cp's OEIS Frontend

A257821 Decimal expansion of the unique real number a>0 such that the real part of li(-a) is zero.

%I A257821 #8 Feb 16 2025 08:33:25
%S A257821 2,4,6,6,4,0,8,2,6,2,4,1,2,6,7,8,0,7,5,1,9,7,1,0,3,3,5,0,7,7,5,9,3,2,
%T A257821 9,5,0,2,9,0,7,8,0,8,7,8,2,7,7,4,0,9,9,8,2,3,7,8,6,0,8,9,8,8,1,6,1,2,
%U A257821 2,4,0,9,4,1,5,0,0,9,1,5,0,7,1,7,1,6,2,7,8,1,5,8,0,4,6,5,5,8,4,7,2,9,3,3,6
%N A257821 Decimal expansion of the unique real number a>0 such that the real part of li(-a) is zero.
%C A257821 As discussed in A257819, the real part of li(z) is a well behaved function for any real z, except for the singularity at z=+1. It has three roots: z=A070769 (the Soldner's constant), z=0, and z=-a. However, unlike in the other two cases, the imaginary part of li(-a) is not infinitesimal in the neighborhood of this root; it is described in A257822.
%H A257821 Stanislav Sykora, <a href="/A257821/b257821.txt">Table of n, a(n) for n = 1..2000</a>
%H A257821 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicIntegral.html">Logarithmic Integral</a>
%H A257821 Wikipedia, <a href="http://en.wikipedia.org/wiki/Logarithmic_integral_function">Logarithmic integral function</a>
%F A257821 Satisfies real(li(-a)) = 0.
%e A257821 2.4664082624126780751971033507759329502907808782774099823786...
%t A257821 RealDigits[a/.FindRoot[Re[LogIntegral[-a]]==0,{a,2},WorkingPrecision->120]][[1]] (* _Vaclav Kotesovec_, May 11 2015 *)
%o A257821 (PARI) li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
%o A257821   if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
%o A257821   return(+Pi*I-eint1(-log(c)))); }
%o A257821   a=-solve(x=-3,-1,real(li(x)))  \\ Better use excess realprecision
%Y A257821 Cf. A070769, A257819, A257822.
%K A257821 nonn,cons
%O A257821 1,1
%A A257821 _Stanislav Sykora_, May 11 2015