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14,134725141734693790457251983562 = ?

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 + It/2]/Pi - It/(2*Pi)Log[Pi] + Log[Zeta[1/2 + It]]/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 + It]]]/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}]]]


Dec
14
comment Please help me evaluate this product involving logarithms.
@AsdrubalBeltran How do you mean? Mathematica evaluates $\log(-\frac{1}{6n+2})$ to $\{-\log (2)+i \pi ,-\log (8)+i \pi ,-\log (14)+i \pi ,-\log (20)+i \pi \}$ for $n=1,2,3,4$
Dec
14
asked Please help me evaluate this product involving logarithms.
Dec
14
revised Where is the fault in this approach for transforming this Dirichlet series?
added 886 characters in body
Dec
14
revised Where is the fault in this approach for transforming this Dirichlet series?
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Dec
14
revised Where is the fault in this approach for transforming this Dirichlet series?
unsurpisingly
Dec
14
asked Where is the fault in this approach for transforming this Dirichlet series?
Dec
9
asked Can there be a Dirichlet series that gives the functional inverse of the Riemann zeta function?
Dec
8
awarded  Caucus
Dec
7
revised Help me generalize what this divisor transform does.
added 874 characters in body
Dec
7
revised Help me generalize what this divisor transform does.
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7
revised Help me generalize what this divisor transform does.
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7
revised Help me generalize what this divisor transform does.
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Nov
22
accepted What is the name for a polynomial with all coefficients equal to 1?
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22
revised What is the name for a polynomial with all coefficients equal to 1?
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Nov
22
comment What is the name for a polynomial with all coefficients equal to 1?
Thanks for the comment Claude.
Nov
22
asked What is the name for a polynomial with all coefficients equal to 1?
Nov
21
comment Are the solutions to $1+1/2^s+1/3^s=0$ known?
Related: mathematica.stackexchange.com/questions/63541/…
Nov
16
revised Help me generalize what this divisor transform does.
added pascal triangle
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14
revised Help me generalize what this divisor transform does.
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Nov
14
asked Help me generalize what this divisor transform does.