OFFSET
1,2
COMMENTS
All solutions are either purely real or purely imaginary. The smallest solution (by absolute value) happens to be purely imaginary.
When expressed in terms of Gauss's Pi function, it is
- The smallest solution to Pi(z) = Pi(-z).
- The smallest `y` such that: Pi(i*y) is purely real or, equivalently, Gamma(i*y) is purely imaginary.
FORMULA
Gamma(1+i*1.8055470716051069...) = Gamma(1-i*1.8055470716051069...) = 0.19754864094576264...
EXAMPLE
1.8055470716051069198763666... .
MAPLE
Digits:= 120:
fsolve(GAMMA(1+z*I)=GAMMA(1-z*I), z=0..3); # Alois P. Heinz, Oct 25 2024
MATHEMATICA
RealDigits[x /. FindRoot[Gamma[1 + x*I] == Gamma[1 - x*I], {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 23 2024 *)
RealDigits[x /. FindRoot[Re[Gamma[I*x]] == 0, {x, 2}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, Oct 25 2024 *)
PROG
(Python)
from mpmath import mp, nstr, factorial, findroot
mp.dps = 120
root = findroot(lambda z: factorial(z)-factorial(-z), 1.8j).imag
A377297 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jwalin Bhatt, Oct 23 2024
STATUS
approved