with Chebyshev polynomials.
  • For Re(s) < 0: use eta(s) = (1 - 2^(1-s)) * zeta(s) with zeta via Riemann-Siegel.
  • Calculator uses modified van Wijngaarden (Borwein), efficient in practice.

Applications

  • Analytic number theory: zeta alternative for critical strip analysis without pole.
  • Statistical physics: Bose-Einstein and Fermi-Dirac distributions with alternating sums.
  • Quantum mechanics: regularized energies via eta(-1) = 1/12 in string theory.
  • Probability theory: alternating distribution generators.
  • Computation: algorithm for zeta without explicit analytic continuation.
  • Math history: Dirichlet, Euler, Riemann, Borwein - 200 years of development.

History of the Dirichlet eta function

The function is named after the German mathematician Peter Gustav Lejeune Dirichlet (1805-1859), one of the founders of analytic number theory. Dirichlet introduced the functions that today bear his name while studying the distribution of prime numbers in arithmetic progressions, work culminating in the famous Dirichlet theorem of 1837. The eta function, defined by the alternating series Σ (−1)^(k+1)/k^s, is the simplest example of a Dirichlet L-function, the family of functions that generalises the Riemann zeta and is at the heart of modern number theory. It is also called the alternating zeta function, for its close link with the Riemann zeta. Its study helped to understand the deep structure of the prime numbers and it continues to be a bridge between complex analysis and arithmetic.

The eta function and the analytic continuation of zeta

One of the most important uses of the eta function is to provide the analytic continuation of the Riemann zeta function. The series defining ζ(s) converges only for Re(s) > 1, but the relation η(s) = (1 − 2^(1−s))·ζ(s) allows the zeta to be extended beyond this limit. Since the alternating series of the eta function converges for all s with Re(s) > 0, it defines ζ(s) = η(s)/(1 − 2^(1−s)) over the whole critical strip 0 < Re(s) < 1, the crucial territory of the Riemann hypothesis. Unlike the zeta, which has a simple pole at s = 1, the eta function is holomorphic over the whole complex plane, with no singularities. This property makes it a valuable numerical and theoretical tool for exploring the zeros of the zeta.

Updated July 2026

Frequently Asked Questions

What is the Dirichlet eta function?

It is the function defined by the alternating series η(s) = 1 − 1/2ˢ + 1/3ˢ − 1/4ˢ + …, that is Σ (−1)^(k+1)/kˢ. It is closely linked to the Riemann zeta function by the relation η(s) = (1 − 2^(1−s))·ζ(s). Unlike the zeta, its series already converges for Re(s) > 0 and the function is holomorphic over the whole complex plane, with no poles. It is also called the alternating zeta function.

What is the relation to the zeta function?

The two functions are linked by the formula η(s) = (1 − 2^(1−s))·ζ(s), from which one derives ζ(s) = η(s)/(1 − 2^(1−s)) for s ≠ 1. This relation is valuable because the eta series converges in a wider region than that of the zeta, allowing ζ(s) to be computed even in the critical strip. For example η(2) = π²/12, while ζ(2) = π²/6: the ratio is exactly 1/2.

What are the special values of the eta function?

Some notable values: η(0) = 1/2, η(1) = ln 2 ≈ 0.693, η(2) = π²/12 ≈ 0.822, η(3) = (3/4)·ζ(3) linked to Apéry's constant, η(4) = 7π⁴/720. For even arguments, η(2n) is always a rational multiple of π^(2n), through the Bernoulli numbers, exactly like the zeta. There are also trivial zeros at negative even integers: η(−2) = 0, η(−4) = 0, etc.

Why does the eta function converge better than the zeta?

Because it is an alternating series: the terms of opposite sign partly cancel, ensuring convergence already for Re(s) > 0, where the zeta series diverges. The zeta converges only for Re(s) > 1. This wider convergence makes the eta function an ideal numerical tool for computing values in the critical strip 0 < Re(s) < 1, where the non-trivial zeros linked to the Riemann hypothesis lie.

What is the Euler-Borwein acceleration?

It is a numerical algorithm that enormously accelerates the convergence of the eta function's alternating series. The direct series converges slowly, especially near s = 1; Borwein's algorithm, based on a transformation with Chebyshev polynomials, allows many correct digits to be obtained with a few dozen terms. It is the method used in this calculator and in scientific libraries to compute η(s) and, through the relation, ζ(s).

What is the link with the Riemann hypothesis?

The eta function provides the analytic continuation of the zeta in the critical strip 0 < Re(s) < 1, precisely the region where the Riemann hypothesis states that all the non-trivial zeros of the zeta lie on the line Re(s) = 1/2. Since the factor (1 − 2^(1−s)) does not vanish in the strip (except in special cases), the zeros of the zeta correspond to the zeros of the eta, making it a tool for studying this famous unsolved problem.

Why does the eta function have no poles?

Unlike the zeta function, which has a simple pole at s = 1 (where the harmonic series diverges), the eta function is holomorphic over the whole complex plane. At s = 1 the eta simply equals ln 2, a finite value, because the alternating series 1 − 1/2 + 1/3 − 1/4 + … converges to ln 2. The factor (1 − 2^(1−s)) in the relation with the zeta vanishes at s = 1, exactly cancelling the zeta's pole.

Who introduced the eta function?

It is attributed to Peter Gustav Lejeune Dirichlet, a 19th-century German mathematician and founder of analytic number theory. Dirichlet studied the L-functions, of which the eta is the simplest case, to prove his famous 1837 theorem on primes in arithmetic progressions. The function is today a standard tool in the analysis of the zeta function and in the study of the zeros linked to the Riemann hypothesis.

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