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Dirichlet Eta Function

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The Dirichlet eta function is the function eta(s) defined by

eta(s)=sum_(k=1)^(infty)((-1)^(k-1))/(k^s)
(1)
=(1-2^(1-s))zeta(s),
(2)

where zeta(s) is the Riemann zeta function. Note that Borwein and Borwein (1987, p. 289) use the notation alpha(s) instead of eta(s). The function is also known as the alternating zeta function and denoted zeta^*(s) (Sondow 2003, 2005).

eta(0)=1/2 is defined by setting s=0 in the right-hand side of (2), while eta(1)=ln2 (sometimes called the alternating harmonic series) is defined using the left-hand side. The function vanishes at each zero of 1-2^(1-s) except s=1 (Sondow 2003).

The eta function is related to the Riemann zeta function and Dirichlet lambda function by

(zeta(nu))/(2^nu)=(lambda(nu))/(2^nu-1)=(eta(nu))/(2^nu-2)
(3)

and

zeta(nu)+eta(nu)=2lambda(nu)
(4)

(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithm function,

eta(x)=-Li_x(-1).
(5)

The value eta(1) may be computed by noting that the Maclaurin series for ln(1+x) for -1<x<=1 is

ln(1+x)=x-1/2x^2+1/3x^3-1/4x^4+....
(6)

Therefore, the natural logarithm of 2 is

ln2=ln(1+1)
(7)
=1-1/2+1/3-1/4+...
(8)
=sum_(n=1)^(infty)((-1)^(n-1))/n
(9)
=eta(1).
(10)

The derivative of the eta function is given by

eta^'(x)=2^(1-x)(ln2)zeta(x)+(1-2^(1-x))zeta^'(x),
(11)

or in the special case x=0, by

lim_(x->0)[d/(dx)eta(x)]=-ln2-zeta^'(0)
(12)
=-ln2+1/2ln(2pi)
(13)
=-ln(sqrt(2/pi))
(14)
=1/2ln(1/2pi).
(15)

This latter fact provides a remarkable proof of the Wallis formula.

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

eta(0)=1/2
(16)
eta(1)=ln2
(17)
eta(2)=1/(12)pi^2
(18)
eta(3)=3/4zeta(3)
(19)
eta(4)=7/(720)pi^4
(20)
eta(5)=(15)/(16)zeta(5).
(21)

It appears in the integral

int_0^1int_0^1([-ln(xy)]^s)/(1+xy)dxdy=Gamma(s+2)eta(s+2)
(22)

(Guillera and Sondow 2005).

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." §16.12 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 206-207, 2003.Sondow, J. "Zeros of the Alternating Zeta Function on the Line R[s]=1." Amer. Math. Monthly 110, 435-437, 2003.Sondow, J. "Double Integrals for Euler's Constant and ln(4/pi) 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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Dirichlet Eta Function

Cite this as:

Weisstein, Eric W. "Dirichlet Eta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DirichletEtaFunction.html

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