A199499 -id:A199499 - cp's OEIS frontend

A275579 Nearest integer to imaginary part of Riemann zeta zeros divided by 2*Pi.

2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26

Offset: 1

Mats Granvik, Aug 02 2016

nonn

This sequence never increases by more than 1. The first differences are given by A275737 starting: 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Guilherme França, André LeClair, Statistical and other properties of Riemann zeros based on an explicit equation for the n-th zero on the critical line, arXiv:1307.8395 [math.NT], 2013-2014, page 13, formula (25).

Cf. A199499, A046654, A275341, A275737.

Table[Round[Im[ZetaZero[n]]/(2*Pi)], {n, 1, 60}]

a(n) = round(im(zetazero(n))/(2*Pi)) = round(A002410(n)/(2*Pi)).

a(n) ~ (n - 11/8)/LambertW(exp(1)^(-1)*(n - 11/8)) (This is the Franca LeClair asymptotic at page 13, formula (25).)

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

Original entry on oeis.org

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

Friedjof Tellkamp, Jan 26 2025

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.

			1.00250471055140713139685378280222475919122794861...

Cf. A058303, A199499.

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

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

cp's OEIS Frontend

A275579 Nearest integer to imaginary part of Riemann zeta zeros divided by 2*Pi.

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