OFFSET
1,3
COMMENTS
Since lim(x->+infinity, zeta(x+i*y)) = 1 (uniformly in y), it follows that Re(zeta(x+i*y)) cannot be zero for arbitrarily large positive x. Hence x_0 exists.
van de Lune (1983) proved that x_0 > 1.192. Arias de Reyna, Brent, and van de Lune (2011) computed x_0 to 500 decimal places.
If Re(z) >= x_0, then Re(zeta(z)) > 0.
Additional references and links for the zeta function are in A002410.
LINKS
J. van de Lune, Table of n, a(n) for n = 1..1000
R. P. Brent, On the distribution of arg zeta(sigma+i*t) in the half-plane sigma > 1/2, lecture slides 2012.
Jan van de Lune, Some observations concerning the zero-curves of the real and imaginary parts of Riemann's zeta function, Math. Cent., Amst., Afd. Zuivere Wiskd. ZW 201/83, 25 p. (1983).
Juan Arias de Reyna, Richard P. Brent, Jan van de Lune, A note on the real part of the Riemann zeta-function, arXiv 2011.
Juan Arias de Reyna, Richard P. Brent, Jan van de Lune, On the sign of the real part of the Riemann zeta-function, arXiv 2012.
FORMULA
x_0 is the (unique) positive real root of the equation sum(p prime, arcsin(1/p^x)) = Pi/2 (van de Lune (1983)).
EXAMPLE
1.1923473371861932028975044274255978834011192308379...
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jonathan Sondow, Dec 23 2011
STATUS
approved