A387052 Decimal expansion of -x, where x is the abscissa of the second local extremum of the Riemann zeta function on the negative real axis.

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

Offset: 1

Author

Artur Jasinski, Aug 15 2025

Keywords

• nonn
• cons
• new

Comments

The Riemann zeta function has zeros for x = -2*n for n >= 1, which means that between -2*(n+1) and -2*n the function has an extremum for each positive integer n.

For the value of zeta(-x_2) see A387065.

It is an open question whether the fractional part of x_n tends to 1 or some unknown constant c < 1 as n tends to infinity.

Links

• Artur Jasinski, Complete list of the first 1000 negative extrema of the Riemann zeta function.

Crossrefs

Cf. A271855, A271856.

Programs

Mathematica

kk = x /. FindRoot[Zeta'[x] == 0, {x, -5}, WorkingPrecision -> 110];
RealDigits[kk, 10, 105][[1]]

PARI

solve(x=-4.95, -4.9, zeta'(x))