Riemann Hypothesis

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The Riemann Hypothesis or the Riemann Hypothesis is an assumption about the zeros of the Riemann zeta function . It was formulated in 1859 by Bernhard Riemann in his work On the number of prime numbers under a given quantity . It is known and proven that the zeta function has real zeros (the so-called “trivial” zeros) and an infinite number of complex zeros with the real part 12 . The Riemann Hypothesis states that there are no other zeros beyond this, i.e. This means that all nontrivial zeros of the zeta function lie on a straight line in the numerical plane parallel to the imaginary axis. Whether the guess is correct or not is one of the most significant unsolved problems in mathematics .

In 1900, David Hilbert added the Riemann Hypothesis to his list of major unsolved mathematical problems . In 2000 the Clay Mathematics Institute (CMI) put the problem on the list of Millennium Problems . The institute in Cambridge (Massachusetts) has awarded a prize of one million US dollars for a conclusive solution to the problem in the form of a mathematical proof .

classification

For every prime number the counting function takes one step up an infinitely long and high staircase. The forecast "snakes" steadily around the counting function. However, for very large values, an ever greater distance between the staircase and the forecast can be expected. However, this distance can be "controlled" by specifying an upper limit. The minimal selectable size of this limit is the subject of the Riemann Hypothesis. If this statement is correct, then the forecast will essentially only be wrong.
The zeros of the zeta function correct the prediction up to an exact term. The more of the infinitely many zeros that are included, the more accurate the approximation. The picture shows the correction of 100 pairs of zeros.

At the center of number theory , that branch of mathematics that deals with the properties of the natural numbers 1, 2, 3, 4 ..., are the prime numbers 2, 3, 5, 7, 11 .... These are distinguished by the property of having exactly two factors , namely 1 and itself. 1 is not a prime number. Already Euclid could show that there are infinitely many prime numbers, which is why the list 2, 3, 5, 7, 11 ... will never end.

The prime numbers are, so to speak, the atoms of the whole numbers, since every positive whole number can be clearly multiplicatively decomposed into such. For example, 21 = 3 · 7 and 110 = 2 · 5 · 11. Despite this elementary property, after several millennia of mathematics history, no simple pattern is known to which the prime numbers are subject in their sequence. Their nature is one of the most significant open questions in mathematics.

Even if the detailed understanding of the sequence 2, 3, 5, 7, 11 ... is unreachable, you can look for patterns if you widen your view. For example, the idea that with the help of statistical methods the behavior of a large number of people (for example with regard to consumption and voting behavior) can often be described with surprising precision, even though a single person is extremely complex. Roughly speaking, this has to do with the fact that increasing amounts of relevant data provide more and more reliable information . In the case of prime numbers, such an expansion leads, among other things, to the question of how many prime numbers there are below a fixed number.

For example, only 4 prime numbers, namely 2, 3, 5 and 7, are smaller than the number 10. In the case of 50 there are already 15 smaller prime numbers, namely

At the end of the 19th century, an astonishingly accurate estimate for the distribution of prime numbers could be proven as a consequence of the prime number theorem. The prime number theorem was already assumed in the 18th century by the 15-year-old Gauss (in the years 1792/93). The estimate was already given before a proof of the prime number theorem by Riemann and appears as a formula that allows the rapid calculation of a predicted value. With this formula, for a given number, the number of prime numbers that are smaller than this number can be estimated in a reasonable time. The formula for the prediction becomes more and more precise in percentage terms, the larger the number is chosen (but with fluctuations). For example, it delivers the forecast 14.97 for the value 50 (there are actually 15 prime numbers, see above), which means that the error is 0.16 percent. It also predicts around 78,527 prime numbers below the number 1,000,000 - in fact there are 78,498. This corresponds to a deviation of 0.037 percent.

