A387378 Decimal expansion of the smallest positive real solution > 0.5 to zeta(z) = zeta(1-z).

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

Offset

2,2

Comments

Using the reflection formula for the zeta function, one can also rewrite the equality in terms of the Gamma function as Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).

There are infinitely many solutions on the real axis and on the critical line.

The solutions on the critical line are the gram points.

There are 12 complex solutions apart from these out of which 3 are unique:

8.990914533614919... + i*4.510594140699146...

13.162787864991035... + i*2.580464971850669...

16.478090665944547... + i*0.679406009477847...

Links

Table of n, a(n) for n=2..121.

Formula

zeta(19.067750847069662...) = zeta(1-19.067750847069662...) = 1.000001820649741...

Smallest positive real root > 0.5 of the equation Gamma(z) = (2^(z-1))*(Pi^z)*sec((Pi*z)/2).

Equals A365281 + 1/2. - Amiram Eldar, Aug 28 2025

Example

19.06775084706966207279...

Mathematica

RealDigits[x /. FindRoot[Zeta[x] == Zeta[1 - x], {x, 19}, WorkingPrecision -> 120]][[1]]

Crossrefs

Cf. A058303, A365281, A377302.

Sequence in context: A381497 A388756 A388555 * A093766 A363368 A097674

Adjacent sequences: A387375 A387376 A387377 * A387379 A387380 A387381

Keyword

nonn,cons

Author

Jwalin Bhatt, Aug 28 2025

Status

approved