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

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, 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, 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

Offset: 1

Author

Stanislav Sykora, May 11 2015

Keywords

• nonn
• cons

Comments

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.

Examples

			2.4664082624126780751971033507759329502907808782774099823786...
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..2000
• Eric Weisstein's World of Mathematics, Logarithmic Integral
• Wikipedia, Logarithmic integral function

Crossrefs

Cf. A070769, A257819, A257822.

Programs

Mathematica

RealDigits[a/.FindRoot[Re[LogIntegral[-a]]==0,{a,2},WorkingPrecision->120]][[1]] (* Vaclav Kotesovec, May 11 2015 *)

PARI

li(z) = {my(c=z+0.0*I); \\ If z is real, convert it to complex
  if(imag(c)<0, return(-Pi*I-eint1(-log(c))),
  return(+Pi*I-eint1(-log(c)))); }
  a=-solve(x=-3,-1,real(li(x)))  \\ Better use excess realprecision

Formula

Satisfies real(li(-a)) = 0.