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Dirichlet Eta Function

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The Dirichlet eta function is the entire function eta(s) whose series representation for Re[s]>0 is

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. The series converges absolutely for Re[s]>1, and the right-hand side gives its analytic continuation, with a removable singularity at s=1. 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), with limiting values used at removable singularities of the quotients. 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)

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
(11)
eta(1)=ln2
(12)
eta(2)=1/(12)pi^2
(13)
eta(3)=3/4zeta(3)
(14)
eta(4)=7/(720)pi^4
(15)
eta(5)=(15)/(16)zeta(5).
(16)

It appears in the integral

int_0^1int_0^1([-ln(xy)]^s)/(1+xy)dxdy=Gamma(s+2)eta(s+2)
(17)

for Re[s]>-2 (Guillera and Sondow 2008).

DirichletEtaPrime

The derivative of the eta function is given by

eta^'(x)=2^(1-x)(ln2)zeta(x)+(1-2^(1-x))zeta^'(x).
(18)

Special cases are given by

eta^'(-1)=3lnA-1/4-(ln2)/3
(19)
=0.2652143709...
(20)
eta^'(0)=1/2ln(pi/2)
(21)
=0.2257913526...
(22)
eta^'(1/2)=zeta(1/2)[1/2(3-sqrt(2))ln2-1/4(sqrt(2)-1)(2gamma+pi+2lnpi)]
(23)
=0.1932888316...
(24)
eta^'(1)=gammaln2-((ln2)^2)/2
(25)
=0.1598689037...
(26)

(OEIS A271533, OEIS A256358, OEIS A265162, and OEIS A091812), where A is the Glaisher-Kinkelin constant, zeta(z) is the Riemann zeta function, and gamma is the Euler-Mascheroni constant. The identity for eta^'(0) 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." Ramanujan J. 16, 247-270, 2008. https://doi.org/10.1007/s11139-007-9102-0.Havil, J. "Real Alternatives." §16.12 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 206-207, 2003.Sloane, N. J. A. Sequences A091812, A256358, A265162, and A271533 in "The On-Line Encyclopedia of Integer Sequences."Sondow, J. "Zeros of the Alternating Zeta Function on the Line Re[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 Resource. https://mathworld.wolfram.com/DirichletEtaFunction.html

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