What is the Riemann xi function?

Series: Explain
Summary: A general-reader, full-length article explaining the Riemann xi function: how Euler's zeta function became Riemann's complex zeta function, why zeta is completed, how xi exposes the s ↔ 1−s symmetry, how capital Xi relates to the critical line, and how the 1859 paper was received and developed historically.
Math Level: GENERAL
Index Excerpt: Euler's zeta function leads to Riemann's complex zeta; completing it produces xi, an entire symmetric function whose zeros are exactly the non-trivial zeta zeros central to the Riemann Hypothesis.

The Riemann zeta function already gives us a remarkable connection between complex analysis and the prime numbers.

So why introduce another function?

That second function is usually written

\xi(s).

It is called the Riemann xi function.

Xi is not a rival to zeta, and it does not replace it. It is built from zeta by adding several carefully chosen factors. The resulting function has a cleaner symmetry, no pole at $s=1$ , and exactly the zeros that matter to the Riemann Hypothesis.

In that sense, xi is best understood as zeta in a more revealing form.

The reason mathematicians make this transformation takes us directly back to Bernhard Riemann, Göttingen, and one extraordinary paper written in 1859\.

A note on the notation

Mathematical notation becomes much easier to follow once the symbols have names.

The two most important ones in this article are Greek letters.

\zeta(s)

( $\zeta$ is the lowercase Greek letter zeta, pronounced roughly “ZEE-tuh”. The expression $\zeta(s)$ is read aloud as “zeta of s”. The letter $s$ is the input to the function.)

The second is

\xi(s).

( $\xi$ is the lowercase Greek letter xi, pronounced “sigh”. The expression $\xi(s)$ is read aloud as “xi of s”.)

Later we will also meet

\Xi(t).

( $\Xi$ is capital xi, again pronounced “sigh”. $\Xi(t)$ is read aloud as “capital xi of t”. Modern mathematicians often use it for the xi function viewed along the critical line.)

These letters are simply conventional names for functions, just as one might write

f(x)

for a function called $f$ .

So throughout this article:

zeta means $\zeta$ ,

xi means $\xi$ ,

and capital xi means $\Xi$ .

When another piece of notation first becomes important, we will identify it in the same way.

The story does not begin with Riemann

There is a historical trap hidden in the name Riemann zeta function.

It can sound as though Bernhard Riemann invented it.

He did not.

The story begins more than a century before Riemann's 1859 paper, with Leonhard Euler.

Euler studied infinite sums such as

1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots.

In modern notation we write this as

\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}.

( $\sum$ is the capital Greek letter sigma, pronounced “SIG-muh”. In mathematics it is used to mean “sum these terms”. Here it tells us to add $1/n^s$ for $n=1,2,3,\ldots$ .)

Euler discovered something extraordinary.

The same function could also be written entirely in terms of prime numbers:

\zeta(s) = \prod_{p\ \mathrm{prime}} \frac{1}{1-p^{-s}}.

( $\prod$ is the capital Greek letter pi, pronounced “pie”. Here it means “multiply these factors”. It plays for multiplication roughly the role that $\sum$ plays for addition.)

This is the Euler product.

The sum runs through all the positive integers.

The product runs only through the primes.

And yet they describe the same function.

That is the great arithmetic fact sitting underneath everything that follows.

The zeta function contains the primes.

For the longer version of this story, see What is the Riemann zeta function?.

What Riemann changed

Euler had found the bridge.

Riemann crossed it.

In 1859 he considered the zeta function not merely for ordinary real inputs but for complex numbers.

Instead of allowing $s$ to be only something like

2,\quad3,\quad4,

Riemann studied inputs of the form

s=\sigma+it.

( $\sigma$ is lowercase sigma, pronounced “SIG-muh”. Here it represents the real part of $s$ . The symbol $i$ is pronounced “eye” and is the imaginary unit, defined by $i^2=-1$ . The number $t$ controls the imaginary part.)

