## Rough plot of the von Mangoldt function

The first of these plots gives a rough picture of the von Mangoldt function.
I first learned it from J. Brian Conreys paper:

The Riemann Hypothesis, J.Brian Conrey

and later understood it from Edwin Chen’s blog:

Prime Numbers and the Riemann Zeta Function, Edwin Chen

But I have simplified it to let “x” be raised to Riemann zeta zero without the other terms
as in Chen’s blog post

``` Clear[n, k, A, a, B, b, kk, ii, i] kk = 50;(*number of Zeta zeros used*) ii = 10;(*reciprocal of x-axis spacing*) n = Table[i, {i, 0, 42, 1/ii}]; A = Table[Re[n^N[ZetaZero[k], 12]], {k, 1, kk}]; a = Table[{-i/ii, Plus @@ A[[All, i]]}, {i, 1, Length[n]}]; a[[ii + 1]] = {-(ii + 1)/ii, 0}; ListLinePlot[-a] ```

Check out Heike’s answer on stack overflow:
Riemann zeta zero spectrum

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### 7 Responses to Rough plot of the von Mangoldt function

1. CK says:

Thank you!
On Jeffrey Stopple’s page at http://www.math.ucsb.edu/~stopple/explicit.html ,

he comments that the VM function “is a little more complicated than the function π(x) which counts only primes, not powers of primes, and gives each prime the weight 1 not log(p)”,

and that “there is a jump by log(3) at 9, no change at 10=2*5, and a jump by log(11) at 11”.

My question is: does this mean that the VM function goes up to 1 at 1, to 2 at 2 3 at 3, levels out, goes to 4 at 7, to 10 at 9 (3 is the log of 9 so goes up 3), then up to 11 at 11 … ? I didn’t understand (not trained in this), but he is in a general sense very clear. Can you help?

• Mats Granvik says:

The von Mangoldt function is a sequences of logarithms beginning:
Log(1), Log(2), Log(3), Log(2), Log(5), Log(1), Log(7), Log(2), Log(3),
Log(1), Log(11), Log(1), Log(13), … and so on. The integer sequence
inside the (natural) logarithms: 1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, …
can be found in the oeis as sequence A014963.

http://oeis.org/

The summatory von Mangoldt function or the partial sums of the
von Mangoldt function:
Log(1)
Log(1) + Log(2)
Log(1) + Log(2) + Log(3)
Log(1) + Log(2) + Log(3) + Log(2)
Log(1) + Log(2) + Log(3) + Log(2) + Log(5)
Log(1) + Log(2) + Log(3) + Log(2) + Log(5) + Log(1)
Log(1) + Log(2) + Log(3) + Log(2) + Log(5) + Log(1) + Log(7)
Log(1) + Log(2) + Log(3) + Log(2) + Log(5) + Log(1) + Log(7) + Log(2)

sums to the following numbers respectively:

Log(1)
Log(2)
Log(6)
Log(12)
Log(60)
Log(60)
Log(420)
Log(840)

which are the values of the summatory von Mangoldt function.

The values of those numbers are:

Log(1) = 0
Log(2) = 0,69314718
Log(6) = 1,79175947
Log(12) = 2,48490665
Log(60) = 4,09434456
Log(60) = 4,09434456
Log(420) = 6,04025471
Log(840) = 6,73340189

This what I think you refer to when you mention the page
by Jeffrey Stopple.

The plot at this page of the von Mangoldt function and the
explicit formula for the summatory von Mangoldt function,
which is also called the Chebyshev function:

http://en.wikipedia.org/wiki/Chebyshev_function

uses the zeta zeros to approximate them.

Search google for “raising to a complex power”
and you should find this formula:

a^(b + I*c) = (a^b)*(Cos(c*Log(a)) + I*Sin(c*Log(a)))

at

http://www.math.toronto.edu/mathnet/questionCorner/complexexp.html

But there are several ways to compute the von Mangoldt function
as a arithmetic sequence. One way is to use the Möbius function
and another way is by describing the fundamental theorem of
arithmetic as a recurrence:

http://math.stackexchange.com/questions/164767/prime-number-generator-how-to-make

The most basic property of the von Mangoldt function is just that,
it encodes the fundamental theorem of arithmetic, so if you are new
to number theory, the von Mangoldt function is a good place to start.

• CK says:

Wow! Thanks! —

In what way does the M function “mirror” the arithmetic theorem?

And hey, can I ask also (kindly), what makes the -2, -4 … zeroes happen?
And what other zeroes are there in the Riemann function — i.e., what do they come from?

• Mats Granvik says:

I don’t know everything in number theory I wish I knew. The fundamental theorem arithmetic however can be seen also from this triangle:

http://oeis.org/A140256

which is just a different formatting than the answer I entered at math stackexchange.
If you look at the terms of the von Mangoldt function more closely, you will see that
they consist of logarithms of only primes. 1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13…
