A380519 Decimal expansion of least x>1 so that Re(x^rho) has a local maximum, with rho as the first zeta zero.

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

Offset: 1

Author

Friedjof Tellkamp, Jan 26 2025

Keywords

• nonn
• cons

Comments

Re(x^rho) = sqrt(x) * cos(Im(rho) * log(x)), with rho = 1/2 + i * 14.134725... (A058303).

Further local maxima at: (this constant) * exp(2*Pi*k/A058303), with integer k.

Examples

			1.00250471055140713139685378280222475919122794861...
		

Crossrefs

Cf. A058303, A199499.

Programs

Mathematica

RealDigits[x /. FindRoot[D[Sqrt[x] Cos[Im@ZetaZero@1 Log[x]], x], {x, 1}, WorkingPrecision -> 100][[1]]][[1]]

Formula

Equals exp(arccos(2*t/sqrt(1+4*t^2))/t), with t = Im(rho).