Table of Chebyshev psi function

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.

asked 10 hours ago

Richard Burke-Ward

5769

add a comment |

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.

asked 10 hours ago

Richard Burke-Ward

5769

add a comment |

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.

asked 10 hours ago

Richard Burke-Ward

5769

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-Ward

5769

asked 10 hours ago

Richard Burke-Ward

5769

asked 10 hours ago

Richard Burke-Ward

5769

asked 10 hours ago

Richard Burke-Ward

5769

asked 10 hours ago

Richard Burke-Ward

5769

add a comment |

1 Answer
1

active

oldest

votes

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}]

You can also directly construct a list of these with Accumulate:

Accumulate@Array[MangoldtLambda, 10]

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}]

enter image description here

edited 20 mins ago

answered 10 hours ago

Roman

2,594717

$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago

add a comment |

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

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}]

You can also directly construct a list of these with Accumulate:

Accumulate@Array[MangoldtLambda, 10]

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}]

enter image description here

edited 20 mins ago

answered 10 hours ago

Roman

2,594717

$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago

add a comment |

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}]

You can also directly construct a list of these with Accumulate:

Accumulate@Array[MangoldtLambda, 10]

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}]

enter image description here

edited 20 mins ago

answered 10 hours ago

Roman

2,594717

$begingroup$
I'll tick this when it lets me! Appreciated.
$endgroup$
– Richard Burke-Ward
10 hours ago

add a comment |

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}]

You can also directly construct a list of these with Accumulate:

Accumulate@Array[MangoldtLambda, 10]

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}]

enter image description here

edited 20 mins ago

answered 10 hours ago

Roman

2,594717

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}]

You can also directly construct a list of these with Accumulate:

Accumulate@Array[MangoldtLambda, 10]

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}]

enter image description here

edited 20 mins ago

answered 10 hours ago

Roman

2,594717

edited 20 mins ago

answered 10 hours ago

Roman

2,594717

answered 10 hours ago

Roman

2,594717

answered 10 hours ago

Roman

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

I'll tick this when it lets me! Appreciated.

– Richard Burke-Ward
10 hours ago

add a comment |

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