The Zeta Function and the Music of the Primes

Q: How are the zeta function and the primes connected?

Euler's product ζ(s) = ∏_p (1 − p^(−s))^(−1) makes a product over all primes p equal to the zeta function. Because the entire information of the primes is packed into this single function, analyzing the zeta function in the complex plane bears directly on the distribution of the primes. The prime number theorem also follows from the properties of this function.

Q: How is the geometry in the figure related to the zeta function?

Geometry is not decoration: it gives the strongest reason to believe the Riemann hypothesis. Instead of the world of numbers Spec Z, consider a curve over a finite field F_q (a purely geometric object); it too carries a zeta function of the same shape, with the points of the curve playing the role of the primes. For this geometric zeta, the analogue of the Riemann hypothesis — that all zeros lie on the critical line — has been completely proved by Weil and Deligne (the proof is thoroughly geometric, via étale cohomology). Another bridge is spectral geometry: on a hyperbolic surface, the Selberg trace formula links the lengths of closed geodesics with the eigenfrequencies of the Laplacian, mirroring the explicit formula (primes ↔ closed geodesics, zeros ↔ eigenfrequencies). The Hilbert–Pólya dream of seeing the zeros as the spectrum of some space lives on this geometric picture.

Number Theory Resonating in the Complex Plane

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Published: 2026-06-16

The primes $2, 3, 5, 7, 11, 13, \dots$ never reveal a rule, however long you stare at them. They refuse to tell you where the next one will land — among the most capricious residents of mathematics.

And yet in the nineteenth century Bernhard Riemann found, beneath that caprice, an order no one had imagined. The sequence of primes was inscribed — like the notes of a score — in the zeros of a single function placed in the complex plane: the zeta function. This article savors, in a relaxed way, how complex analysis came to touch the very heart of number theory.

The Bridge Euler Built

The story begins before Riemann, with Leonhard Euler. He noticed that the sum over all natural numbers $\displaystyle \zeta(s) = \sum_{n=1}^{\infty} n^{-s}$ can be written as a product over all primes.

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p:\,\text{prime}} \frac{1}{1 - p^{-s}}.$$

The left side speaks of every natural number; the right side, of the primes alone. This equality holds because every natural number factors uniquely into primes — the fundamental theorem of arithmetic. So this single identity translates the very fact that "the primes are the atoms of multiplication" into the language of analysis.

The whole information of the primes is packed into one function. Examine it carefully with the tools of complex analysis, and the secrets of the primes should be within reach — this idea is the bridge Euler built, and the bridge Riemann crossed.

A note: zeta is the transform of a "comb on a log scale"

Asking "what is zeta a transform of?" opens up the signal-processing view one more notch. The key is that $n^{-s}=e^{-s\ln n}$.

The most natural reading is the Laplace transform of a "comb" (a Dirac comb): $$\zeta(s)=\sum_{n=1}^{\infty} e^{-s\ln n}=\mathcal{L}\!\left[\sum_{n=1}^{\infty}\delta(t-\ln n)\right](s).$$ It is the Laplace transform of a comb of unit impulses placed at $t=\ln 1,\ln 2,\ln 3,\dots$ (top panel of Figure 1). To smooth it, one can also write $\zeta(s)=s\displaystyle\int_0^\infty \lfloor e^{t}\rfloor\,e^{-st}\,dt$ with a step function.
For the comb of primes, take $-\zeta'/\zeta$: $$-\frac{\zeta'}{\zeta}(s)=\sum_{n}\Lambda(n)\,n^{-s}=\mathcal{L}\!\left[\sum_{p^{k}}\ln p\,\cdot\,\delta(t-\ln p^{k})\right](s).$$ Here $\Lambda$ is the von Mangoldt function ($\Lambda(n)=\ln p$ if $n=p^{k}$ is a prime power, and $0$ otherwise). An impulse of height $\ln p$ stands at the logarithm of each prime power $p^{k}$ — the primes themselves line up as a pulse train on a logarithmic scale (bottom panel of Figure 1).
The cleanest integral form is the Mellin transform (the "multiplicative twin" of Laplace): $\Gamma(s)\,\zeta(s)=\displaystyle\int_0^\infty \frac{t^{s-1}}{e^t-1}\,dt$. The integrand's $\dfrac{1}{e^t-1}=\displaystyle\sum_{n\ge1}e^{-nt}$ is the Bose–Einstein distribution from physics (the one in Planck's black-body radiation), and the formula falls out simply by summing $n^{-s}\Gamma(s)=\displaystyle\int_0^\infty t^{s-1}e^{-nt}\,dt$ over $n$.

