A271855 - OEIS

A271855

Decimal expansion of -x_1 such that the Riemann function zeta(x) has at real x_1 < 0 its first local extremum.

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

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OFFSET

1,1

COMMENTS

For real x < 0, zeta(x) undergoes divergent oscillations, passing through zero at every even integer value of x. In each interval (-2n,-2n-2), n = 1, 2, 3, ..., it attains a local extreme (maximum, minimum, maximum, ...). The location x_n of the n-th local extreme does not match the odd integer -2n-1. Rather, x_n > -2n-1 for n = 1 and 2, and x_n < -2n-1 for n >= 3. This entry defines the location x_1 of the first maximum. The corresponding value is in A271856.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Eric Weisstein's World of Mathematics, Riemann Zeta Function.

FORMULA

zeta(x_1) = A271856.

EXAMPLE

-2.7172628292045741015705806616765284124247518539174926559440...

MATHEMATICA

RealDigits[x /. FindRoot[Zeta'[x] == 0, {x, 0}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Apr 11 2026 *)

PROG

(PARI) \\ This function was tested up to n = 11600000:

zetaextreme(n) = {solve(x=-2.0*n, -2.0*n-1.9999999999, zeta'(x))}