Dirichlet Eta Function

The Dirichlet eta function is the entire function whose series representation for is

			(1)
			(2)

where is the Riemann zeta function. The series converges absolutely for , and the right-hand side gives its analytic continuation, with a removable singularity at . 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), with limiting values used at removable singularities of the quotients. 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

			(7)
			(8)
			(9)
			(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

			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

It appears in the integral

(17)

for (Guillera and Sondow 2008).

The derivative of the eta function is given by

(18)

Special cases are given by

			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

(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.