For a realpositive,
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.
For real ,
the Riemann-Siegel theta function is defined as
(2)
(3)
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).
Berry, M. V. "The Riemann-Siegel Expansion for the Zeta Function: High Orders and Remainders." Proc. Roy. Soc. London A450,
439-462, 1995.Borwein, J. M.; Bradley, D. M.; and Crandall,
R. E. "Computational Strategies for the Riemann Zeta Function." J.
Comput. Appl. Math.121, 247-296, 2000.Brent, R. P.
"On the Zeros of the Riemann Zeta Function in the Critical Strip." Math.
Comput.33, 1361-1372, 1979.Edwards, H. M. Riemann's
Zeta Function. New York: Dover, 2001.Karatsuba, A. A. and
Voronin, S. M. The
Riemann Zeta-Function. Hawthorn, NY: de Gruyter, 1992.Odlyzko,
A. M. "The th Zero of the Riemann Zeta Function and 70 Million of
Its Neighbors." Preprint.Sloane, N. J. A. Sequences
A036282, A114721,
A114865, and A114866
in "The On-Line Encyclopedia of Integer Sequences."Titchmarsh,
E. C. The
Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.van
de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros
of the Riemann Zeta Function in the Critical Strip. IV." Math. Comput.46,
667-681, 1986.Vardi, I. Computational
Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 143, 1991.
,
the Riemann-Siegel 
function is defined by
-function (Karatsuba and Voronin 1992, Borwein et al. 1999).
The top plot superposes 
(thick line) on 
, where 
is the Riemann zeta
function.
,
the Riemann-Siegel theta function 
is defined as
![I[lnGamma(1/4+1/2it)]-1/2tlnpi](/images/equations/Riemann-SiegelFunctions/Inline11.svg)
![arg[Gamma(1/4+1/2it)]-1/2tlnpi.](/images/equations/Riemann-SiegelFunctions/Inline14.svg)
has local extrema at 
(OEIS A114865
and A114866).
such that
,
1, ... are known as Gram points (Edwards 2001, pp. 125-126).
about 0 is given by
![1/2[-lnpi+psi(1/4)]t-1/(48)psi_2(1/4)t^3+1/(3840)psi_4(1/4)t^5-1/(645120)psi_6(1/4)t^7](/images/equations/Riemann-SiegelFunctions/Inline25.svg)
![-1/4[2gamma+pi+2ln(8pi)]t+1/(24)[pi^3+28zeta(3)]t^3-[(pi^5)/(96)+(31zeta(5))/(10)]+...](/images/equations/Riemann-SiegelFunctions/Inline28.svg)
by
th Zero of the Riemann Zeta Function and 70 Million of
Its Neighbors." Preprint.Sloane, N. J. A. Sequences