Zeta zero approximations

7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 \pi \right]

9 \pi -\text{Log}\left[-3 e^{\pi /2}-3 e^{2 \pi /2}-e^{3 \pi /2}+3 e^{4 \pi /2 }\right]

11 \pi -\text{Log}\left[-1+3 e^{4 \pi /2}+e^{6 \pi /2}\right]

13 \pi -\text{Log}\left[2-e^{\pi /2}+3 e^{2 \pi /2}+2 e^{3 \pi /2}+2 e^{4 \pi /2}-2 e^{5 \pi /2}+3 e^{6 \pi /2}\right]

15 \pi -\text{Log}\left[1-e^{\pi /2}+e^{2 \pi /2}-4 e^{3 \pi /2}+2 e^{4 \pi /2}+5 e^{5 \pi /2}+e^{7 \pi /2}+e^{9 \pi /2}\right]

In[447]:= N[
7*Pi – Log[
2*Pi + Exp[5/2*Pi] + 3/2*Exp[-3/2*Pi] + 5/2*Exp[-5/2*Pi] +
7/2*Exp[-7/2*Pi]], 90]
7*Pi – Log[
2*Pi + Exp[5/2*Pi] + 3/2*Exp[-3/2*Pi] + 5/2*Exp[-5/2*Pi] +
7/2*Exp[-7/2*Pi]]
N[9*Pi – Log[
Exp[4/2*Pi]*3 – Exp[3/2*Pi] – Exp[2/2*Pi]*3 – Exp[1/2*Pi]*3], 90]
9*Pi – Log[Exp[4/2*Pi]*3 – Exp[3/2*Pi] – Exp[2/2*Pi]*3 – Exp[1/2*Pi]*3]
N[11*Pi – Log[Exp[6/2*Pi] + Exp[4/2*Pi]*3 – 1], 90]
11*Pi – Log[Exp[6/2*Pi] + Exp[4/2*Pi]*3 – 1]
N[13*Pi –
Log[Exp[6/2*Pi]*3 – Exp[5/2*Pi]*2 + Exp[4/2*Pi]*2 + Exp[3/2*Pi]*2 +
Exp[2/2*Pi]*3 – Exp[1/2*Pi] + 2], 90]
13*Pi – Log[
Exp[6/2*Pi]*3 – Exp[5/2*Pi]*2 + Exp[4/2*Pi]*2 + Exp[3/2*Pi]*2 +
Exp[2/2*Pi]*3 – Exp[1/2*Pi] + 2]
N[15*Pi –
Log[Exp[9/2*Pi] + Exp[7/2*Pi] + Exp[5/2*Pi]*5 + Exp[4/2*Pi]*2 –
Exp[3/2*Pi]*4 + Exp[2/2*Pi] – Exp[1/2*Pi] + Exp[0/2*Pi]], 90]
15*Pi – Log[
Exp[9/2*Pi] + Exp[7/2*Pi] + Exp[5/2*Pi]*5 + Exp[4/2*Pi]*2 –
Exp[3/2*Pi]*4 + Exp[2/2*Pi] – Exp[1/2*Pi] + Exp[0/2*Pi]]

Out[447]= \
14.1347251415462971625332949457130250888808428761125331718801906227734\
522626031127266673111

Out[448]=
7 \[Pi] –
Log[7/2 E^(-7 \[Pi]/2) + 5/2 E^(-5 \[Pi]/2) + 3/2 E^(-3 \[Pi]/2) +
E^(5 \[Pi]/2) + 2 \[Pi]]

Out[449]= \
21.0220647317531170031433976766645381602165975607485034136361666286850\
112342614339440360907

Out[450]=
9 \[Pi] –
Log[-3 E^(\[Pi]/2) – 3 E^\[Pi] – E^(3 \[Pi]/2) + 3 E^(2 \[Pi])]

Out[451]= \
25.0109121181194454425895620012384712403356051145851908039782928528267\
355833273049906375471

Out[452]= 11 \[Pi] – Log[-1 + 3 E^(2 \[Pi]) + E^(3 \[Pi])]

Out[453]= \
30.4248954527601648230070306069243251177298536494717015917080755061626\
004025280687729937055

Out[454]=
13 \[Pi] –
Log[2 – E^(\[Pi]/2) + 3 E^\[Pi] + 2 E^(3 \[Pi]/2) + 2 E^(2 \[Pi]) –
2 E^(5 \[Pi]/2) + 3 E^(3 \[Pi])]

