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An open research record on the Riemann Hypothesis.

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Who was Bernhard Riemann?

Series
People Behind the Mathematics
Summary
A general-reader portrait of Bernhard Riemann: his short mathematical life, the 1859 paper on prime numbers, the zeta function, and the conjecture that became the Riemann Hypothesis.
Math Level
GENERAL
Index Excerpt
Georg Friedrich Bernhard Riemann; Riemann's 1859 paper; On the Number of Primes Less Than a Given Quantity

There is a slightly strange thing about the name Riemann.

Spend enough time around mathematics and it begins to appear everywhere: the Riemann Hypothesis, the Riemann zeta function, Riemann surfaces, Riemannian geometry, the Riemann integral.

They are not the work of a mathematical dynasty. They lead back, overwhelmingly, to one person: Georg Friedrich Bernhard Riemann, a German mathematician who lived for just 39 years.

ASCII PORTRAIT // Bernhard Riemann

ASCII portrait of Bernhard Riemann, based on the 1863 photographic portrait held by Smithsonian Institution Libraries, Scientific Identity collection, accession SIL14-R003-02.

Bernhard Riemann, 1863. Terminal-rendered interpretation after the Smithsonian Institution Libraries portrait, accession SIL14-R003-02.

And the famous hypothesis that now bears his name occupied only a small part of an extraordinary mathematical life.

A mathematician who changed several subjects

Riemann was born in 1826 in the Kingdom of Hanover, in what is now Germany. His father was a Lutheran minister, and Riemann entered the University of Göttingen intending to study theology. [1]

Mathematics drew him elsewhere. With his father's permission he transferred to mathematics, later studying in Berlin before returning to Göttingen. His doctoral work helped reshape the study of functions of a complex number, and his later work ranged across analysis, integration and geometry.

In 1854 he gave a lecture on geometry that opened the way to what we now call Riemannian geometry: a framework for thinking about curved spaces of more than two dimensions. [1]

Decades later, that geometry became part of the mathematical language of Einstein's general theory of relativity.

Riemann had not designed it for Einstein. Einstein had not yet been born.

That is one of the recurring features of fundamental mathematics: an idea developed because it answers one set of questions can later become the right language for something its creator could not have anticipated.

ASCII FIGURE // Bernhard Riemann: a short mathematical life

A timeline showing six points in Bernhard Riemann's life, from his birth in 1826 to his death in 1866, including his doctorate, geometry lecture and 1859 paper on prime numbers.

Riemann's best-known conjecture appeared late in a remarkably short mathematical career.

Six pages about prime numbers

In 1859 Riemann turned his attention to prime numbers.

His manuscript is just six pages long. The resulting paper was titled “On the Number of Primes Less Than a Given Quantity”. Its subject was essentially a counting question: given a large number xx, how many primes lie below it? [2]

That question is formalised by the prime-counting function. Mathematicians already understood that primes become less common as numbers grow. What they wanted was a deeper description of their distribution.

Riemann approached the problem through a function that had already been studied by Euler:

ζ(s)=1+12s+13s+14s+⋯\zeta(s)=1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots

This is the Riemann zeta function.

At first sight, an infinite sum involving 1,2,3,4,…1,2,3,4,\ldots does not look like a machine for understanding prime numbers. But through the Euler product, the same function can be written in a form built directly from primes.

Riemann's leap was to study the zeta function as a function of a complex variable, and to extend it by analytic continuation far beyond the region where the original infinite series straightforwardly converges.

(A complex variable allows the input to have both a real and an imaginary part. Instead of exploring a function along a single number line, we can explore it across a two-dimensional complex plane.)

That change of viewpoint revealed a remarkable connection: the distribution of the primes is tied to the places where the zeta function has zeros.

The conjecture inside the paper

Some zeros of the zeta function occur at the negative even integers,

−2,  −4,  −6,  −8,…-2,\;-4,\;-6,\;-8,\ldots

These are called the trivial zeros.

The more mysterious non-trivial zeros lie in a region of the complex plane called the critical strip.

Riemann's conjecture was that all of those non-trivial zeros lie on the vertical line

Re⁡(s)=12.\operatorname{Re}(s)=\frac12.

That is the critical line, and the conjecture is what we now call the Riemann Hypothesis.

(The hypothesis is not saying that the zeros themselves equal one-half. A non-trivial zero is generally a complex number. The claim is that its real part is one-half.)

The statement is remarkably compact. Proving it is not.

Why was Riemann looking at zeros at all?

This is perhaps the most useful historical point to understand.

Riemann did not begin with an arbitrary vertical line in the complex plane and wonder whether infinitely many mysterious points might sit on it. He was trying to understand the primes.