A complex input gives the zeta function an entire two-dimensional plane to explore.

This is where the object we now call the Riemann zeta function acquires its modern character.

Riemann extended zeta, by analytic continuation, far beyond the region where Euler's original infinite sum directly converges.

The resulting function is defined throughout the complex plane except at

s=1,

where it has a pole.

Once Riemann could study zeta across the complex plane, its zeros became accessible.

That was the doorway into the Riemann Hypothesis.

But it was not the question Riemann had originally set out to answer.

Göttingen, 1859

The year 1859 was a remarkable one in Riemann's life.

He was 32 when the year began and turned 33 in September.

His academic career had already been extraordinary.

ASCII FIGURE // Riemann on the road to 1859

1826        born in Breselenz
  |
1851        doctorate at Goettingen
  |         complex functions and Riemann surfaces
  |
1854        habilitation
  |         famous lecture on the foundations of geometry
  |
1857        extraordinary professor at Goettingen
  |         major paper on Abelian functions
  |
1859        Dirichlet dies
  |         Riemann succeeds him in the Goettingen chair
  |         elected corresponding member of the Berlin Academy
  |
  +-------> writes his paper on the distribution of primes
            published November 1859

The prime-number paper appeared at a moment when Riemann had just reached the highest point of his academic career.

Riemann had completed his doctorate at Göttingen in 1851 under Gauss.

His 1854 habilitation included the celebrated lecture that would eventually transform geometry.

In 1857 he published major work on Abelian functions and was appointed extraordinary professor at Göttingen.

Then, in May 1859, Peter Gustav Lejeune Dirichlet died.

Dirichlet had occupied the Göttingen chair previously held by Gauss.

On 30 July 1859, Riemann was appointed as Dirichlet's successor.

Only days later, he was elected a corresponding member of the Berlin Academy of Sciences. Kummer, Borchardt and Weierstrass had proposed him, describing him not merely as a promising mathematician but as an already mature and independent investigator who had significantly advanced mathematics. (mathshistory.st-andrews.ac.uk)

The election gave Riemann an occasion to communicate work to the Academy.

He chose prime numbers.

His resulting paper was called

Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse,

usually translated as

On the Number of Primes Less Than a Given Magnitude.

It was submitted in October and published in November 1859\. It was only a few pages long, and it would be the only paper on number theory that Riemann ever published. (claymath.org)

[1]

Riemann was trying to understand the primes

This matters because history is often told backwards.

Today Riemann's name is inseparable from the Riemann Hypothesis. That can give the impression that he began with the conjecture about zeros on the line

\operatorname{Re}(s)=\frac12.

He did not.

( $\operatorname{Re}(s)$ is read “the real part of s”. It means the left-right coordinate of the complex number $s$ . Thus $\operatorname{Re}(s)=1/2$ describes all points whose horizontal coordinate is one-half.)

Riemann's paper was about counting primes.

Mathematicians already knew that primes became less common as numbers grew larger.

Gauss and Legendre had discovered remarkably good approximations to the rate at which their density falls.

But why did those approximations work?

How accurately could the primes be counted?

What governed the difference between the smooth approximation and the actual, irregular sequence of primes?

Riemann attacked those questions by taking Euler's zeta function into the complex plane.

That move eventually revealed a startling connection:

the distribution of the primes is controlled, in part, by the zeros of zeta.

The zeros were not an unrelated curiosity.

They entered because Riemann was trying to understand prime counting.

And xi entered because Riemann wanted a better mathematical object in which to study those zeros.

Why isn't zeta already good enough?

The zeta function is extraordinarily powerful.

Analytically, however, it carries some baggage.

First, Euler's original infinite series

\zeta(s) = \sum_{n=1}^{\infty}n^{-s}

works directly only when

\operatorname{Re}(s)>1.

Riemann can extend the function beyond that region by analytic continuation, but the extended function has a pole at

s=1.

