The Dirichlet eta function is the function 
defined by
where 
is the Riemann zeta function. Note that Borwein
and Borwein (1987, p. 289) use the notation 
instead of 
. The function is also known as the alternating zeta function
and denoted 
(Sondow 2003, 2005).
is defined by setting 
in the right-hand side of (2), while
(sometimes called the alternating harmonic
series) is defined using the left-hand side. The function vanishes at each zero
of 
except 
(Sondow 2003).
The eta function is related to the Riemann zeta function and Dirichlet lambda function by
 |
(3) |
and
 |
(4) |
(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithm function,
 |
(5) |
The value 
may be computed by noting that the Maclaurin series
for 
for 
is
 |
(6) |
Therefore, the natural logarithm of 2 is
Values for even integers are related to the analytical values of the Riemann zeta function. Particular values are given in Abramowitz and Stegun (1972, p. 811), and include
It appears in the integral
![int_0^1int_0^1([-ln(xy)]^s)/(1+xy)dxdy=Gamma(s+2)eta(s+2)](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FDirichletEtaFunction%2FNumberedEquation5.svg){kind=small} |
(17) |
(Guillera and Sondow 2005).
The derivative of the eta function is given by
 |
(18) |
Special cases are given by
(OEIS A271533, OEIS A256358, OEIS A265162, and OEIS A091812),
where 
is the Glaisher-Kinkelin constant,
is the Riemann zeta function, and 
is the Euler-Mascheroni
constant. The identity for 
provides a remarkable proof of the Wallis
formula.
See also
Dedekind Eta Function, Dirichlet Beta Function, Dirichlet Lambda Function, Hadjicostas's Formula, Riemann Zeta Function, Zeta Function
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References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 807-808, 1972.Borwein, J. M. and Borwein,
P. B. Pi
& the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, 1987.Guillera, J. and Sondow, J. "Double Integrals
and Infinite Products for Some Classical Constants Via Analytic Continuations of
Lerch's Transcendent." 16 June 2005. http://arxiv.org/abs/math.NT/0506319.Havil,
J. "Real Alternatives." ยง16.12 in Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 206-207,
2003.Sloane, N. J. A. Sequences A271533,
A256358, A265162,
and A091812 in "The On-Line Encyclopedia
of Integer Sequences."Sondow, J. "Zeros of the Alternating
Zeta Function on the Line ![R[s]=1](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FDirichletEtaFunction%2FInline78.svg){kind=small}
." Amer. Math. Monthly 110, 435-437,
2003.Sondow, J. "Double Integrals for Euler's Constant and 
and an Analog of Hadjicostas's
Formula." Amer. Math. Monthly 112, 61-65, 2005.Spanier,
J. and Oldham, K. B. "The Zeta Numbers and Related Functions." Ch. 3
in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 25-33, 1987.
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Cite this as:
Weisstein, Eric W. "Dirichlet Eta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirichletEtaFunction.html