So zeta turns out to be the transform of "a comb of integers (or primes) lined up on a logarithmic scale." Take that comb as the input and the zeros as frequencies, and its inverse transform is the wave of the next section (Figure 2) — entrance and exit joined by a single thread.

A graph of the comb on a logarithmic scale that the zeta function transforms — Figure 1: The "comb" that zeta transforms. The **top** is $\zeta(s)=\mathcal{L}\bigl[\sum_{n}\delta(t-\ln n)\bigr]$ — a train of unit impulses placing each natural number at $t=\ln n$ (on a log scale, so they crowd together to the right). The **bottom** is the comb of prime powers corresponding to $-\zeta'/\zeta(s)$, with a pulse of height $\ln p$ at $t=\ln p^{k}$ (red = prime $p$, orange = prime power $p^{k}$). The primes themselves appear as a pulse train on a logarithmic scale.

Analytic Continuation — Widening a Function's World

However, the sum above converges only where the real part of $s$ exceeds $1$ ($\operatorname{Re}(s) > 1$). The mystery of the primes sleeps somewhere else entirely. What Riemann used to get there is analytic continuation.

Recall the "rigidity" of holomorphic functions seen in the basic level. An ordinary (real) smooth function can be nudged near a single point without affecting things far away — it is, in a sense, "soft." A holomorphic function is the opposite: its behavior on the tiniest region constrains the whole function. The identity theorem says it outright: if two holomorphic functions agree on a small region (or on a sequence with an accumulation point), they agree completely throughout the domain. This strong constraint is the "rigidity." So the zeta function, at first definable only on the limited region $\operatorname{Re}(s) > 1$, extends — preserving holomorphy, in exactly one way — to almost the entire complex plane, leaving behind a single pole, at $s=1$.

In this widened world, the zeta function vanishes effortlessly at the negative even integers $s = -2, -4, -6, \dots$. Because these are well understood, they are called the trivial zeros. The real problem is the zeros hidden elsewhere — the non-trivial zeros. It is precisely these notes that play the music of the primes.

A note: calling the zeros "notes" is no metaphor

The function that measures the distribution of the primes — the Chebyshev function $\psi(x) = \displaystyle\sum_{p^k \le x} \ln p$ (roughly, "a weighted count of the primes and their powers up to $x$") — can be written using the zeros of the zeta function not as an approximation but exactly, via the explicit formula (the von Mangoldt formula):

$$\psi(x) = x \;-\; \sum_{\rho} \frac{x^{\rho}}{\rho} \;-\; \ln(2\pi) \;-\; \tfrac{1}{2}\ln\!\left(1 - x^{-2}\right).$$

The leading roles are the smooth main term $x$ and the sum over all non-trivial zeros $\rho$, namely $\displaystyle\sum_{\rho} \frac{x^{\rho}}{\rho}$. If a zero lies on the critical line it can be written $\rho = \tfrac{1}{2} + i\gamma$, so that $x^{\rho} = \sqrt{x}\,x^{i\gamma} = \sqrt{x}\,\bigl(\cos(\gamma \ln x) + i\sin(\gamma \ln x)\bigr)$. That is, each individual zero produces an oscillation (a wave) wrapped in $\sqrt{x}$, and its frequency is set by the zero's height $\gamma$ (the imaginary part). Pairing it with the conjugate zero $\bar\rho$ turns the wave into a clean cosine. The distribution of the primes is thus the superposition of waves produced by countless zeros — an ensemble you can almost hear. "The music of the primes" is simply this fact, put into words.

A graph of the reconstruction of psi(x) via the explicit formula and the waveform of the oscillation produced by the zeros — Figure 2: The explicit formula $\psi(x)=x-\sum_{\rho} x^{\rho}/\rho-\cdots$, reconstructed by truncating at the first few non-trivial zeros. The **top** shows the Chebyshev function $\psi(x)$ (a staircase that jumps by $\ln p$ at each prime power $p^k$) being approximated by the waves of the zeros; the **bottom** shows the oscillating part with the linear main term removed — the "sound" alone, the wave produced by each zero $\rho=\tfrac{1}{2}+i\gamma$. The more zeros you add (10 → 30 → 100), the more the wave converges to the steps of the primes. The sawtooth appearance comes from $\psi(x)$ jumping up by $\ln p$ at each prime power while sloping down between them by the main term $-x$ — each tooth corresponds to one prime power.