The zeros entered because they appear in formulas describing how the actual distribution of primes departs from its average behaviour. The Prime Number Theorem describes the broad trend; the zeta zeros are bound up with the finer fluctuations around it.

So the hypothesis belongs to a larger idea:

the apparently irregular sequence of prime numbers can be studied through the analytic behaviour of a function.

That bridge between arithmetic and analysis is much of the reason the 1859 paper became so important. The hypothesis is its most famous conjecture, but the deeper legacy is the landscape around it.

Did Riemann know it was true?

No surviving proof by Riemann is known.

His paper presents the statement as a conjectural claim rather than a proved theorem. Since 1859, vastly more zeros have been checked, and every verified example so far is consistent with the hypothesis. But checking individual cases, however many, is not the same thing as proving a statement about all of them.

That is why the problem remains open.

For the idea itself, see What is the Riemann Hypothesis?. For the distinction between enormous numerical evidence and proof, see If the Riemann Hypothesis has worked so far, why do we still need a proof?.

Why does Riemann's name appear so often here?

Riemann Console is concerned with the Riemann Hypothesis, so some appearances of his name are inevitable. But the connection runs deeper than the title of the problem.

The public research repeatedly encounters mathematical objects descended from the framework surrounding Riemann's work: the zeta function, its zeros, the functional equation, the completed xi function, and analytic descriptions of the distribution of primes.

That becomes concrete in public research objects such as Completed Sidebands and Spectral Positivity, where completed-Xi constructions and their transforms are part of the mathematical machinery, and Local Real-Zero Geometry Does Not Force Nonlocal Positivity, which studies what real-zero structure can and cannot force.

Those are modern research questions, not things Riemann himself wrote down. The link is lineage: later mathematics keeps building new objects from structures that enter through the zeta function and its zeros.

Some other names recur on this site for the same reason: Fourier, Bessel, Laguerre, Pólya and Bochner each mark a mathematical idea or theorem that has become part of the working language.

That is the purpose of this series: to put people back behind the names.

More than the hypothesis

It would be misleading to remember Riemann only for an unsolved problem.

By the time he died in 1866, aged 39, his ideas had already changed several areas of mathematics. Later generations developed them much further, and his name became attached to concepts across analysis, geometry and number theory.

There is an appealing irony here.

The Riemann Hypothesis is famous because it is something Riemann did not prove.

His broader importance comes from an extraordinary amount that he did discover.

Follow the thread

If this is your starting point, the natural next stops are What is the Riemann Hypothesis?, the Riemann zeta function, and the site's Research map.

The one sentence to remember

Bernhard Riemann was a 19th-century mathematician whose attempt to understand the distribution of prime numbers led him to connect primes with the zeros of the zeta function — and to the still-unproved conjecture that those non-trivial zeros all lie on the critical line.

References

  1. Detlef Laugwitz. “Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics.” Birkhäuser Boston (1999). doi:10.1007/978-0-8176-4777-3
  2. 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

Glossary connections

  1. [..]Analytic continuationGLOSSARY · STANDARD MATHEMATICS
  2. [..]Completed xi functionGLOSSARY · RIEMANN HYPOTHESIS
  3. [..]Complex numberGLOSSARY · STANDARD MATHEMATICS
  4. [..]Complex planeGLOSSARY · STANDARD MATHEMATICS
  5. [..]Critical lineGLOSSARY · RIEMANN HYPOTHESIS
  6. [..]Critical stripGLOSSARY · RIEMANN HYPOTHESIS
  7. [..]Distribution of primesGLOSSARY · RIEMANN HYPOTHESIS
  8. [..]Euler productGLOSSARY · RIEMANN HYPOTHESIS
  9. [..]Functional equationGLOSSARY · RIEMANN HYPOTHESIS
  10. [..]Non-trivial zeroGLOSSARY · RIEMANN HYPOTHESIS
  11. [..]Prime Number TheoremGLOSSARY · RIEMANN HYPOTHESIS
  12. [..]Prime numberGLOSSARY · RIEMANN HYPOTHESIS
  13. [..]Prime-counting functionGLOSSARY · RIEMANN HYPOTHESIS
  14. [..]Real partGLOSSARY · STANDARD MATHEMATICS
  15. [..]Riemann HypothesisGLOSSARY · RIEMANN HYPOTHESIS
  16. [..]Riemann zeta functionGLOSSARY · RIEMANN HYPOTHESIS
  17. [..]Trivial zeroGLOSSARY · RIEMANN HYPOTHESIS
  18. [..]Zero of a functionGLOSSARY · STANDARD MATHEMATICS