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Who was Leonhard Euler?

Series
People Behind the Mathematics
Summary
A beginner-friendly portrait of Leonhard Euler: his life across Basel, St Petersburg and Berlin, the Basel problem, and the discovery that connects the zeta function to prime numbers through the Euler product.
Math Level
GENERAL
Index Excerpt
Leonhard Euler; Euler product; Basel problem; Letters to a German Princess; Variae observationes circa series infinitas

Euler is one of those names that mathematics seems determined to put in your path.

You can encounter an Euler product while studying prime numbers, Euler's formula while learning about complex numbers, Euler angles while thinking about rotation, Euler's number ee in growth and logarithms, and Euler's name attached to equations, lines, functions, graphs and theorems across subjects that, at first sight, do not seem to have much to do with one another.

If this is your first encounter with him, however, none of that is useful yet. The important starting point is much simpler: Leonhard Euler was an eighteenth-century Swiss mathematician who spent most of his working life far from Switzerland, first in St Petersburg, then in Berlin, then in St Petersburg again, and whose habit of looking for connections between apparently different problems helped shape the mathematical language we still use today. [1]

ASCII PORTRAIT // Leonhard Euler

ASCII portrait of Leonhard Euler, based on Jakob Emanuel Handmann's 1753 pastel portrait in the Kunstmuseum Basel, using a tight face-priority crop for increased facial detail.

Leonhard Euler, after Jakob Emanuel Handmann, 1753 (Kunstmuseum Basel). Terminal-rendered interpretation of the historical portrait.

A childhood near Basel

Euler was born in Basel in 1707, but he grew up in Riehen, a village just outside the city, where his father, Paul Euler, was a Protestant minister. Paul had studied some mathematics himself under Jakob Bernoulli, one of a celebrated family of mathematicians based in Basel, and he taught his son elementary mathematics alongside the other subjects expected in a minister's household. [1]

The original plan was not for Leonhard to become a mathematician at all. He entered the University of Basel while still very young, completed a master's degree in philosophy, and began studying theology because his father expected him to enter the church; mathematics, however, kept drawing him back, and Johann Bernoulli — Jakob's younger brother, and one of the strongest mathematicians in Europe — recognised enough ability in the young Euler to encourage him, guide his reading and eventually help persuade his father that mathematics was where his future lay. [1]

That small network matters. Mathematics in Euler's youth was not an anonymous body of knowledge arriving through textbooks and websites; it travelled through letters, lectures, personal introductions and families of scholars, and Euler's relationship with the Bernoullis would help carry him from a quiet Swiss upbringing into a much wider European scientific world.

From Riehen to St Petersburg

In 1727, aged twenty, Euler left Switzerland for St Petersburg, where the new Imperial Academy of Sciences had been established only a few years earlier. The journey took him away from the places and language of his childhood and into a city that was itself still being built into a scientific capital, but he did not arrive alone in an intellectual sense: Daniel Bernoulli, Johann's son and already Euler's friend, was there, as were mathematicians and scientists including Jakob Hermann, Christian Goldbach and Joseph-Nicolas Delisle. [1]

It was an unusually fertile group. Euler and Daniel Bernoulli shared interests in mathematics and mechanics; Goldbach became a long-term correspondent and helped draw Euler further into questions about whole numbers and primes; Delisle worked in astronomy and geography. Around them, the Academy expected mathematics to be useful as well as abstract, so Euler's working life ranged across maps, ships, mechanics, astronomy, acoustics and practical state projects as well as the number theory for which he is now remembered.

He married Katharina Gsell in St Petersburg in 1734, and family life grew around the work. They had thirteen children, although only five survived infancy, and the picture that emerges from contemporary and later accounts is not of a mathematician sealed away from ordinary life but of a man calculating, writing and thinking amid a busy household, carrying an enormous correspondence and moving repeatedly between theoretical questions and the practical demands of an academy. [1]

ASCII FIGURE // Euler's working life moved across three European centres

A simple timeline showing Euler's childhood near Basel, his first period in St Petersburg, twenty-five years in Berlin, and his return to St Petersburg.

Euler's career moved from Basel and Riehen to St Petersburg, then Berlin, then back to St Petersburg.

