For every real ,
when ,
including at the threshold itself. Thus the supremum
in the definition is not attained (van de Lune 1983, pp. 9-11). The defining
equation can also be written using the prime
zeta function
as
(4)
Arias de Reyna et al. (2012, pp. 32-33) used this representation to compute
to at least 500 decimal places.
Arias de Reyna, J.; Brent, R. P.; and van de Lune, J. "A Note on the Real Part of the Riemann Zeta-Function." In Leven
met getallen: liber amicorum ter gelegenheid van de pensionering van Herman te Riele
(Ed. J. A. J. van Vonderen). Amsterdam, Netherlands: CWI, pp. 30-36,
2012. Revised 15 May 2014. https://arxiv.org/abs/1112.4910.Sloane,
N. J. A. Sequence A201559 in "The
On-Line Encyclopedia of Integer Sequences."van de Lune, J. Some
Observations Concerning the Zero-Curves of the Real and Imaginary Parts of Riemann's
Zeta Function. Report ZW 201/83. Amsterdam, Netherlands: Mathematisch Centrum,
1983. https://ir.cwi.nl/pub/6554.
![sigma_0=sup{sigma>1: exists t in R, Re[zeta(sigma+it)]=0}.](/images/equations/vandeLuneConstant/NumberedEquation1.svg)
is the unique real solution greater than 1 of
denotes the principal value of the inverse
sine. Numerically,
,
![Re[zeta(sigma+it)]>0](/images/equations/vandeLuneConstant/Inline4.svg)
when 
,
including at the threshold itself. Thus the supremum
in the definition is not attained (van de Lune 1983, pp. 9-11). The defining
equation can also be written using the prime
zeta function 
as
to at least 500 decimal places.