Listen to the Music of the Primes

Taking the waveform in the bottom panel of Figure 2 (the oscillating part), removing the $\sqrt{x}$ amplitude envelope, and sounding each zero $\rho=\tfrac{1}{2}+i\gamma$ at a frequency equal to its height $\gamma$ — you can play the result below.

Because $x$ is mapped to time, the frequency $\gamma/x$ falls steadily (about 6.3 octaves overall, down to the audible floor of 20 Hz), and as the pitch drops the higher partials descend one after another into the audible range. This texture, where partials in non-integer ratios with one another densely overlap into a sound-mass, echoes Iannis Xenakis's glissando clouds for strings (Metastaseis, Pithoprakta) and granular synthesis; Xenakis himself brought number theory into composition through his "sieve theory" built from residue classes.

A note: flip the zeros into "poles" and they sound

In the language of signals, what produces sound (a mode) under the inverse Laplace transform is a pole of the function. But for $\zeta(s)$ a zero is a "vanishing point," not a pole (its only pole is at $s=1$). So if we turn the zeros into the denominator, flipping them into poles, the thing finally begins to sound.

$\dfrac{1}{\zeta(s)} = \displaystyle\sum_{n=1}^{\infty}\frac{\mu(n)}{n^s}$ ($\mu$ is the Möbius function). Zeros and poles swap, and inverting produces, at each zero, an oscillation of the form $x^{\rho}=\sqrt{x}\,e^{i\gamma\ln x}$ — on the log scale $t=\ln x$ it reads $e^{t/2}\cos(\gamma t)$. Its partial sum is the Mertens function $M(x)=\displaystyle\sum_{n\le x}\mu(n)\approx\sum_{\rho} \frac{x^{\rho}}{\rho\,\zeta'(\rho)}$.
$-\dfrac{\zeta'}{\zeta}(s)=\displaystyle\sum_{n}\frac{\Lambda(n)}{n^s}$ also has poles at the zeros $\rho$ of $\zeta$. Moreover its residue (the residue of complex analysis — the coefficient on $\frac{1}{s-\rho}$ near the pole) equals the order of the zero, which for a simple zero is cleanly $1$ — it does not vary from zero to zero, unlike the residue $1/\zeta'(\rho)$ of $1/\zeta(s)$. So the pole standing at each zero $\rho$ produces exactly one note $x^{\rho}/\rho$ with equal weight. Their sum (the inverse transform) is the explicit formula above, $\psi(x)=x-\displaystyle\sum_{\rho}\frac{x^{\rho}}{\rho}-\ln(2\pi)-\tfrac{1}{2}\ln\!\left(1-x^{-2}\right)$ — the standard "music of the primes" seen in Figure 2.

Either note is $x^{\rho} = x^{\operatorname{Re}\rho}\,e^{i\gamma\ln x}$. The Riemann hypothesis — "every zero has $\operatorname{Re}\rho=\tfrac{1}{2}$" — is nothing but the claim that every note sounds within the same $\sqrt{x}$ envelope, with none growing unduly loud (equivalent, for the Mertens function, to $M(x)=O(x^{1/2+\varepsilon})$).

The Critical Line and the Riemann Hypothesis

Where are the non-trivial zeros? Every one examined so far lines up, neat and well-behaved, on the vertical line of real part exactly $\tfrac{1}{2}$ — the critical line.

What makes $\tfrac{1}{2}$ special in the first place is zeta's functional equation — the symmetry relating $\zeta(s)$ and $\zeta(1-s)$ — which forces the non-trivial zeros to lie symmetrically about the line $\operatorname{Re}(s)=\tfrac{1}{2}$. The Riemann hypothesis says that all the zeros sit, without exception, exactly on that axis of symmetry.

The Riemann hypothesis asserts that this is no accident but a necessity: that "all non-trivial zeros lie on the line of real part $\tfrac{1}{2}$." It was posed in 1859. Since then, an enormous number of zeros have been verified by computation and deep theory has been built up, yet it remains neither proved nor disproved. It is one of the Clay Mathematics Institute's Millennium Prize Problems, and many mathematicians call it "the most important open problem."

Why so weighty? If all the zeros lie on the critical line, the fluctuation of the prime distribution is held as small as it can possibly be — the error term in the prime number theorem can be estimated with the utmost precision. The claim is that the capricious melody of the primes is in fact tuned to the highest standard. Looking further into the analysis of the Riemann zeta function, or into the xi function that lays bare zeta's symmetry, makes the elegance of this tuning stand out all the more.

What Do Mathematicians Sense Here?