Berlin, books and a teacher who explained things

By 1741 Euler's reputation had spread far beyond Russia, and Frederick the Great invited him to Berlin, where a new academy was being developed. Euler would spend twenty-five years there. His responsibilities were broad enough to make the modern distinction between pure mathematician, scientist, engineer and administrator feel rather artificial: he supervised scientific work, advised on practical problems, dealt with maps and calendars, wrote about planetary motion, artillery, ships, light, sound and fluids, and at the same time produced hundreds of mathematical papers and several major books. [1]

One of the most revealing things he wrote in Berlin, though, was not aimed at professional mathematicians. Between 1760 and 1762 he wrote 234 letters, now known as Letters to a German Princess, as a continuation of lessons he had been giving within the Brandenburg-Schwedt household; surviving testimony from Nicolas Fuss identifies the elder daughter, Friederike Charlotte, who was about fourteen when the correspondence began, as the principal recipient. Euler moved from elementary ideas about size and measurement into light, sound, gravity, tides, electricity, magnetism, philosophy and other subjects, and the letters were later published and widely translated. [2]

That matters for the way we should read Euler today. He could work at an extremely high mathematical level, but he also understood that explanation has to begin where the learner is, not where the expert has arrived. So, before we get to the Euler product — the idea that matters most directly for Riemann Console — we are going to do the same thing.

Start with the building blocks: what is a prime?

A prime number is a whole number greater than 1 that can be divided exactly only by 1 and by itself.

The first few primes are

2,  3,  5,  7,  11,  13,  17,…2,\;3,\;5,\;7,\;11,\;13,\;17,\ldots

Numbers such as 6, 12 and 30 are not prime, because they can be built by multiplying smaller whole numbers together:

6=2×3,12=2×2×3,30=2×3×5.6=2\times3,\qquad 12=2\times2\times3,\qquad 30=2\times3\times5.

Writing a whole number as a multiplication of primes is called its prime factorisation.

(A factor is simply a number being multiplied. In 12=2×2×312=2\times2\times3, the three factors are 22, 22 and 33.)

There is a deeper rule underneath this. Every whole number greater than 1 can be broken into primes in essentially one way: the order may change, but the collection of prime factors cannot. That fact is called the Fundamental Theorem of Arithmetic.

So primes are not merely unusual numbers scattered along the number line. They are the multiplication building blocks of all positive whole numbers.

An endless sum

Now take a different-looking idea.

Suppose we square each positive whole number and then take its reciprocal:

1,14,19,116,125,…1,\quad \frac{1}{4},\quad \frac{1}{9},\quad \frac{1}{16},\quad \frac{1}{25},\ldots

(To square a number means to multiply it by itself: 42=4×4=164^2=4\times4=16. A reciprocal turns a non-zero number upside down as a fraction: the reciprocal of 1616 is 1/161/16.)

Then add those reciprocals:

1+14+19+116+125+⋯1+\frac14+\frac19+\frac1{16}+\frac1{25}+\cdots

The dots mean that the addition never reaches a final term. This kind of never-ending addition is called an infinite series.

(An infinite series can contain infinitely many terms and still approach a finite total. “There are infinitely many things to add” and “the answer is infinitely large” are not the same statement.)

This particular problem became famous in Euler's home city and is now known as the Basel problem. Mathematicians knew the sum settled towards a finite number; the difficult question was whether that number had a clean exact form. Euler found one:

1+14+19+116+⋯=π26.1+\frac14+\frac19+\frac1{16}+\cdots=\frac{\pi^2}{6}.

Today we would describe the left-hand side as the value ζ(2)\zeta(2) of the Riemann zeta function. Euler was studying the underlying series more than a century before Riemann's 1859 paper.

(The symbol ζ\zeta is the Greek letter zeta. Writing ζ(2)\zeta(2) simply means “the value of the zeta function when its input is 2”.)

Euler finds primes inside the sum

Here comes the surprising step.

The infinite sum above appears to involve every positive whole number: 1, 2, 3, 4, 5, 6 and so on. Primes do not seem to have been singled out.

Euler discovered that the same structure can be rebuilt by multiplying together one factor for each prime. In a 1737 paper, published in 1744, product expansions of this kind appeared explicitly in his work. [3]

For the squared-reciprocal example, the bridge can be written as

1+122+132+142+⋯=11−12211−13211−15211−172⋯1+\frac1{2^2}+\frac1{3^2}+\frac1{4^2}+\cdots = \frac{1}{1-\frac{1}{2^2}} \frac{1}{1-\frac{1}{3^2}} \frac{1}{1-\frac{1}{5^2}} \frac{1}{1-\frac{1}{7^2}} \cdots

The left side adds one term for every positive whole number.