It also has the trivial zeros

-2,-4,-6,-8,\ldots

and a more complicated functional equation expressing its symmetry.

The zeros that matter to the Riemann Hypothesis are the other ones: the non-trivial zeros.

They lie inside the critical strip,

0<\operatorname{Re}(s)<1.

The zeta function therefore contains the information Riemann wants.

But it does not present that information in its cleanest possible form.

This is where completion enters.

Completing zeta

Modern mathematicians usually define the Riemann xi function by

\xi(s) = \frac12 s(s-1) \pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s).

The formula is dense at first glance, but every factor is there for a reason.

We already know the object at the far right:

\zeta(s).

That is zeta of $s$ .

Immediately beside it is

\Gamma\left(\frac{s}{2}\right).

( $\Gamma$ is the capital Greek letter gamma, pronounced “GAM-uh”. $\Gamma(s/2)$ is read aloud as “gamma of s over two”. The gamma function is a separate mathematical function. It extends the factorial idea into a much wider setting and appears naturally in the functional equation of zeta.)

Then we have

\pi^{-s/2}.

( $\pi$ is pi, pronounced “pie”, the familiar constant beginning $3.14159\ldots$ . Here it appears raised to a power depending on $s$ .)

And at the front:

\frac12s(s-1).

None of these pieces has been added arbitrarily.

Together they perform what mathematicians call a completion of the zeta function.

The point is not to replace zeta.

It is to package zeta together with the analytic factors that expose its natural symmetry.

The standard modern definition above is the one used, for example, by the NIST Digital Library of Mathematical Functions. (dlmf.nist.gov)

What has the completion actually fixed?

A useful intermediate object is

\pi^{-s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s).

This combination already exposes the fundamental reflection between $s$ and $1-s$ .

But it still has singular behaviour at the two symmetric points

s=0 \qquad\text{and}\qquad s=1.

Multiplying by

s(s-1)

removes those remaining singularities.

The factor $1/2$ is a conventional normalisation.

What emerges is something mathematically special.

The function

\xi(s)

is entire.

An entire function is a complex function with no poles or other singularities anywhere in the finite complex plane.

That is a major simplification.

Zeta has a pole at $1$ .

Xi does not.

Zeta has the trivial zeros at the negative even integers.

Those trivial zeros are absorbed by the completion rather than becoming zeros of xi.

What remain as the zeros of $\xi(s)$ are precisely the non-trivial zeros of zeta.

So xi removes much of the analytic housekeeping while preserving exactly the zero set at the heart of the Riemann Hypothesis. (dlmf.nist.gov)

This is one of the key distinctions to remember:

zeta carries the arithmetic connection to the primes; xi repackages zeta so that the important analytic structure is cleaner.

The payoff: a much simpler symmetry

Written directly for zeta, the functional equation is not especially transparent.

Written for xi, it becomes

\xi(s)=\xi(1-s).

(Read this aloud as “xi of s equals xi of one minus s.” It says that reflecting the input $s$ across the midpoint $1/2$ leaves the value of xi unchanged.)

This is the payoff of the completion.

The complicated-looking definition produces an extremely simple symmetry.

ASCII FIGURE // Completing zeta

     arithmetic connection to the primes
                  |
                  v
               zeta(s)
                  |
                  |  add the natural completion factors
                  |  gamma, powers of pi, s(s-1)
                  v
                xi(s)
             /           \
            /             \
   entire function     same non-trivial
   no pole at s=1      zeros that RH studies
            \             /
             \           /
                  |
                  v

             xi(s) = xi(1-s)

        reflection around Re(s) = 1/2

Xi is not a rival to zeta. It is zeta packaged with additional factors so that its central symmetry becomes much cleaner.

The line

\operatorname{Re}(s)=\frac12

is therefore not an arbitrary line that happens to occur in a famous conjecture.

It is the fixed centre of the reflection

s\longleftrightarrow1-s.

For example,

0.3+10i

is reflected to

0.7-10i.

