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For a real positive ,
the Riemann-Siegel
function is defined by
(1)
|
This function is sometimes also called the Hardy function or Hardy -function (Karatsuba and Voronin 1992, Borwein et al. 1999).
The top plot superposes
(thick line) on
, where
is the Riemann zeta
function.
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For real ,
the Riemann-Siegel theta function
is defined as
(2)
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(3)
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The function
has local extrema at
(OEIS A114865
and A114866).
Values
such that
(4)
|
for ,
1, ... are known as Gram points (Edwards 2001, pp. 125-126).
The series expansion of about 0 is given by
(5)
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(6)
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(7)
|
(OEIS A067626), and about by
(8)
|
(OEIS A036282 and A114721; Edwards 2001, p. 120).
These functions are implemented in the Wolfram Language as RiemannSiegelZ[z] and RiemannSiegelTheta[z].