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I'm trying to make a function to calculate the Log sum of primes over a limited range $1/2n$ to $n$ or Chebychev theta function over limited range $1/2n$ to $n$. This will be used only for even numbers.

To start off I'm trying to modify a function that works:

LogSumPrime[n_] := Total[Log[Table[Prime[i], {i, PrimePi[n]}]]];

this will output as desired for LogSumPrime[12]:

Log[2] + Log[3] + Log[5] + Log[7] + Log[11]

So far so good. Now modifying to get range $1/2n$ to $n$:

LogSumAllUpperPrime[n_] := 
  Total[Log[Table[(Prime[i + PrimePi[n] - PrimePi[n/2] + 1]), 
  {i, PrimePi[n] - PrimePi[n/2]}]]];

for LogSumAllUpperPrime[12], it is fine:

Log[7] + Log[11]

for LogSumAllUpperPrime[6], it is not fine:

Log[5]

It should be Log[3] + Log[5]. Now the problem stems from half of 6 being odd and the limits not starting for this at 3, while for 12 it starts at 6 and the problem does not matter.

I do not understand Mathematica, so I cannot see how to solve this problem efficiently. If it were C, I'd just see if the bit is 1 or 0 to determine whether its even, but that may not be the best remedy here.

I would like to do this quite efficiently as I will be working with quite large numbers and the Table approach is said to be reasonably efficient.

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    $\begingroup$ If primePi was input correctly then it depends on how that function is defined. If it was intended to be PrimePi[n/2] then possibly use of Floor or Ceiling will remedy the issue. $\endgroup$
    – Daniel Lichtblau
    Commented May 28, 2018 at 14:49
  • $\begingroup$ @Daniel Lichtblau I thought this initially too but this is not an issue since only even numbers are used so it made no difference when I tried so I removed it from the code. The actual problem is when half the n is an odd number. $\endgroup$
    – onepound
    Commented May 28, 2018 at 15:07
  • $\begingroup$ Please post the definition of primePi $\endgroup$
    – m_goldberg
    Commented May 28, 2018 at 15:12
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    $\begingroup$ Or when n/2 is prime... Could use Table[Prime[i + PrimePi[n/2] - Boole[PrimeQ[n/2]]], {i, PrimePi[n] - PrimePi[n/2] + Boole[PrimeQ[n/2]]}] $\endgroup$
    – Daniel Lichtblau
    Commented May 28, 2018 at 17:49
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    $\begingroup$ Asymptotically this approaches n/2. It has been a long time since I saw this nifty function (like, three decades). I had forgotten all about it, and the relation to PNT. $\endgroup$
    – Daniel Lichtblau
    Commented May 29, 2018 at 2:46

1 Answer 1

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Perhaps this is what you mean

LogSumPrime[n_] := Total@Log@Prime@Range@PrimePi@n

LogSumAllUpperPrime[n_] := 
 Total@Log@Prime@Range[Max[1, PrimePi[n/2]], PrimePi@n]

{#, LogSumPrime[#], LogSumAllUpperPrime[#]} & /@ Range[2, 12, 2] // 
  Prepend[#, Style[#, Bold] & /@ {"n", LogSumPrime, LogSumAllUpperPrime}] & //
  Grid[#, Frame -> All] &

enter image description here

EDIT: Based on your comment, you apparently intend

LogSumAllUpperPrime[n_] := Total@Log@Select[Prime@Range@PrimePi@n, # >= n/2 &]

{#, LogSumPrime[#], LogSumAllUpperPrime[#]} & /@ Range[2, 12, 2] // 
  Prepend[#, Style[#, Bold] & /@ {"n", LogSumPrime, LogSumAllUpperPrime}] & //
  Grid[#, Frame -> All] &

enter image description here

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  • $\begingroup$ not quite I think LogSumAllUpperPrime[] with 8 and 12 should start at Log[5] and Log[7] otherwise it includes partial sums < n/2 $\endgroup$
    – onepound
    Commented May 28, 2018 at 16:04
  • $\begingroup$ yes that's it, thanks. $\endgroup$
    – onepound
    Commented May 28, 2018 at 18:14

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