The Riemann-Siegel formula is an asymptotic formula discovered (but not published) by Riemann for computing the Riemann-Siegel
function 
, or equivalently the Riemann
zeta function on the critical line. The formula
was subsequently discovered in an archive of Riemann's papers by C. L. Siegel
(Edwards 2001, p. 136; Derbyshire 2004, pp. 257 and 263) and published
by Siegel (1932).
For real 
, write
![Z(t)=2sum_(k=1)^(nu(t))1/(sqrt(k))cos[theta(t)-tlnk]+R(t),](/images/equations/Riemann-SiegelFormula/NumberedEquation1.svg) |
(1)
|
where 
, 
is the floor function,
and 
and 
are the Riemann-Siegel functions. The
correction 
has the asymptotic expansion
 |  |  |
(2)
|
 |  | ![[omega^k]{exp[i(1/2tln(t/(2pi))-1/2t-1/8pi-theta(t))]×[y^0][(sum_(j=0)^(infty)A_j(y)omega^j)(sum_(j=0)^(infty)(psi^((j))(p))/(j!)y^j)]}](/images/equations/Riemann-SiegelFormula/Inline13.svg) |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  | ![(cos[2pi(p^2-p-1/(16))])/(cos(2pip))](/images/equations/Riemann-SiegelFormula/Inline22.svg) |
(6)
|
as 
(Siegel 1932, Edwards 2001). Here 
lies in 
, 
,
and the recurrence for 
applies for 
.
The coefficient notation ![[y^0]](/images/equations/Riemann-SiegelFormula/Inline29.svg)
extracts a constant Laurent
series coefficient, and ![[omega^k]](/images/equations/Riemann-SiegelFormula/Inline30.svg)
refers to the formal expansion after substituting
and expanding 
asymptotically. The quotient defining 
and its derivatives are interpreted by analytic
continuation at 
and 
,
where 
. The first few terms 
are given by
 |  |  |
(7)
|
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(8)
|
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(9)
|
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(10)
|
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(11)
|
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(12)
|
The numerators and denominators are 1, 
,
1, 1, 
, 
, 
, 1, 19, 11, 1, 
, 
, ... (OEIS A050276)
and 1, 96, 64, 18432, 64, 3840, 5308416, 128, ... (OEIS A050277),
respectively.
It is based on evaluation of the integral
 |  |  |
(13)
|
 |  | ![(cos[2pi(p^2-p-1/(16))])/(cos(2pip)),](/images/equations/Riemann-SiegelFormula/Inline67.svg) |
(14)
|
also denoted 
, where 
is an infinite straight line of slope 1, directed from
northeast infinity to southwest infinity, which crosses the imaginary axis strictly
between 0 and 
(Edwards 2001, p. 147).
Assuming the Riemann hypothesis, Riemann's prime-counting formula (Riemann 1859) gives the large-
approximation
 |
(15)
|
for real 
, where 
is the prime counting
function and 
is the logarithmic
integral. The sum is truncated over initial segments 
of the positive ordinates 
of nontrivial zeros 
of the Riemann zeta function, in increasing
order and counting multiplicity. The left side is the negative of the normalized
overcount of 
as an estimator for the prime counting function
(Borwein and Bailey 2003, p. 68; Granville and Martin 2004).
In 1939, Alan Turing designed a gear-driven analog machine intended to tabulate the cosine terms in the Riemann-Siegel sum and thereby assist a search for zeros of the Riemann zeta function. The surviving blueprint was never built and contains several mechanical errors, but it anticipates Turing's later computer calculations of zeta zeros (Casselman 2006).