The right side multiplies one factor for every prime.

(A product is what we get from multiplication, just as a sum is what we get from addition. An infinite product is multiplication continued through an endless list of factors.)

Those two descriptions are equal because every whole number has a unique prime factorisation. The primes are therefore able to generate, through multiplication, exactly the same integers that appear one by one in the sum.

ASCII FIGURE // Euler's bridge from all integers to the primes

A diagram showing that a sum indexed by all positive integers and a product indexed only by primes can encode the same information because every positive integer has a unique prime factorisation.

Prime factorisation is the hidden mechanism that lets Euler move between the two sides.

Why does the product really generate every number?

It helps to slow this down with one example.

First notice what one prime factor contains. For the prime 2,

11−14=1+14+116+164+⋯ ,\frac{1}{1-\frac14}=1+\frac14+\frac1{16}+\frac1{64}+\cdots,

while for the prime 3,

11−19=1+19+181+⋯ .\frac{1}{1-\frac19}=1+\frac19+\frac1{81}+\cdots.

When the full product is multiplied out, one term is chosen from each prime's bracket. Choosing 1/161/16 from the 2-bracket and 1/91/9 from the 3-bracket gives 1/1441/144, which is 1/1221/12^2; choosing 1 from every other prime's bracket leaves that value unchanged.

(The identities above are examples of a geometric series: repeatedly multiplying by the same fraction produces 1+r+r2+r3+⋯=1/(1−r)1+r+r^2+r^3+\cdots=1/(1-r) when the terms shrink quickly enough. Here the important point is simply that each prime's factor contains all the powers of that prime that the product may need.)

Take 12\. Its prime factorisation is

12=22×3.12=2^2\times3.

If our series uses squared reciprocals, then

1122=(122)2(132).\frac1{12^2} = \left(\frac1{2^2}\right)^2 \left(\frac1{3^2}\right).

In the prime-product description, the factor belonging to 2 supplies the powers of 2, the factor belonging to 3 supplies the powers of 3, and the factors belonging to the other primes contribute nothing to this particular term. Multiplying the chosen pieces produces 1/1221/12^2.

The same happens for 18, 25, 70, 1001 and every other positive whole number. Because each number has one prime factorisation, each reciprocal term is generated once.

That is the heart of the Euler product.

When mathematicians write the idea compactly, they often use

ζ(s)=∏p11−p−s.\zeta(s)=\prod_p\frac{1}{1-p^{-s}}.

(The letter ss is an input to the zeta function. A negative exponent such as p−sp^{-s} means take the reciprocal: p−s=1/psp^{-s}=1/p^s; for example, 2−2=1/42^{-2}=1/4. The symbol ∏\prod means “multiply these factors together”, and the small pp underneath tells us to run through the prime numbers. The compact formula is not a new idea; it is shorthand for the longer prime-by-prime multiplication shown above.)

The formula is valid in the region where this infinite product converges. At this stage, however, the important idea is not the technical boundary. It is the change of viewpoint: a function written as a sum over all positive integers can also be written as a product built entirely from primes.

That is a very deep connection hiding inside an elementary fact about multiplication.

From Euler to Riemann

Euler did not formulate the Riemann Hypothesis; he died in 1783, seventy-six years before Riemann published the six-page paper in which the hypothesis appeared.

What Euler supplied was part of the mathematical road leading towards it.

The Euler product makes the link between the zeta function and prime numbers explicit. Riemann's later step was to take this function, treat its input as a complex number, extend it by analytic continuation, and investigate its zeros. Those zeros, in turn, are tied to the distribution of the primes.

(A complex number extends the ordinary number line by adding a second component involving ii, where i2=−1i^2=-1. You do not need complex numbers to understand Euler's prime product; they matter here because Riemann later allowed the zeta function to accept this larger kind of input.)

So the historical progression is not

Euler had an idea, then Riemann happened to study something with the same name.

It is closer to this:

ASCII FIGURE // From Euler's primes to Riemann's zeros

A conceptual chain from unique prime factorisation through Euler's product representation of the zeta function to Riemann's complex extension and study of zeta zeros.

Riemann's 1859 work grows out of a prime-zeta connection that Euler had already exposed.

