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

1,9

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).

Links

Table of n, a(n) for n=1..40.

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

Formula

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

Example

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

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}]

Crossrefs

Cf. A002410.

Sequence in context: A229194 A304790 A306486 * A293865 A337078 A024322

Adjacent sequences: A236209 A236210 A236211 * A236213 A236214 A236215

Keyword

nonn

Author

Jonathan Sondow, Jan 25 2014

Status

approved