A201559 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.

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, 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, 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

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

Cf. A002410.

Sequence in context: A010161 A222226 A104539 * A300015 A246499 A199002

Adjacent sequences: A201556 A201557 A201558 * A201560 A201561 A201562

Keyword

nonn,cons

Author

Jonathan Sondow, Dec 23 2011

Status

approved