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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
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
KEYWORD
nonn,cons
AUTHOR
Jonathan Sondow, Dec 23 2011
STATUS
approved

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Last modified June 23 22:09 EDT 2024. Contains 373661 sequences. (Running on oeis4.)