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Table of Chebyshev psi function
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$begingroup$
This should be easy, but for some reason I'm struggling. The Chebyshev psi function is given as the sum from 0 to x of the von Mangoldt function MangoldtLambda[x]. I want to tabulate it. Ideally, I'd like to express the function using Sum, but I can't find the right form. This doesn't work:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {x, 0, k}]],
{x, 0, 10}]]
This produces a series that makes no sense at all:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {k, 0, x}]],
{x, 0, 10}]]
Accumulate doesn't work either:
TableForm[Table[Accumulate[N[MangoldtLambda[x], 5]], {x, 0, 10}]]
Clearly I'm suffering from a failure of imagination, but I'd appreciate help.
functions table summation
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
$endgroup$
add a comment |
$begingroup$
This should be easy, but for some reason I'm struggling. The Chebyshev psi function is given as the sum from 0 to x of the von Mangoldt function MangoldtLambda[x]. I want to tabulate it. Ideally, I'd like to express the function using Sum, but I can't find the right form. This doesn't work:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {x, 0, k}]],
{x, 0, 10}]]
This produces a series that makes no sense at all:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {k, 0, x}]],
{x, 0, 10}]]
Accumulate doesn't work either:
TableForm[Table[Accumulate[N[MangoldtLambda[x], 5]], {x, 0, 10}]]
Clearly I'm suffering from a failure of imagination, but I'd appreciate help.
functions table summation
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
$endgroup$
add a comment |
$begingroup$
This should be easy, but for some reason I'm struggling. The Chebyshev psi function is given as the sum from 0 to x of the von Mangoldt function MangoldtLambda[x]. I want to tabulate it. Ideally, I'd like to express the function using Sum, but I can't find the right form. This doesn't work:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {x, 0, k}]],
{x, 0, 10}]]
This produces a series that makes no sense at all:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {k, 0, x}]],
{x, 0, 10}]]
Accumulate doesn't work either:
TableForm[Table[Accumulate[N[MangoldtLambda[x], 5]], {x, 0, 10}]]
Clearly I'm suffering from a failure of imagination, but I'd appreciate help.
functions table summation
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
$endgroup$
This should be easy, but for some reason I'm struggling. The Chebyshev psi function is given as the sum from 0 to x of the von Mangoldt function MangoldtLambda[x]. I want to tabulate it. Ideally, I'd like to express the function using Sum, but I can't find the right form. This doesn't work:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {x, 0, k}]],
{x, 0, 10}]]
This produces a series that makes no sense at all:
TableForm[
Table[With[{k = 10}, Sum[N[MangoldtLambda[x], 5], {k, 0, x}]],
{x, 0, 10}]]
Accumulate doesn't work either:
TableForm[Table[Accumulate[N[MangoldtLambda[x], 5]], {x, 0, 10}]]
Clearly I'm suffering from a failure of imagination, but I'd appreciate help.
functions table summation
functions table summation
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
asked 10 hours ago
Richard Burke-WardRichard Burke-Ward
5769
5769
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
There seems to be a duplicate use of the symbol x in your formulas.
ChebyshevPsi[x_] := Sum[MangoldtLambda[y], {y, x}]
Table[ChebyshevPsi[x], {x, 10}]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
You can also directly construct a list of these with Accumulate:
Accumulate@Array[MangoldtLambda, 10]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
Plot the deviation of the Chebyshev $psi$ function from $x$:
ListLinePlot[MapIndexed[{#2[[1]], #1 - #2[[1]]} &,
Accumulate@Array[MangoldtLambda, 1000]], Frame -> True,
FrameLabel -> {x, ψ[x] - x}]
answered 10 hours ago
RomanRoman
2,594717
$endgroup$
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
add a comment |
Your Answer
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
There seems to be a duplicate use of the symbol x in your formulas.
ChebyshevPsi[x_] := Sum[MangoldtLambda[y], {y, x}]
Table[ChebyshevPsi[x], {x, 10}]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
You can also directly construct a list of these with Accumulate:
Accumulate@Array[MangoldtLambda, 10]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
Plot the deviation of the Chebyshev $psi$ function from $x$:
ListLinePlot[MapIndexed[{#2[[1]], #1 - #2[[1]]} &,
Accumulate@Array[MangoldtLambda, 1000]], Frame -> True,
FrameLabel -> {x, ψ[x] - x}]
answered 10 hours ago
RomanRoman
2,594717
$endgroup$
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
add a comment |
$begingroup$
There seems to be a duplicate use of the symbol x in your formulas.
ChebyshevPsi[x_] := Sum[MangoldtLambda[y], {y, x}]
Table[ChebyshevPsi[x], {x, 10}]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
You can also directly construct a list of these with Accumulate:
Accumulate@Array[MangoldtLambda, 10]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
Plot the deviation of the Chebyshev $psi$ function from $x$:
ListLinePlot[MapIndexed[{#2[[1]], #1 - #2[[1]]} &,
Accumulate@Array[MangoldtLambda, 1000]], Frame -> True,
FrameLabel -> {x, ψ[x] - x}]
answered 10 hours ago
RomanRoman
2,594717
$endgroup$
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
add a comment |
$begingroup$
There seems to be a duplicate use of the symbol x in your formulas.
ChebyshevPsi[x_] := Sum[MangoldtLambda[y], {y, x}]
Table[ChebyshevPsi[x], {x, 10}]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
You can also directly construct a list of these with Accumulate:
Accumulate@Array[MangoldtLambda, 10]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
Plot the deviation of the Chebyshev $psi$ function from $x$:
ListLinePlot[MapIndexed[{#2[[1]], #1 - #2[[1]]} &,
Accumulate@Array[MangoldtLambda, 1000]], Frame -> True,
FrameLabel -> {x, ψ[x] - x}]
answered 10 hours ago
RomanRoman
2,594717
$endgroup$
There seems to be a duplicate use of the symbol x in your formulas.
ChebyshevPsi[x_] := Sum[MangoldtLambda[y], {y, x}]
Table[ChebyshevPsi[x], {x, 10}]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
You can also directly construct a list of these with Accumulate:
Accumulate@Array[MangoldtLambda, 10]
{0, Log[2], Log[2] + Log[3], 2 Log[2] + Log[3],
2 Log[2] + Log[3] + Log[5], 2 Log[2] + Log[3] + Log[5],
2 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7],
3 Log[2] + 2 Log[3] + Log[5] + Log[7]}
Plot the deviation of the Chebyshev $psi$ function from $x$:
ListLinePlot[MapIndexed[{#2[[1]], #1 - #2[[1]]} &,
Accumulate@Array[MangoldtLambda, 1000]], Frame -> True,
FrameLabel -> {x, ψ[x] - x}]
answered 10 hours ago
RomanRoman
2,594717
edited 16 mins ago
answered 10 hours ago
RomanRoman
2,594717
answered 10 hours ago
RomanRoman
2,594717
answered 10 hours ago
RomanRoman
2,594717
2,594717
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
add a comment |
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago
add a comment |
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