A112933 - OEIS

%I #15 Aug 11 2019 12:28:50

%S 1,-1,-1,1,-1,1,-1,-1,2,-2,1,1,2,-1,-3,1,-1,3,1,-3,-1,-3,2,4,-1,2,1,

%T -3,-1,2,-4,-3,1,7,1,-1,4,1,-3,-2,-5,2,1,-6,-3,2,6,2,-1,-3,1,5,-1,-1,

%U 7,2,-3,-13

%N Constant terms arising in an asymptotic formula for 1/(zeta(s)-1) as s --> infinity.

%H Andrei Vieru, <a href="http://arxiv.org/abs/1306.0496">Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions</a>, arXiv:1306.0496 [math.GM], 2015.

%e 1/(zeta(s)-1)=2^s-(4/3)^s-1+(8/9)^s-(4/5)^s+(2/3)^s-(16/27)^s-(4/7)^s+2*(8/15)^s-2*(4/9)^s+(2/5)^s+(32/81)^s+2*(8/21)^s-(4/11)^s-3*(16/45)^s+o((16/45)^x) and here sequence gives constant terms before rational powers :1,-1,-1,1,-1,1,-1,-1,2,-2,1,1,2,-1,-3,...

%t nmax = 20; lz = ConstantArray[0, nmax]; lc = ConstantArray[0, nmax]; ax = 0; Do[le = Exp[Limit[Log[Abs[(1/(Zeta[x] - 1) - ax)]]/x, x -> Infinity]]; ls = Limit[(1/(Zeta[x] - 1) - ax)/le^x, x -> Infinity]; ax = ax + ls*le^x; lz[[j]] = le; lc[[j]] = ls;, {j, 1, nmax}]; lc (* _Vaclav Kotesovec_, Aug 11 2019 *)

%Y Cf. A112931, A112932.

%K more,sign

%O 0,9

%A _Benoit Cloitre_, Oct 06 2005

%E a(15)-a(33) computed by Andrei Vieru, added by _Vaclav Kotesovec_, Aug 11 2019

%E Terms a(34) and beyond from _Vaclav Kotesovec_, Aug 11 2019