The real coordinates $0.3$ and $0.7$ sit equally far from $1/2$ .

And exactly on the middle line,

\operatorname{Re}(s)=\frac12,

the reflection fixes the real coordinate itself.

This does not prove the Riemann Hypothesis.

But it explains why one-half is built so deeply into the geometry of the problem.

From xi to capital Xi

There is another transformation that makes the central line easier to study.

A point on the critical line can be written as

s=\frac12+it.

So instead of allowing $s$ to range across the whole complex plane, we can restrict it deliberately to that line.

Modern notation commonly defines

\Xi(t) = \xi\left(\frac12+it\right).

( $\Xi(t)$ is read “capital xi of t”. We have taken the complex xi function and evaluated it only at points lying on the critical line.)

For real $t$ , this new function has especially useful properties.

It is real-valued.

And it is even:

\Xi(t)=\Xi(-t).

That means its graph can be treated as an ordinary real-variable object extending symmetrically in both directions.

Most importantly, the Riemann Hypothesis can now be phrased in a striking new way.

Instead of saying:

every non-trivial zero of $\zeta(s)$ has real part $1/2$ ,

we can say:

every zero of $\Xi(t)$ is real.

These are two formulations of the same conjecture. (encyclopediaofmath.org)

That is a remarkable transformation.

We began with a question about the distribution of prime numbers.

We moved to a complex function.

We completed that function.

We restricted the completed function to its symmetry line.

And the Riemann Hypothesis became a question about whether the zeros of a particular entire function are all real.

A notation trap: Riemann did not use our symbols in quite the same way

This is worth making explicit because otherwise reading Riemann's original paper can be confusing.

Today we commonly write

\xi(s)

for the completed function on the complex $s$ -plane, and

\Xi(t) = \xi\left(\frac12+it\right)

for the corresponding function along the critical line.

Riemann's own 1859 notation was different.

He used the symbol $\xi$ for essentially the object that modern authors usually denote by capital $\Xi$ — the completed function after expressing the critical-line coordinate in terms of $t$ . (encyclopediaofmath.org)

So if Riemann's original $\xi(t)$ does not look quite like a modern textbook's $\xi(s)$ , the discrepancy is historical notation rather than different mathematics.

A useful modern family tree is:

ASCII FIGURE // Zeta, xi and capital Xi

Euler
  |
  |  infinite series + product over primes
  v
zeta(s)
  |
  |  Riemann lets s range over the complex plane
  |  and analytically continues the function
  v
complex zeta(s)
  |
  |  complete it with gamma, pi and s(s-1)
  v
xi(s)                         modern notation
  |
  |  put s = 1/2 + it
  v
Xi(t)
  |
  +------ RH becomes:
          are all the zeros of Xi real?

Riemann's 1859 notation:
his xi(t) is essentially the object
modern authors usually call Xi(t)

Modern notation separates the completed function on the complex plane, xi(s), from its critical-line form, capital Xi(t). Riemann's own notation predates this modern convention.

This is the chronology worth keeping in view.

Riemann did not invent zeta and then invent a competing function called xi.

Euler had already discovered the arithmetic object.

Riemann extended it into complex analysis.

Completion then exposed its natural symmetry.

And the xi function gave the crucial non-trivial zeros a cleaner home.

The hypothesis appears almost in passing

Once Riemann had this machinery, he studied the zeros.

He derived an asymptotic estimate for how many of them should occur up to a given height and considered the completed function in a form corresponding to modern $\Xi$ .

Then comes one of the most consequential sentences in the history of mathematics.

Riemann observed that the roots he was considering appeared to be real and wrote that it was

“very probable that all the roots are real.”

He immediately added that a rigorous proof would of course be desirable, but that after some preliminary attempts he had temporarily put the search aside because such a proof was not necessary for the immediate purpose of his investigation. (link.springer.com)

[1]

That immediate purpose was the distribution of primes.

This is an important correction to the way the story is sometimes remembered.