Out[455]= \
32.9350618199689987097953015374911208470972884058585555099783653166032\
622776454859421331614

Out[456]=
15 \[Pi] –
Log[1 – E^(\[Pi]/2) + E^\[Pi] – 4 E^(3 \[Pi]/2) + 2 E^(2 \[Pi]) +
5 E^(5 \[Pi]/2) + E^(7 \[Pi]/2) + E^(9 \[Pi]/2)]

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inverted von Mangoldt function plot as sum of cosines


Clear[nn, k, n, a, res];
res = 100;
Monitor[a =
N[Table[Sum[
MangoldtLambda[n]*1/n*
Sum[Cos[-nn*(k - 1)/n*2*Pi], {k, 1, n}], {n, 1, nn}], {nn, 1,
res, 1/res}]];, N[nn]]
g1 = ListLinePlot[a, DataRange -> {1, res}];
Show[g1, ImageSize -> Full]

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Twelve digits

N[Log[2/3*Exp[-5/2*Pi] + Exp[7*Pi – Log[7/2*Exp[-7/2*Pi] + 5/2*Exp[-5/2*Pi] + 3/2*Exp[-3/2*Pi] + Exp[5/2*Pi] + 2*Pi]]], 15]
N[Im[ZetaZero[1]], 15]

14.1347251417344…
14.1347251417347…

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Periodic sequences from cosine sums.

Mathematica:

In[292]:= len = 24;
nn = 1;
Table[Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]
nn = 2;
Table[Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]
nn = 3;
Table[Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]
nn = 4;
Table[Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]

Out[294]= {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, \
1, 1, 1, 1, 1}

Out[296]= {0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, \
2, 0, 2, 0, 2}

Out[298]= {0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, 0, 3, 0, \
0, 3, 0, 0, 3}

Out[300]= {0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, \
4, 0, 0, 0, 4}

In[301]:= len = 24;
nn = 1;
Table[n/nn^2*Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]
nn = 2;
Table[n/nn^2*Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]
nn = 3;
Table[n/nn^2*Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]
nn = 4;
Table[n/nn^2*Sum[Cos[n*(k – 1)/nn*2*Pi], {k, 1, nn}], {n, 1, len}]

Out[303]= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, \
18, 19, 20, 21, 22, 23, 24}

Out[305]= {0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, 7, 0, 8, 0, 9, 0, \
10, 0, 11, 0, 12}

Out[307]= {0, 0, 1, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, \
0, 7, 0, 0, 8}

Out[309]= {0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, \
5, 0, 0, 0, 6}

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Dirichlet series converging to zero

Mathematica


Clear[j, a1, cc, OnePlusB, n, dd, a]
OnePlusB = (1 + N[Sum[(-1)^j*(3*j)^(-1/2), {j, 1, Infinity}], 120])
a1 = N[Sum[
1/Sqrt[i] - 1/Sqrt[1 + i] - 2/Sqrt[2 + i] - 1/Sqrt[3 + i] + 1/Sqrt[
4 + i] + 2/Sqrt[5 + i], {i, 1, \[Infinity], 6}], 500]
Monitor[cc = Table[a1*OnePlusB^n, {n, 0, 1000000}];, n]
dd = 2 + Total[cc]
a1 = N[Sum[
1/Sqrt[i] - 1/Sqrt[1 + i] - dd/Sqrt[2 + i] - 1/Sqrt[3 + i] + 1/
Sqrt[4 + i] + dd/Sqrt[5 + i], {i, 1, \[Infinity], 6}], 500]

Dirichlet series converging to zero

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Logarithmic square root waves

Mathematica


Clear[A, B, n, k, T, nn]
nn = 100;
Monitor[A =
Table[Table[N[Re[k^ZetaZero[n]], 12], {k, 1, nn}], {n, 1, nn}];, n]
ArrayPlot[A, ImageSize -> Full]

logarithmic square root waves

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von Mangoldt function and Riemann zeta zeros

Mathematica


Clear[A, B, n, k, T, nn]
nn = 24;

A = Table[Table[N[Re[k^ZetaZero[n]], 12], {k, 1, nn}], {n, 1, nn}];

MatrixForm[A]
T[n_, k_] :=
T[n, k] =
If[n < 1 || k n, T[k, n],
If[n > k, T[k, Mod[n, k, 1]], -Sum[T[n, i], {i, n - 1}]]]]];

B = Table[Table[T[n, k]/n, {k, 1, nn}], {n, 1, nn}];

MatrixForm[B]
Total[A]
Total[N[B, 10]]

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