Mathematicians who face this problem share a certain feeling: that "the zeros of the zeta function are not merely an analytic object but the shadow of something deeper."

For instance, the statistics of the spacings between zeros resemble, uncannily, the statistics of eigenvalues of random matrices that appear in quantum physics. Beneath the primes — a purely arithmetic object — the same face as a physical spectrum shows through. The quest to capture the zeros as eigenvalues of some operator (the Hilbert–Pólya dream) still draws researchers in.

What is here is the fact that complex analysis is not mere calculation but a language that secretly binds fields together. Analysis, number theory, geometry, and physics intersect at the single point of the zeta function. Researchers at the frontier stand at that crossing, feeling out the contour of a great structure no one has yet named.

A 3D figure of four planes representing analysis, number theory, geometry, and physics intersecting at a single point, the zeta function — Figure 3: An image of four fields — analysis, number theory, geometry, and physics — intersecting at the single point of the zeta function $\zeta$. Each colored plane stands for a field, and the glowing point at the center represents $\zeta$ (the origin shared by every plane).

A note: a paper of just nine pages

Riemann set down the core of this theory in a single paper of about nine pages, in 1859. Into a bold outline with the proofs left out, he packed analytic continuation, the functional equation, and the relation between zeros and primes — ideas that would become the backbone of later number theory. Short, unfinished, and yet looking impossibly far ahead. It is a rare case in the history of mathematics of "one paper giving birth to a whole field."

Coda — Music Still Playing

The sequence of primes, which looked so capricious, was inscribed as notes in the zeros of a single function placed in the complex plane. A melody that could never be heard over the reals resounds for the first time through the ear of complex analysis. And the last line of that score — whether all the notes line up on a single line — has stayed blank for more than a century and a half.

The journey of complex analysis began with the hesitation of imaginary numbers, passed through the rigidity of holomorphic functions and the magic of residues, and arrived at last at the music of the primes. Beyond here lies a score no one has yet been able to write in full. The one who hears out its continuation may, perhaps, be you, reading this.

Frequently Asked Questions

How are the zeta function and the primes connected?

Euler's product $\zeta(s) = \prod_p (1 - p^{-s})^{-1}$ makes a product over all primes $p$ equal to the zeta function. Because the entire information of the primes is packed into this single function, analyzing the zeta function in the complex plane bears directly on the distribution of the primes. The prime number theorem also follows from the properties of this function.

What does the Riemann hypothesis assert?

It asserts that the non-trivial zeros that appear when the zeta function is analytically continued to the whole complex plane all lie on the line of real part $\tfrac{1}{2}$ (the critical line). If true, the error in the distribution of the primes is bounded as tightly as possible. Posed over a century and a half ago, it has still been neither proved nor disproved — one of the greatest open problems in mathematics.

How is the geometry in the figure related to the zeta function?

Geometry is not decoration: it gives the strongest reason to believe the Riemann hypothesis is true. Instead of the world of numbers $\mathrm{Spec}\,\mathbb{Z}$, consider a curve over a finite field $\mathbb{F}_q$ (a purely geometric object); it too carries a zeta function of the same shape, with the "points" of the curve playing the role of the primes. For this geometric zeta, the analogue of the Riemann hypothesis — that all zeros lie on the critical line — has been completely proved by Weil and Deligne (the proof is thoroughly geometric, via étale cohomology). The real existence of a world where "translated into the language of geometry, the conjecture was a theorem" is the foothold for believing the original.

A second bridge is spectral geometry. On a curved surface (a hyperbolic surface), the Selberg trace formula links "the lengths of closed geodesics" with "the eigenfrequencies of the Laplacian," mirroring the explicit formula for the primes — primes ↔ closed geodesics, zeros ↔ the surface's eigenfrequencies (spectrum). The Hilbert–Pólya dream — that the zeros might be the frequencies at which some curved space rings — lives on this geometric picture. The "music of the primes" heard in the body of the article is supported, from behind, by geometry.

Has the Riemann hypothesis been proved yet?

No. As of 2026 it remains open. An enormous number of zeros have been verified by computation to lie on the critical line, but there is no proof that all of them do. It is one of the Clay Mathematics Institute's Millennium Prize Problems, with a one-million-dollar prize for a solution.

Why is the Riemann hypothesis important?

If the Riemann hypothesis is true, the fluctuation in the distribution of the primes — the error term in the prime number theorem — is bounded as tightly as possible. Moreover, countless theorems about the zeros of zeta have been proved conditionally on it, so a proof would settle a broad swath of number theory at once, with wide repercussions including applications of primes in cryptography.