The logical journey was not

\text{Riemann Hypothesis} \longrightarrow \text{zeta function}.

It was much closer to

\text{prime numbers} \longrightarrow \text{Euler's zeta connection} \longrightarrow \text{complex zeta} \longrightarrow \text{completion} \longrightarrow \text{xi} \longrightarrow \text{zeros} \longrightarrow \text{Riemann Hypothesis}.

The conjecture emerged from the investigation.

It was not its starting point.

Did the mathematical world immediately realise what Riemann had done?

Not entirely.

Looking backwards from the twenty-first century, the 1859 paper seems almost impossibly dense with important ideas.

It contains the beginning of the modern complex-analytic study of prime distribution, the relationship between primes and zeta zeros, an explicit formula connecting the two, zero-counting ideas and the conjecture that became the Riemann Hypothesis.

It is tempting to imagine an immediate mathematical sensation.

The historical evidence suggests a slower process.

A recent historical study of the development of prime-number theory notes that Riemann's prime-number paper was not immediately received in that way. Among the early written responses was work by Hermann Kinkelin, but Riemann himself published nothing further in this direction. (link.springer.com)

There was also substantial mathematical work still to be done.

Riemann's paper was astonishingly compressed.

Several deep claims were stated with arguments that later mathematicians would need to place on firmer rigorous foundations.

The architecture was there.

Some of its foundations still had to be completed.

Riemann never returned publicly to the problem

Riemann's health deteriorated badly in the years that followed.

He married Elise Koch in 1862 and spent considerable periods in Italy seeking a climate more favourable to his health.

On 20 July 1866, he died of tuberculosis near Lake Maggiore.

He was 39\. (link.springer.com)

His 1859 paper remained his only published work devoted specifically to number theory.

The programme it opened did not disappear.

But some of its most important consequences would take decades to mature.

The thirty-year slow burn

The afterlife of the 1859 paper is almost as interesting as the paper itself.

ASCII FIGURE // From Riemann's paper to the Prime Number Theorem

1859   RIEMANN
       prime-number paper
       zeta, zeros, completion, explicit formula, RH
         |
1866   Riemann dies, aged 39
         |
         |        more than three decades
         |
1893   HADAMARD
       major advance in entire-function factorisation
       and the product structure surrounding xi
         |
1895   VON MANGOLDT
       rigorous development of important parts
       of Riemann's zero-counting and explicit formula
         |
1896   HADAMARD + DE LA VALLEE POUSSIN
       independent proofs of the Prime Number Theorem
         |
         v
       analytic number theory develops around
       the landscape Riemann opened

Riemann's prime-number paper was not an instant finished theory. Several of its deepest ideas became rigorous foundations for analytic number theory over the following decades.

In 1893 Jacques Hadamard made a major breakthrough in the theory of entire functions and supplied rigorous factorisation machinery of exactly the kind needed around Riemann's completed function.

In 1895 Hans von Mangoldt established rigorous versions of important parts of Riemann's explicit-formula and zero-counting programme.

Then, in 1896, Hadamard and Charles-Jean de la Vallée Poussin independently proved the Prime Number Theorem.

That theorem says, broadly, that the number of primes below a large number $x$ behaves like

\frac{x}{\log x}.

It was the great asymptotic prime-counting statement towards which the nineteenth-century work of Gauss, Legendre, Dirichlet and Riemann had been pointing.

And the successful proofs travelled through the complex-analytic territory Riemann had opened in 1859\. (claymath.org)

The immediate prime-counting problem was eventually conquered.

The conjecture Riemann had set aside because it was not essential to his immediate purpose was not.

Why xi still matters

It would be easy to treat xi as a historical curiosity: an elegant repackaging that happened to appear in Riemann's argument.

It is much more than that.

Xi belongs to an enormous mathematical tradition concerned with entire functions and the geometry of their zeros.

It makes the fundamental zeta symmetry transparent:

\xi(s)=\xi(1-s).

