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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, 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 (list; constant; graph; refs; listen; history; text; internal format)
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

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Last modified October 27 11:21 EDT 2021. Contains 348274 sequences. (Running on oeis4.)