A possible tool for proving this formula is the Riemann zeta function . It takes advantage of the fact that it expresses the law of the unique prime factorization in the language of analysis . So the properties of the prime numbers are stored hidden in this function. The decisive features that allow conclusions to be drawn about the prime numbers are the zeros of the zeta function, i.e. all points at which it takes on the value 0. These generate a correction term for the above formula, which converts them into an exact expression. The resulting exact formula knows the distribution of the prime numbers down to the last detail. However, this does not mean that the questions about the prime numbers have been solved: the computational effort increases sharply with increasing values ​​and practical calculations with this formula are therefore not effective. In contrast, modern primality tests are better suited for numerical research . However, the exact formula is of theoretical interest: it contains the margin of error between the simple prediction and the actual prime number distribution. It is assumed that this error (within the spectrum of all possibilities) is the smallest possible. Within the exact formula, which should output the number of prime numbers under the number , terms are also added up, whereby the zeros are. If the real part of is now larger, this also increases the size of , which would mean that the distance between the estimate of the prime number theorem and the actual distribution would also be larger. It can be shown that the real part of infinitely many values ​​of is equal , which is why the error will definitely have a minimum order of magnitude The Riemann hypothesis now says, however, that there are no further zeros that behave differently than those previously known in the critical strip.

The decoding of the error is not relevant to the numerics . Rather, pure mathematics strives to find out the previously hidden reason why the error (if applicable) is as small as possible. Mathematicians hope to gain a fundamental insight into the nature of numbers behind the formal justification of this regularity.

The Riemann zeta function

Riemann zeta function in the complex plane, horizontally and vertically . A row of white spots marks the zeros at
Click here for a complete display of the preview image .

The Riemann zeta function is a complex-valued function that works for a complex number with a real part through the infinite sum

is defined.

One of the most important properties of the Riemann zeta function is its connection with the prime numbers . It establishes a relationship between complex analysis and number theory (see analytical number theory ) and forms the starting point of the Riemann Hypothesis. The following expression, which goes back to Leonhard Euler (1748), represents the connection in a formulaic way as

where represents an infinite product over all prime numbers. The expression follows directly from the theorem about the uniqueness of prime number decomposition and the summation formula for the geometric series .

The function can be  clearly analytically continued beyond the original convergence range of Euler's sum or product formula to the entire complex level - with the exception of . A meromorphic function is obtained

where are the gamma function and the Bernoulli numbers . At the point it has a simple pole . The other singularities in this representation can all be removed because the whole function has a simple zero at each of these positions .

Riemann Hypothesis

Magnitude of the zeta function on the critical line Re (s) = 1/2
Function values ​​of the zeta function on the critical straight line Re (s) = 1/2

In the following, the Riemann zeta function is considered in an analytical continuation. In this form, the zeta function has so-called “trivial zeros”, which result from the set of poles of the gamma function, minus the set of poles of the expression in brackets by cancellation. It is the set of negative even numbers

A central finding of Riemann's famous work from 1859 was the finding that all possible nontrivial zeros are in the so-called critical strip

must be located.

The famous - and to this day neither refuted nor proven - conjecture by Bernhard Riemann states that all non-trivial zeros are on the middle straight line

lie.

Riemann came up with his conjecture while examining the product of the zeta function with the gamma function

,

which is invariant when swapping with , that is, it satisfies the functional equation :

Riemann himself used and received for everyone :

The straight line in the complex number plane with the real part 1/2 is therefore also invariant in this reflection. Riemann himself writes about the zeros:

"[...] and it is very likely that all roots are real. Strict proof of this would certainly be desirable; I have meanwhile left the search for it aside for the time being, after a few fleeting unsuccessful attempts, since it seemed unnecessary for the next purpose of my investigation. "

By “real roots” Riemann meant that for one in the critical strip the equation

only for real , that is , to be solved.

From the position of the zeros of the zeta function, independent of the Riemann assumption, important statements about the prime number distribution can be made; the prime number theorem is equivalent to the statement that the zeta function has no zeros on the straight line , and every expansion of the zero-free regions into the critical strip leads to an improvement in the error term in the prime number theorem up to the Riemann conjecture.

meaning

The nontrivial zeros and prime numbers

A significant discovery by Riemann was the connection between prime numbers and the zeros of his zeta function. In his work he was concerned with finding an analytical expression for the prime number function . As a starting point, he used the formula

which fundamentally underpins the relationship between prime numbers and the zeta function. This can be converted into the following expression by taking the logarithm and suitable power series :

