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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

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


1d
revised What is the radius of convergence for the power series of the Riemann zeta function at $x_0=0$?
rephrased title
1d
revised What is the radius of convergence for the power series of the Riemann zeta function at $x_0=0$?
changed "A wise person" to "someone".
1d
asked What is the radius of convergence for the power series of the Riemann zeta function at $x_0=0$?
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 $4$?
Aug
11
comment What is the closed form for this sequence, powers of $4$?
Thank you for the comments.
Aug
11
asked What is the closed form for this sequence, powers of $4$?
Aug
3
comment Is this double limit for logarithms true?
No there are no integration limits but Mathematica knows that it becomes hyper geometric series which simplified is a PolyGamma expression.
Aug
2
asked Is this double limit for logarithms true?
Aug
1
accepted Is this formula for the harmonic numbers true?
Aug
1
comment Is this formula for the harmonic numbers true?
I got the same answer again. The sequence of Harmonic numbers.
Aug
1
comment Is this formula for the harmonic numbers true?
Wait I will shut down Mathematica and run the code again.
Aug
1
comment Is this formula for the harmonic numbers true?
What is a range? Mathematica did this integration that I copied to latex, and pasted here.
Aug
1
revised Is this formula for the harmonic numbers true?
added differences command to program
Aug
1
comment Is this formula for the harmonic numbers true?
Any ways integrate with respect to $s$ first, then take the limit.
Aug
1
comment Is this formula for the harmonic numbers true?
I think it is $s$ which is a real number.
Aug
1
asked Is this formula for the harmonic numbers true?
Jul
31
revised Successive ratios of a sequence, is this limit true?
added number of divisors of n
Jul
31
revised Similarity of two limits related to the sum of divisors $\sigma(n)$ and the harmonic numbers $H_n$
added tag riemann hypothesis