A286707 a(n) = (RiemannSiegelTheta(n)+im(log(zeta(1/2+i*n))))/Pi.

-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 13, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 24, 24, 24, 24, 25, 25, 26, 27, 27, 27, 28, 28

Offset: 1

Author

Mats Granvik, May 13 2017

Keywords

• sign

Comments

a(n) is not the same as A135297(n) - 1.

Links

• André LeClair, The Riemann Hypothesis for physicists, page 30.
• Raymond Manzoni, Mathematics Stack Exchange: Riemann Zeta function - number of zeros.

Crossrefs

Cf. A135297.

Programs

Mathematica

a = Table[Round[((Im[LogGamma[1/4 + I*t/2]]/Pi - t/(2*Pi)*Log[Pi] + Im[Log[Zeta[1/2 + I*t]]]/Pi))], {t, 1, 100}]
a = Table[Round[((Floor[Im[LogGamma[1/4 + I*t/2]]/Pi - t/(2*Pi)*Log[Pi] + 1] + (Sign[Im[Zeta[1/2 + I*t]]] - 1)/2))], {t, 1, 100}]
a = Table[Round[((RiemannSiegelTheta[t] + Im[Log[Zeta[1/2 + I*t]]])/Pi)], {t, 1, 100}]
a = Table[Round[((Floor[RiemannSiegelTheta[t]/Pi + 1]) + (Sign[Im[Zeta[1/2 + I*t]]] - 1)/2)], {t, 1, 100}]
a = Table[Round[(Floor[t/(2*Pi)*Log[t/(2*Pi*Exp[1])] + 7/8] + (Sign[Im[Zeta[1/2 + I*t]]] - 1)/2)], {t, 1, 100}]
a = Table[Round[(t/(2*Pi)*Log[t/(2*Pi*Exp[1])] + 7/8 + (Im[Log[Zeta[1/2 + I*t]]])/Pi - 1)], {t, 1, 100}]

Formula

a(n) = im(LogGamma (1/4 + I*n/2))/Pi - n/(2*Pi)*log (Pi) + im(log(zeta(1/2 + I*n)))/Pi

a(n) = floor(im(LogGamma (1/4 + I*n/2))/Pi - n/(2*Pi)*log(Pi) + 1) + (sign(im(zeta (1/2 + I*n))) - 1)/2

a(n) = (RiemannSiegelTheta(n) + im(log (zeta (1/2 + I*n))))/Pi

a(n) = (floor(RiemannSiegelTheta(n)/Pi + 1)) + (sign(im (zeta(1/2 + I*n))) - 1)/2

a(n) = n/(2*Pi)*log[n/(2*Pi*Exp(1))] + 7/8 + (im(log (zeta (1/2 + I*n))))/Pi - 1 - BigO(n^(-1))

a(n) = floor(n/(2*Pi)*log(n/(2*Pi*exp(1))) + 7/8) + (sign(im(zeta (1/2 + I*n))) - 1)/2