A242069 -id:A242069

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page 1

A242070

Decimal expansion of the supremum of all real s such that zeta'(s+i*t) = 0 for some real t.

+10
2

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

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..100.

J. Arias de Reyna, J. van de Lune, Some bounds and limits in the theory of Riemann's zeta function. arXiv:1107.5134 [math.NT]

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 28.

FORMULA

The unique solution y > 1 of the equation zeta'(y)/zeta(y) = -2^(y + 1)*log(2)/(4^y - 1).

EXAMPLE

2.81301402025289836752725540121668696384614056054026221526643874...

MATHEMATICA

y /. FindRoot[Zeta'[y]/Zeta[y] == -2^(y + 1)*Log[2]/(4^y - 1), {y, 2}, WorkingPrecision -> 100] // RealDigits // First

CROSSREFS

Cf. A242069.

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Aug 14 2014

STATUS

approved

A246844

Decimal expansion of t_0, the lower bound of the conjectured first interval [t_0, t_1] where the real part of zeta(1+i*t) is negative.

+10
1

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

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

OFFSET

6,1

LINKS

Table of n, a(n) for n=6..104.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.

EXAMPLE

682112.8913382399411595568288044300347117777561378753092...

MATHEMATICA

t0 = t /. FindRoot[Re[Zeta[1 + I*t]] == 0, {t, 682112.891 }, WorkingPrecision -> 120]; RealDigits[t0, 10, 99] // First

CROSSREFS

Cf. A242069, A242070, A246843, A246845.

KEYWORD

nonn,cons

AUTHOR

Jean-François Alcover, Sep 05 2014

STATUS

approved

A246845

Decimal expansion of t_1, the upper bound of the conjectured first interval [t_0, t_1] where the real part of zeta(1+i*t) is negative.

+10
1

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

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

OFFSET

6,1

LINKS

Table of n, a(n) for n=6..104.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.

EXAMPLE

682112.9442504917624390226743949369073828564481103491515...

MATHEMATICA

t1 = t /. FindRoot[Re[Zeta[1 + I*t]] == 0, {t, 682112.944 }, WorkingPrecision -> 120]; RealDigits[t1, 10, 99] // First

CROSSREFS

Cf. A242069, A242070, A246843, A246844.