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Riemann's Explicit Formula

The Von Mangoldt function Ψ(x) counts, for each x>1, the powers of prime numbers p which are less than x, each one weighted by the natural logarithm log(p). This 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). None the less, Ψ(x) is easier to work with analytically. A. E. Ingham, in his classic book "The Distribution of Prime Numbers", says "It happens that, [the function] which arises most naturally from the analytical point of view is the one most remote from the original problem, namely Ψ(x). For this reason it is usually most convenient to work in the first instance with Ψ(x), and to [then] deduce results about π(x). This is a complication which seems inherent in the subject, and the reader should familiarize himself at the outset with the function Ψ(x), which is to be regarded as the fundamental one."

The movie shows the graph of Ψ(x) over larger and larger ranges of x values. On the smallest scale one can see the jump at each prime number or power of a prime number. For example, there is a jump by log(3) at 9, no change at 10=2*5, and a jump by log(11) at 11. On the larger scales one can see the function seems to be very nearly approximated by the line y=x.

This is a theorem; the fact that the Riemann zeta function has a pole at s=1 means that Ψ(x) is asymptotic to x, and from this one can deduce the Prime Number Theorem, that π(x) is asymptotic to x/log(x). But even more is true; Riemann's Explicit Formula gives the exact value of the function Ψ(x) in terms of the zeros of the zeta function. The deviation from the "main term", x, is given as a sum of 'quasi-periodic' functions


corresponding to each zero β+iγ of ζ(s).

The movie above shows the contribution of the first 100 zeros to Ψ(x), added one at a time. The further out you go on the x axis, the more zeros you need to get a good fit. The movie below show the contribution of the same first 100 zeros to Ψ(x), but now for x between 100 and 120. The exact graph of Ψ(x) is shown in red for comparison purposes.

Music of the Primes

If the Riemann Hypothesis is true, each zero has real part β=1/2. This could be factored out of each term in the explicit formula, writing

Ψ(x)=x+Sqrt[x] times a sum of periodic functions.

This says the distribution of the prime numbers is amazingly regular.

A physicist will think of a sum of periodic functions as a superposition of waves, a vibration or sound. This is what the physicist Sir Michael Berry meant by "... we can give a one-line nontechnical statement of the Riemann hypothesis: The primes have music in them."

Using the Audio package in Mathematica it is not hard to create a sound file in which each note has the same amplitude and frequency as the corresponding term in the explicit formula, coming from a single zero of the zeta function. In this sound file there are the contributions of the first 100 zeros, added one at a time, in intervals of .2 seconds. Finally all 100 zeros play together for ten seconds. This sound is best listened to with headphones or external speakers. For maximum effect, play it LOUD.

You can download this as an MP3 file.