Harmonic Primes

[Natalie Omahony]

Thiswas proved by Leonhard Euler in 1737,[1] and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers and Nicole Oresme's 14th-century proof of the divergence of the sum of the reciprocals of the integers (harmonic series).

Such infinite products are today called Euler products. The product above is a reflection of the fundamental theorem of arithmetic. Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

Euler considered the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the natural logarithm of each side, then he used the Taylor series expansion for log x as well as the sum of a converging series:

It is almost certain that Euler meant that the sum of the reciprocals of the primes less than n is asymptotic to log log n as n approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by Franz Mertens in 1874.[3] Thus Euler obtained a correct result by questionable means.

Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as log log n. The proof is due to Ivan Niven,[4] adapted from the product expansion idea of Euler. In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes.

Another proof rewrites the expression for the sum of the first n reciprocals of primes (or indeed the sum of the reciprocals of any set of primes) in terms of the least common denominator, which is the product of all these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime does divide the denominator. Thus the expression is irreducible and is non-integer.

Fortunately, it is not necessary to wait upon the answers to these admittedly very interesting questions before finding interesting uses for the prime harmonic series as material for scale generation. For example, through the magic of octave reduction, the first n members of the prime harmonic series can produce an n-atonic scale:

"The Prime Guitar is designed to play prime partials 17-199. The lowest string is tuned to the 17th partial of a series, and the successive strings are then tuned to partials 23, 37, 47, 67, and 89 to preserve the basic tonal spread of a standard set of guitar strings. Since every single note is a unique prime limit pitch, it was not practical to color code the frets in the same way as the previous guitars. On the other hand, it is very difficult to orient one's self on a neck with such irregular fret spacing without any color coding, so I settled on a simple alternating three color scheme that allows for easy visual spotting of where the frets are located.

"Six Primes is composed using the first six prime numbers 2, 3, 5, 7, 11, and 13 to govern both its tuning and temporal structure, including harmony, rhythmic subdivisions, and form. I wrote this music to explore the limits of using the same integer ratios to simultaneously provide melodic, harmonic, and rhythmic materials. The piano must be retuned in just intonation using the tuning system factor of 2, three ratios with highest prime of 3, and two ratios each with highest primes of 5, 7, 11, and 13. This creates a great diversity of interval relationships: whereas 12-tone equal temperament has just twelve distinct intervals, this tuning has 75.

I understand that applying the Mellin Transform to the partial sum of the van Mangoldt function Λ(x) makes it complex-analytic and related to zeta, so we can study the result's asymptotic behaviors and then take the Mellin Inversion ride back to the sum of primes. This makes sense to me on a surface level.

I think it would make more immediate intuitive sense to me if applying the Mellin Transform to a function representative of the group of positive integers gave prime distribution, considering that I conceptualize the primes as some kind of multiplicative bases, but it is confusing to me that in fact we should be going the opposite way.

I've been wracking my head with this problem for a few weeks and also have read up on everything I could on the Mellin Transform and the PNT, but no search of mine turned up yet an exact answer to this particular confusion.

If anyone could provide any tips/directions/hints or reading recommendations that might reveal some fundamental intuition about why it is valid to apply the Mellin Transform to prime distribution, I'd really appreciate it.

One can indeed motivate its presence in the theory of prime numbers in this form. However, we're not interested in diagonalizing dilations themselves but rather operators constructed as sums of dilations. You can consider the operator $$ f\mapsto \sum_n=1^\infty f(nx)$$ (which is rarely well-defined near zero but often well-defined away from $0$) which is the composition of the operators $$ f\mapsto \sum_k=1^\infty f(p^k x)$$ for prime $p$. Each of these can be obtained as the formal exponential of the operator $$f \mapsto \sum_k=1^\infty f(p^k x)/k$$ (which you might prefer to think of as the integral of the flow $$\fracddt f(x,t) =\sum_k=1^\infty f(p^k x,t)/k$$ from $t=0$ to $t=1$) and so the whole operator is the formal exponential of $$ f\mapsto \sum_p \sum_k=1^\infty f(p^k x)/k. $$

It is this last operator which has the property that understanding it would clearly tell us a lot about how to count the primes, and therefore diagonalizing it would be a reasonable choice, motivating the Mellin transform. The reason for using this operator, rather than anything else, to study sums over primes is that exponentiating it gives a nice operator expressed in terms of sums over all numbers that we can hope to understand using additive Fourier analysis or other tools.

If you have any sum, you wonder what you can do to it. (That's maths right?:D) Whenever you have a sum you can do Fourier analysis on, you can try applying Fourier's Inversion Theorem - for the primes, this is all Mellin inversion, i.e. Perron's formula. For divisor functions, it's Voronoi's summation formula. In each case you started with a sum and now have a new sum - the dual sum.

Maybe the new sum is better, maybe it's not. If it is better and you get new results, then later people can look at it more philosophically/deeply and try to figure out why it worked, what really goes on, etc., probably in the hope of generalising it to other situations, but in practice the reason why it was attempted in the first place was maybe just to see what happens.

So you consider the Dirichlet series just to see what you can say about the new sum after applying an inversion theorem - a contour integral for anything you're looking at the Dirichlet series of. Of course, it was Riemann's insight to realise you can move the contour by extending $\zeta (s)$.

Additive structure means $z^n z^m=z^n+m$, multiplicative structure means $n^-sm^-s=(nm)^-s$. The Mellin transform of $z^n$ (after the change of variables $z\mapsto e^-x$) is just $n^-s$ (up to a factor $\Gamma(s)$). In particular, the Mellin transform of a generating function $\sum_n a_n z^n$ gives the (generalized) zeta function $\sum_n a_n n^-s$.

It's well known that the harmonic series diverges. We may thin it somewhat by removing all terms with denominators divisible by 2. What is left over is the series of the reciprocals of the odd whole numbers:

which should be understood as a manipulation of partial sums of the two series. The partial sums of the reciprocals of the odd numbers are greater than the corresponding sums of the harmonic series divided by 2. Since the latter diverge, so do the former.

I'll use the classic argument from Hardy and Wright to show that the remaining series, i.e. the series of the reciprocals of the primes is still divergent. From here it will follow that the number of primes cannot be finite. (This prooof is also discussed at length in Ingenuity in Mathematics, and a different proof can be found in, say Proofs from THE BOOK.)

Many people have commented over the ages on the similarities between mathematics and music. Leibniz once said that "music is the pleasure the human mind experiences from counting without being aware that it is counting". But the similarity is more than mere numerical. The aesthetics of a musical composition have much in common with the best pieces of mathematics, where themes areestablished, then mutate and interweave until we find ourselves transformed at the end of the piece to a new place. Just as we listen to a piece of music over and over, finding resonances we missed on first listening, mathematicians often get the same pleasure in rereading proofs, noticing the subtle nuances that make the piece hang together so effortlessly.

The one advantage that music has over mathematics is the physical connection that our body has with the sound of a composition. The hairs on the back of my neck never fail to stand on end when I hear Schubert's Death and the Maiden Quartet. The tingle factor stimulated by some pieces of music is something mathematics is rarely able to match. This is why those without any musicaltraining can respond to a concert performance, whereas only after years of mathematical training does one eventually have the ears to listen to the great mathematical compositions.

It is one of the failings of our mathematical education that few even realise that there is such wonderful mathematical music out there for them to experience beyond schoolroom arithmetic. In school we spend our time learning the scales and time signatures of this music, without knowing what joys await us if we can master these technical exercises. Very few would have the patience to learn thepiano if they were denied the pleasure of hearing Rachmaninov.

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