|
%I #10 Apr 23 2022 14:54:55
%S 1,0,0,0,0,2,7,8,5,7,6,3,3,0,6,6,4,4,0,7,3,0,2,1,5,0,9,1,8,5,7,3,6,2,
%T 1,7,7,8,2,9,7,1,0,0,9,1,4,0,5,3,3,3,0,4,7,8,7,9,7,3,1,9,2,8,4,5,8,6,
%U 4,7,3,5,4,1,6,6,6,1,2,9,3,5,2,6,5,0,0
%N Decimal expansion of the pearl of the Riemann zeta function.
%C Let Z be the Riemann zeta function, and consider its sequence of nontrivial zeros with nonnegative imaginary part, {r(m)}, so that for every m >= 1, Z(r(m)) = 0, 0 <= Re(r(m)) <= 1, and 0 <= Im(r(m)), and for every k > m, Im(r(m)) < Im(r(k)), or Im(r(m)) = Im(r(k)) and Re(r(m)) < Re(r(k)).
%C Let i be the imaginary unit, and define the sequence {b(m)} as follows: b(1) = Z((r(1)-1/2)/i), b(2) = Z((r(1)-1/2)/i + Z((r(2)-1/2)/i)), b(3) = Z((r(1)-1/2)/i + Z((r(2)-1/2)/i + Z((r(3)-1/2)/i))), and so on. If this sequence converges, we call its limit the pearl of Z.
%C Suppose that the Riemann Hypothesis is true. Then the sequence {b(m)} is real. On the interval [2,oo), Z is decreasing, positive, and bounded above by 2, so {b(2*m-1)} is decreasing and bounded below by 0, and hence, it converges to a real value, say A. Moreover, {b(2*m)} is increasing and b(2*m) <= b(2*m+1), and by repeated application of the mean value theorem, b(2*m+1) - b(2*m) <= Z(Im(r(2*m+1))) * |Z'(Im(r(1)))|^(2*m) <= 2*(4/100000)^(2*m), so {b(2*m)} also converges to A, and {a(n)} is the decimal expansion of this value.
%C We don't know if the existence of a real pearl of Z implies the Riemann Hypothesis.
%C More generally, the definition of pearl works for Dirichlet L-functions, giving rise to analogous constants, not necessarily real.
%H Eduard Roure Perdices, <a href="/A352495/b352495.txt">Table of n, a(n) for n = 1..5000</a>
%e 1.00002785763306644073021509185736217782971009140533304787973192845864...
%t RealDigits[Re[res = Fold[Zeta[#1 + #2] &, 0, Reverse[(ZetaZero[Range[10]] - 1/2)/I]]], 10, 100][[1]]
%Y Cf. A072449, A099874, A099876, A099877, A105546, A105817, A151558, A239349, A277313, A278812.
%K nonn,cons
%O 1,6
%A _Eduard Roure Perdices_, Mar 18 2022
|
