A201559 - OEIS

%I #26 Aug 28 2021 05:44:38

%S 1,1,9,2,3,4,7,3,3,7,1,8,6,1,9,3,2,0,2,8,9,7,5,0,4,4,2,7,4,2,5,5,9,7,

%T 8,8,3,4,0,1,1,1,9,2,3,0,8,3,7,9,9,9,4,3,0,1,3,7,1,9,4,9,2,9,9,0,5,2,

%U 4,5,8,6,4,8,4,8,3,0,1,3,9,2,4,0,8,4,9,9,8,6,3,8,3,7,8,8,3,6,2,4

%N Decimal expansion of x_0 = sup{x: there exists y with Re(zeta(x+i*y)) = 0}, where zeta(z) = sum(n>0, 1/n^z) is the Riemann zeta function.

%C 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.

%C 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.

%C If Re(z) >= x_0, then Re(zeta(z)) > 0.

%C Additional references and links for the zeta function are in A002410.

%H J. van de Lune, <a href="/A201559/b201559.txt">Table of n, a(n) for n = 1..1000</a>

%H R. P. Brent, <a href="http://carma.newcastle.edu.au/alfcon/pdfs/Richard_Brent-alfcon.pdf">On the distribution of arg zeta(sigma+i*t) in the half-plane sigma > 1/2</a>, lecture slides 2012.

%H Jan van de Lune, <a href="http://oai.cwi.nl/oai/asset/6554/6554A.pdf">Some observations concerning the zero-curves of the real and imaginary parts of Riemann's zeta function</a>, Math. Cent., Amst., Afd. Zuivere Wiskd. ZW 201/83, 25 p. (1983).

%H Juan Arias de Reyna, Richard P. Brent, Jan van de Lune, <a href="http://arxiv.org/abs/1112.4910">A note on the real part of the Riemann zeta-function</a>, arXiv 2011.

%H Juan Arias de Reyna, Richard P. Brent, Jan van de Lune, <a href="http://arxiv.org/abs/1205.4423">On the sign of the real part of the Riemann zeta-function</a>, arXiv 2012.

%F x_0 is the (unique) positive real root of the equation sum(p prime, arcsin(1/p^x)) = Pi/2 (van de Lune (1983)).

%e 1.1923473371861932028975044274255978834011192308379...

%Y Cf. A002410.

%K nonn,cons

%O 1,3

%A _Jonathan Sondow_, Dec 23 2011