About the integral

Riemann was able to bring the expression into a closed form. For this, he led the number-theoretic function with

a, where the Heaviside function symbolizes. This adds up the fraction for every prime power that is less than . A simple example would be

It is also a step function . So a pure integral expression for is:

Riemann was a master of Fourier analysis and thus achieved a milestone in analytical number theory with the next transformation. Using an inverse Mellin transformation, he deduced an analytical expression for :

with one . In the next steps of his work, Riemann referred to the product representation of the Riemann function named after him , which is defined by:

This product representation runs over all non-trivial zeros of the zeta function and has the form of a polynomial factorized to infinity (similar to the factoring of the sine or cosine ):

From this one obtains a literally nontrivial second expression for :

The last part of Riemann's work deals on the whole only with the substitution of this second expression for in the equation

Despite difficult evaluation, Riemann came to the result

where is the integral logarithm . With the connection between and , inferred from the Möbius inversion (with the Möbius function ) , namely

a deep connection was created between prime numbers and the zeros of the zeta function.

Note: In the case of a numerical calculation of with Riemann's formula, the expression in the sum should be replaced by, where the (complex) integral exponential function denotes, since when evaluating over the main branch of the complex logarithm does not always apply and thus the result would be falsified.

Inferences

From the Riemann Hypothesis, for example, an estimation of the remainder of the prime number theorem follows ( Helge von Koch 1901)

Koch's result is equivalent to the Riemann Hypothesis. It can also be written as

for a constant , and is a slightly weaker form

for any .

Many other results of analytical number theory, but also those for the fast primality tests that are important in cryptography , can so far only be proven or carried out assuming the Riemann hypothesis. In the complex zeros of the zeta function, as Michael Berry wrote, the fluctuations around the rough asymptotically logarithmic distribution of the prime numbers described by the prime number theorem are coded. If you know the exact distribution, you can also make more precise statements about the probability of how many prime numbers are to be found in a range.

The real reason why many mathematicians have searched so intensely for a solution is - apart from the fact that this is the last as yet unproven statement in Riemann's famous essay - that an otherwise very chaotic function (e.g. B. Voronin's universality theorem : the zeta function can arbitrarily approximate any analytical non-zero function within a circle of radius 1/4) probably hides the tip of the iceberg of a fundamental theory, just as the Fermat conjecture is the parameterization of elliptic curves by module functions hid, part of the Langlands program .

history

The Riemann Hypothesis was mentioned as early as 1859 by Bernhard Riemann in a famous paper that laid the foundations of analytical number theory . In doing so, he wrote "that although strict evidence would be desirable, after a few cursory attempts he had temporarily stopped exploring it, since it would be dispensable for the next purpose of his investigation." as Carl Ludwig Siegel found out in the 1930s when he examined Riemann's estate. In addition, nothing was found in his unpublished writings. The mathematician and mathematician Harold Edwards formulated some speculations as to how Riemann could have come to his conjecture without significant numerical evidence.

In 1903 Jørgen Pedersen Gram published numerical approximate values ​​for the first 15 zeros in the critical area. They support (but do not prove) the Riemann Hypothesis, as do all other zeros that were found later and the number of which exceeded the 100 million mark in the early 1980s. In 2001 it was shown with the help of mainframes that the first ten billion zeros of the complex zeta function all satisfy the Riemann Hypothesis, i.e. i.e., they are all on the straight line with the real part .

Another milestone in the numerical search was the Zeta-Grid project started in August 2001. With the help of the distributed computing method , in which many thousands of Internet users took part, around 1 trillion zeros were found after three years. The project has since been discontinued.

The two French mathematicians Gourdon and Demichel started a new experiment with the method of Odlyzko and Schönhage in 2004 and checked the first 10 billion zeros in October 2004 without finding a counterexample. Although all calculations are numerical methods, they show exactly and not only approximately that the examined zeros are on the critical straight line.

