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A236212 Floor of the value of Riemann's xi function at n. 1

%I #4 Jan 25 2014 16:38:59

%S 0,0,0,0,0,0,1,1,2,3,5,8,13,21,36,63,113,206,386,736,1433,2849,5773,

%T 11919,25059,53613,116658,258032,579856,1323273,3065246,7204159,

%U 17172291,41498712,101635485,252180415,633710357,1612310803,4151993262,10819115820

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

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

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

%H J. Sondow and C. Dumitrescu, <a href="http://arxiv.org/abs/1005.1104">A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis</a>, Period. Math. Hungar. 60 (2010), 37-40.

%H E. Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/Xi-Function.html">Xi Function</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_Xi_function">Riemann Xi function</a>

%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>

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

%e xi(1) = 1/2, so a(1) = [0.5] = 0.

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

%t 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}]

%Y Cf. A002410.

%K nonn

%O 1,9

%A _Jonathan Sondow_, Jan 25 2014

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