It removes the pole of zeta at $s=1$ .

It absorbs the trivial zeros into the completion.

It retains precisely the non-trivial zeros that matter to RH.

And in the real-variable form

\Xi(t),

it turns the Riemann Hypothesis into a real-zero problem.

Those changes do not solve RH.

They reorganise it.

That distinction is important.

A change of representation can make a mathematical structure dramatically clearer without making the underlying theorem easy to prove.

Xi is a particularly beautiful example.

Zeta and xi side by side

It is useful to put the three objects next to one another.

The Riemann zeta function

\zeta(s)

pronounced “zeta of s”,

begins historically with Euler's series and prime product.

Its Euler product exposes the deep connection with prime numbers.

Riemann extends it through the complex plane.

It has a pole at $s=1$ .

It has trivial and non-trivial zeros.

Its non-trivial zeros are the zeros involved in RH.

The modern Riemann xi function

\xi(s) = \frac12s(s-1)\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s),

pronounced “xi of s”,

is built from zeta.

It is not a separate arithmetic object competing with zeta.

It is a completion of zeta.

It is entire.

Its zeros correspond precisely to zeta's non-trivial zeros.

Its symmetry is beautifully simple:

\xi(s)=\xi(1-s).

The modern capital-Xi function

\Xi(t) = \xi\left(\frac12+it\right),

pronounced “capital xi of t”,

takes xi along the critical line.

For real $t$ , it is real and even.

And RH can be stated as:

all the zeros of $\Xi(t)$ are real.

What completion does not do

Because xi is cleaner than zeta, it is worth being precise about what has and has not been gained.

Completion removes technical clutter.

It exposes symmetry.

It gives us an entire function.

It isolates the zeros that matter.

But it does not tell us why every one of those zeros should lie on the critical line.

The equation

\xi(s)=\xi(1-s)

tells us that zeros are reflected symmetrically around that line.

It does not say that every zero must sit on the line itself.

Two zeros could, in principle, sit equally far to either side and still respect the symmetry.

That distinction is one of the reasons the Riemann Hypothesis remains difficult.

Symmetry explains why the critical line is special.

RH says something much stronger about it.

For that larger story, continue with What is the Riemann Hypothesis?.

The sequence to remember

The relationship between the objects can be summarised like this:

ASCII FIGURE // From primes to the Riemann Hypothesis

PRIMES
  |
  | Euler discovers that zeta is built from them
  v
zeta(s)
  |
  | Riemann takes zeta into the complex plane
  v
COMPLEX ZETA
  |
  | complete it
  v
xi(s)
  |
  | restrict to s = 1/2 + it
  v
Xi(t)
  |
  | ask where its zeros are
  v
RIEMANN HYPOTHESIS

A compact route through the objects in the article: primes lead to zeta, Riemann takes zeta into the complex plane, completion produces xi, restriction to the critical line produces capital Xi, and RH asks where the relevant zeros lie.

Zeta carries the arithmetic connection to the primes.

Xi reorganises zeta into a cleaner, entire and symmetric function.

Capital Xi lets us view that completed function directly along the critical line.

And the Riemann Hypothesis asks whether the zeros relevant to the problem all live there.

The historical irony

Riemann's 1859 paper was written to investigate the distribution of prime numbers.

The tools he introduced helped transform that subject.

By 1896, the broad prime-counting law that motivated much of the story had been proved.

But one observation inside Riemann's tiny paper survived all of that progress.

He thought it was very probable that all the relevant roots were real.

He could not prove it.

He did not need the proof for his immediate purpose.

So he put it aside.

More than a century and a half later, mathematicians have not been able to put it aside with him.

The immediate problem Riemann attacked was eventually solved.

The sentence he could not prove became the larger problem inherited by everyone who came after him.

References

Bernhard Riemann. “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the Number of Primes Less Than a Given Quantity).” Monatsberichte der Berliner Akademie / Clay Mathematics Institute manuscript collection (1859). Source