including some “ones”. That is exactly what the fundamental theorem of arithmetic says, that every number can be factored into primes.

Regarding the zeros of the Riemann zeta function, I don’t know much about them.

• CK says:

Fair enough. So where do the 1s come from after the first one? And why the 2 for 11? I mean, wouldn’t it be log of (I can see) 1 for 1, 2 for 2, 3 for 3 then wouldn’t it be log of 5 for 5, 7 for 7, actual 3 as log of 9, then log of 11 for 11 …? I think I’m missing something. (Sorry.)

And yes, the fundamental theorem of arith. says that we can factor every number into primes. But where is that showing in this? Does the 1, 2, 3, 2, 5, 1, 7 … etc. tell you something about the numbers which factor into each prime or power of primes? Does it tell you something about how many primes go in?

Thank you for helping. I think I’m getting parts of this, but these are the things I don’t see the connection between. I AM interested, at least!!! 🙂 And you’re very nice to help. Thanks!

2. CK says:

Oh — and to what base is the VM function logarithm here? Is it to base of the current p or to the base 10 or e or what? (If I missed that, sorry.)

3. Mats Granvik says:

The logarithm is the natural logarithm with base e = 2.718281828459…, always in number theory. To repeat the fundamental theorem of arithmetic. The von Mangoldt function is:

Log(1), Log(2), Log(3), Log(2), Log(5), Log(1), Log(7), Log(2), Log(3)

Then place a “0” in between every term of the sequence (and at the beginning):
0, Log(1), 0, Log(2), 0, Log(3), 0, Log(2), 0, Log(5), 0, Log(1), 0, Log(7), 0, Log(2), 0, Log(3)

And then place a “0” after every “0” that is already in there:
0, 0, Log(1), 0, 0, Log(2), 0, 0, Log(3), 0, 0, Log(2), 0, 0, Log(5), 0, 0, Log(1), 0, 0, Log(7), 0, 0, Log(2), 0, 0, Log(3)

And continue by placing a “0” after every “0, 0” that is already in there:
0, 0, 0, Log(1), 0, 0, 0, Log(2), 0, 0, 0, Log(3), 0, 0, 0, Log(2), 0, 0, 0, Log(5), 0, 0, 0, Log(1), 0, 0, 0, Log(7), 0, 0, 0, Log(2), 0, 0, 0, Log(3)

and so on.

Putting these new sequences as columns next to each other we get a matrix of numbers.
The row sums of that matrix will then be the Logarithm of the natural numbers,
Log(1), Log(2), Log(3), Log(4), Log(5), Log(6), Log(7), Log(8), Log(9), …

To get a glimpse of the over all pattern in this matrix you might want to view
this plot where black dots are equal to one, and white dots are equal to zero:

(click at the image once to zoom in)

This image is perhaps misleading, but my point is that there are certain
arithmetic sequences that when summed over this matrix also called the
divisor plot, which will yield simple sequences.

One is the Möbius function:
1, −1, −1, 0, −1, 1, −1, 0, 0, 1, −1, 0, −1, 1, 1, 0, −1, 0, −1, 0, 1, 1, −1, 0, 0, …
which when summed over the divisorplot gives the sequence:
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

Then there is the Euler totient function:
1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8,…
which when summed over the divisorplot gives the natural numbers:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,…

And finally the von Mangoldt function:
Log(1), Log(2), Log(3), Log(2), Log(5), Log(1), Log(7), Log(2), Log(3), …
which when summed over the divisorplot gives the natural logarithm
of the natural numbers:
Log(1), Log(2), Log(3), Log(4), Log(5), Log(6), Log(7), Log(8), Log(9), …

Where do the Log(1) terms come from in the von Mangoldt function?

The conventional way of defining the von Mangoldt function,
also called Lambda(n), is:

Lambda(n) = Log(p) if n = p^k for some prime p and integer k >= 1
and Lambda(n) = 0 other wise.

So if a natural number is not a power of any prime number then
the value of the von Mangoldt function is Log(1) = 0.

Example:
Lambda(6) = Lambda(2*3) = Log(1) = 0
Lambda(10) = Lambda(2*5) = Log(1) = 0
Lambda(24) = Lambda(2*2*2*3) = Log(1) = 0

while:
Lambda(8) = Lambda(2*2*2) = Log(2)
Lambda(9) = Lambda(3*3) = Log(3)
Lambda(16) = Lambda(2*2*2*2) = Log(2)
Lambda(25) = Lambda(5*5) = Log(5)

or the numbers right at the beginning:
Lambda(2) = Lambda(2) = Log(2)
Lambda(3) = Lambda(3) = Log(3)
Lambda(4) = Lambda(2*2) = Log(2)
Lambda(5) = Lambda(5) = Log(5)

Another way to put it: Log(1) = 0 is the
only number that fits in if the row sums
are to be Log(n) where “n” is a natural number.

Regarding how many primes factors go in to
a number on average, I don’t know. This could
be a known result though.