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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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The derivative of the eta function is given by
(11)
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or in the special case , by
(12)
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(13)
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(14)
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(15)
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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
(16)
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(17)
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(18)
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(19)
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(20)
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(21)
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It appears in the integral
(22)
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(Guillera and Sondow 2005).