A236212 Floor of the value of Riemann's xi function at n.

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 21, 36, 63, 113, 206, 386, 736, 1433, 2849, 5773, 11919, 25059, 53613, 116658, 258032, 579856, 1323273, 3065246, 7204159, 17172291, 41498712, 101635485, 252180415, 633710357, 1612310803, 4151993262, 10819115820

Offset: 1

Author

Jonathan Sondow, Jan 25 2014

Keywords

• nonn

Comments

On the interval [1, infinity), the xi function takes real values and is (strictly) increasing, so a(n) <= a(n+1) for n >= 1.

Same as floor of the value of the xi function at 1-n, because of the functional equation xi(1-s) = x(s).

Examples

			xi(1) = 1/2, so a(1) = [0.5] = 0.
xi(8) = (4*Pi^4)/225 = 1.7317…, so a(8) = [1.7] = 1.
		

Links

• J. Sondow and C. Dumitrescu, A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis, Period. Math. Hungar. 60 (2010), 37-40.
• E. Weisstein's MathWorld, Xi Function
• Wikipedia, Riemann Xi function
• Index entries for zeta function

Crossrefs

Cf. A002410.

Programs

Mathematica

xi[ s_] := If[ s == 1, 1/2, (s/2)*(s - 1)*Pi^(-s/2)*Gamma[ s/2]*Zeta[ s]]; Table[ Floor[ xi[ n]], {n, 40}]

Formula

a(n) = [xi(n)] for n > 0.