For the next part of that story, see Who was Bernhard Riemann? and What is the Riemann Hypothesis?.

A life of calculation, even as sight failed

Euler's eyesight deteriorated over many years. Problems were already serious during his first period in St Petersburg; later a cataract affected his remaining useful eye, and after an operation in 1771 gave only temporary improvement he became almost totally blind. The chronology is more complicated than the familiar story that Euler simply “worked himself blind”, and modern biographical accounts treat the causes with more caution. [1]

What is well established is that loss of sight did not end his mathematical work. By then Euler had developed an exceptional ability to calculate and organise mathematics mentally, but just as importantly he worked with other people: his sons and colleagues helped, and the young Swiss mathematician Nicolas Fuss became a particularly important assistant, taking dictation and participating in the mathematical work rather than acting as a merely mechanical secretary. [1]

That collaborative setting is worth remembering because the familiar portrait of the solitary genius can flatten the reality of eighteenth-century science. Euler's output depended not only on his own unusual memory and mathematical fluency, but on academies, correspondence, printers, assistants, family networks and colleagues spread across Europe.

Why does Euler's name appear so often here?

On Riemann Console, the most important Euler connection is the Euler product.

The Riemann zeta function can be approached as a series over the positive integers and, in the appropriate region, as a product over the primes. That bridge is the reason a function from analysis can carry such detailed arithmetic information.

But Euler's relevance is broader than one formula. His work helped make functions, infinite series and symbolic analysis central mathematical tools; he worked extensively in number theory; and his notation and methods became part of the language later mathematicians inherited. Riemann entered a mathematical world that Euler had helped reshape. [1]

This is why the names in this series connect to one another. Riemann did not work in isolation from mathematical history, and neither did Euler. The story passes backwards through the Bernoullis and earlier number theory, then forwards through Riemann and into the mathematics that eventually reaches Fourier, Bessel, Laguerre, Pólya, Bochner and the modern structures encountered elsewhere on this site.

More than one famous formula

It is tempting to remember a mathematician through the shortest thing attached to their name, especially when the name appears on so many formulas that it begins to look like a label rather than a person.

Euler's actual interests were far less tidy. He wrote about the motion of the Moon and planets, the behaviour of fluids, optics, acoustics, shipbuilding, mechanics, maps, music theory and the mathematics of whole numbers; he worked for academies in two empires, corresponded across Europe, taught younger scholars, raised a large family and continued producing mathematics through political upheaval, fire, illness and severe loss of sight. [1]

For Riemann Console, though, one idea is enough to remember first.

Euler showed that the world of all positive integers and the world of the primes are not two separate stories. Through the product that bears his name, they can be two descriptions of the same mathematical object.

Follow the thread

If this is your starting point, the natural next stops are prime number, prime factorisation, the Euler product, the Riemann zeta function, and Who was Bernhard Riemann?.

The one sentence to remember

Leonhard Euler was an eighteenth-century mathematician whose work reached across mathematics and physics, and whose discovery that a sum over all positive integers can be rewritten as a product over the prime numbers exposed one of the central bridges on which Riemann's later study of the zeta function was built.

References

  1. Ronald S. Calinger. “Leonhard Euler: Mathematical Genius in the Enlightenment.” Princeton University Press (2016). doi:10.2307/j.ctv7h0smb
  2. Johan C.-E. Stén. “Leonhard Euler’s Letters to a German Princess: A Milestone in the History of Physics Textbooks and More.” The Mathematical Intelligencer 43 (2021). 99–102. doi:10.1007/s00283-021-10052-2
  3. Leonhard Euler. “Variae observationes circa series infinitas (Various observations about infinite series).” Commentarii academiae scientiarum Petropolitanae / Euler Archive 9 (1744). 160–188. Source

Glossary connections

  1. [..]Analytic continuationGLOSSARY · STANDARD MATHEMATICS
  2. [..]Complex numberGLOSSARY · STANDARD MATHEMATICS
  3. [..]Euler productGLOSSARY · RIEMANN HYPOTHESIS
  4. [..]Fundamental Theorem of ArithmeticGLOSSARY · STANDARD MATHEMATICS
  5. [..]Prime factorisationGLOSSARY · RIEMANN HYPOTHESIS
  6. [..]Prime numberGLOSSARY · RIEMANN HYPOTHESIS
  7. [..]Riemann zeta functionGLOSSARY · RIEMANN HYPOTHESIS