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 A352495 Decimal expansion of the pearl of the Riemann zeta function. 1
 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, 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, 4, 7, 3, 5, 4, 1, 6, 6, 6, 1, 2, 9, 3, 5, 2, 6, 5, 0, 0 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS 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)). 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. 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. We don't know if the existence of a real pearl of Z implies the Riemann Hypothesis. More generally, the definition of pearl works for Dirichlet L-functions, giving rise to analogous constants, not necessarily real. LINKS Eduard Roure Perdices, Table of n, a(n) for n = 1..5000 EXAMPLE 1.00002785763306644073021509185736217782971009140533304787973192845864... MATHEMATICA RealDigits[Re[res = Fold[Zeta[#1 + #2] &, 0, Reverse[(ZetaZero[Range] - 1/2)/I]]], 10, 100][] CROSSREFS Cf. A072449, A099874, A099876, A099877, A105546, A105817, A151558, A239349, A277313, A278812. Sequence in context: A329406 A019731 A021363 * A141721 A129603 A263809 Adjacent sequences: A352492 A352493 A352494 * A352496 A352497 A352498 KEYWORD nonn,cons AUTHOR Eduard Roure Perdices, Mar 18 2022 STATUS approved

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Last modified March 25 23:47 EDT 2023. Contains 361529 sequences. (Running on oeis4.)