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N[2*Pi/ProductLog[Log[2]], 30]

Reduce[2*Pi/Log[1/n] == Im[ZetaZero[1]], n]

N[%, 30]

Reduce[2*Pi/Log[2]/n == Im[ZetaZero[1]], n]

N[%, 30]

ProductLog[Log[2]]/Log[2]

N[%, 30]

LambertW(k)/k by tetration for natural numbers.

Table[Limit[ Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}], s -> 1], {n, 1, 12}]

Table[Limit[ Zeta[s] Total[1/Divisors[n]^(s - 1)*MoebiusMu[Divisors[n]]], s -> 1], {n, 1, 32}]

https://oeis.org/A177885

Table[Im[LogGamma[ZetaZero[n]/2]/Pi - I*Im[ZetaZero[n]]/(2*Pi)*Log[Pi] + Log[Zeta[ZetaZero[n]]]/Pi + I], {n, 1, 12}]

Plot[Im[LogGamma[1/4 + I*t/2]/Pi - I*t/(2*Pi)*Log[Pi] + Log[Zeta[1/2 + I*t]]/Pi + I], {t, 0, 60}, ImageSize -> Large]

Round[Chop[ N[Table[Im[LogGamma[1/4 + I*t/2]]/Pi - t/(2*Pi)*Log[Pi] + Im[Log[Zeta[1/2 + I*t]]]/Pi + 1, {t, 0, 100}]]]]

From this answer: http://math.stackexchange.com/a/442686/8530

by Raymond Manzoni.

This Excel Spreadsheet formula uses Andre LeClaire's formula to approximate the Riemann zeta zeros:

=IF(OR(ROW()=1; COLUMN()=1);0; IF(ROW()>=COLUMN();EXP(-(1-11/8/(COLUMN()-1))/EXP(1)*SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1; COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4)));0))

(European dot-comma)

you need to divide the result with: /2/PI()/EXP(1) and take the reciprocal. tetration this is.

The von Mangoldt function matrix:

=IF(OR(ROW()=1; COLUMN()=1); 1; IF(ROW()>=COLUMN();-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));-SUM(INDIRECT(ADDRESS(COLUMN()-ROW()+1;ROW(); 4)&":"&ADDRESS(COLUMN()-1; ROW(); 4); 4))))

=REPLACE(A1;FIND(".";A1);1;",")

http://pastebin.com/u/MatsGranvik

Clear[x]

x = x /. FindRoot[ 2*Pi/ProductLog[Log[x^x]] == Im[ZetaZero[1]], {x, 1.5}, WorkingPrecision -> 100]

Log[x]

(2*Pi)/Log[x]

Divisibility recurrence:

=IF(OR(COLUMN()=1); 1; IF(ROW()>=COLUMN();SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN()-1; 4)&":"&ADDRESS(ROW()-1; COLUMN()-1; 4); 4))-SUM(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));0))

Logarithm recurrence:

=IF(OR(COLUMN()=1); 0; IF(ROW()>=COLUMN();PRODUCT(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN()-1; 4)&":"&ADDRESS(ROW()-1; COLUMN()-1; 4); 4))-PRODUCT(INDIRECT(ADDRESS(ROW()-COLUMN()+1;COLUMN(); 4)&":"&ADDRESS(ROW()-1; COLUMN(); 4); 4));1))

Clear[nn];

nn = 12

f[n_, s_] = ((s + 1)^(n - 1) + s - 1)/s;

TableForm[ FullSimplify[ Table[Integrate[Integrate[f[n, s], {n, 1, 2}], {s, 0, k}], {k, 0, nn}]]]

Table[Limit[f[n, s], s -> 0], {n, 1, nn}]

Table[Limit[D[f[n, s], s], s -> 0], {n, 1, nn}]

Table[Limit[Integrate[f[-n, s], s], s -> 0], {n, 1, nn}]

FullSimplify[ Differences[Table[Limit[Sum[f[-n, s], s], s -> 0], {n, -1, nn}]]]

FullSimplify[ Differences[ Table[Limit[Sum[f[-n, s/ZetaZero[1]], s], s -> 0], {n, -1, nn}]]]


2d
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23
revised Algorithm for reversion of power series?
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Sep
19
accepted Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.
Sep
18
revised Prove that this sequence of fractions is returned unchanged after the divisor recurrence, the matrix inverse, and the sum over divisors.
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18
asked Prove that this sequence of fractions is returned unchanged after the divisor recurrence, the matrix inverse, and the sum over divisors.
Sep
18
accepted What is the conventional notation for these logic statements?
Sep
14
revised Do these series converge to the Mangoldt function?
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Sep
13
asked Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.
Sep
13
answered Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.
Aug
23
accepted Why is a complex number plus infinity equal to infinity?
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23
asked Why is a complex number plus infinity equal to infinity?
Aug
14
accepted What is the function describing the minimal surface of this object?
Aug
11
comment What is the closed form for this sequence, powers of $4$?
@G.T.R The idea is to find the general formula for the row sums of variations of a recurrence in a lower triangular array giving the squares 1,4,9,16,25, as row sums, similar to the natural numbers here: math.stackexchange.com/questions/873899/… Then I want to integrate it and find a expression for partials sums of reciprocals of squares. But before that I need a formula, this sequence in the question is the row sums for recurrence sums multiplied by -2, when odd numbers are the first column in the lower triangular array.
Aug
11
accepted What is the closed form for this sequence, powers of $4$?