2,117 reputation
33
bio website mobiusfunction.wordpress.com
location
age
visits member for 3 years, 10 months
seen 19 hours ago

14,134725141734693790457251983562 = ?

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

A = Table[Table[If[Mod[n, k] == 0, k, 0], {k, 1, 12}], {n, 1, 12}]; B = Table[ Table[If[Mod[k, n] == 0, MoebiusMu[n], 0], {k, 1, 12}], {n, 1, 12}]; MatrixForm[A.B]

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.

Andre LeClaire's approximation of 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}]]]

Table[Limit[(-1 + n s (1 + s) + (2 + s)^n)/((1 + s)^2), s -> -1], {n, 1, nn}]


This user has not answered any questions
This user has not participated in any tags
This user has not asked any questions
Mathematics 2,117 rep 11127
Mathematica 370 rep 17
MathOverflow 193 rep 7
Stack Overflow 153 rep 6
Signal Processing 136 rep 4

52 Votes Cast

all time   by type   month   week  
52 up 37 question 1 1
0 down 15 answer