The Dirichlet eta function is the function defined by
(1)
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(2)
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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)
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and
(4)
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(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithm function,
(5)
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The value may be computed by noting that the Maclaurin series for for is
(6)
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Therefore, the natural logarithm of 2 is
(7)
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(8)
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(9)
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(10)
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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)
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(12)
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(13)
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(14)
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(15)
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(16)
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It appears in the integral
(17)
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(Guillera and Sondow 2005).
The derivative of the eta function is given by
(18)
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Special cases are given by
(19)
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(20)
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(21)
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(22)
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(23)
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(24)
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(25)
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(26)
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(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.