Table of Chebyshev psi function

Question

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.

Answer 1

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}]