Many famous mathematicians have tried the Riemann Hypothesis. Jacques Hadamard claimed in 1896 in his work Sur la distribution des zéros de la fonction ζ (s) et ses conséquences arithmétiques in which he proved the prime number theorem , without further elaboration , that the then recently deceased Stieltjes had proven the Riemann conjecture without publishing the proof . In 1885, Stieltjes claimed in an article in the Compte Rendu of the Académie des sciences to have proven a theorem about the asymptotic behavior of the Mertens function, from which the Riemann Hypothesis follows (see below). The famous British mathematician Godfrey Harold Hardy used to send a telegram before crossing the English Channel in bad weather, claiming that he had found evidence, following the example of Fermat , who passed on to posterity on the margin of a book that he had for his guess is evidence that is unfortunately too long to fit on the edge. His colleague John Edensor Littlewood in Cambridge in 1906 as a student even received the Riemann hypothesis as a function-theoretical problem from his professor Ernest William Barnes , without any connection to the prime number distribution - Littlewood had to discover this connection for himself and proved in his fellowship dissertation that the prime number theorem derives from the The hypothesis follows, which has long been known in continental Europe. As he admitted in his book A mathematician's miscellany , this did not shed a good light on the state of mathematics in England at the time. Littlewood soon made important contributions to analytical number theory in connection with the Riemann hypothesis. The problem was declared in 1900 by David Hilbert in his list of 23 mathematical problems as a problem of the century, whereby Hilbert himself classified it as less difficult than, for example, the Fermat problem: In a 1919 lecture he expressed the hope that a proof would still be admitted would be found during his lifetime, in the case of the Fermat conjecture, perhaps during the lifetime of the youngest audience; he found the evidence of transcendence in his list of problems the most difficult - a problem that was solved in the 1930s by Gelfond and Theodor Schneider . Many of the problems on Hilbert's list have now been solved, but the Riemann Hypothesis withstood all attempts. Since no proof for the Riemann Hypothesis was found in the 20th century, the Clay Mathematics Institute again declared this project to be one of the most important mathematical problems in 2000 and offered a price of one million US dollars for a conclusive proof, but not for a counterexample.

There are also conjectures for other zeta functions analogous to the Riemann conjecture, some of which are also well supported numerically. In the case of the zeta function of algebraic varieties (the case of the function fields) over the complex numbers, the conjecture was proven in the 1930s by Helmut Hasse for elliptic curves and in the 1940s by André Weil for Abelian varieties and algebraic curves (also over finite fields) . Weil also formulated the Weil conjectures , which also include an analogue of the Riemann hypothesis, for algebraic varieties (also higher dimensions than curves) over finite fields. The proof was provided by Pierre Deligne after the development of modern methods of algebraic geometry in the Grothendieck School in the 1970s .

More recent attempts at evidence or refutation

In 1945, Hans Rademacher claimed to have disproved the presumption and caused quite a stir in the USA. Shortly before publication in the Transactions of the American Mathematical Society , Carl Ludwig Siegel found a mistake. Alan Turing also thought the assumption was wrong. He occupied himself intensively with the calculation of zeros of the zeta function and, shortly before his involvement in deciphering work in Bletchley Park , tried to build a mechanical machine that should help him to find at least one hypothetical (and thus refuting) zero.

Louis de Branges de Bourcia dealt with the problem for decades. In 1985 (shortly after his proof of the Bieberbach conjecture ) he presented a proof based on his theory of the Hilbert spaces of whole functions, in which Peter Sarnak found an error. In 1989, on the occasion of a series of lectures at the Institut Henri Poincaré , he presented further evidence, which he himself soon recognized as faulty. In 2004 he published a new piece of evidence which was critically examined. Years before, however , Eberhard Freitag had given a counterexample for an assertion made in the evidence, so that the evidence is now viewed as false.

Generalized Riemann Hypothesis

The following assertion is usually referred to as a generalized or general Riemann Hypothesis :

The analytical continuation of the Dirichlet series for any Dirichlet character ( series)
has only zeros on the straight line in the critical strip

From the generalized Riemann Hypothesis, the Riemann Hypothesis follows as a special case. Andrew Granville was able to show that the (strong) Goldbach hypothesis is essentially equivalent to the generalized Riemann hypothesis.

For a generalized version for L functions of the Selberg class see L function .

Related conjectures and equivalent formulations

In analytic number theory there are further conjectures that are related to the Riemann conjecture. The Mertens Hypothesis says

for everyone . Here is the Möbius function and the so-called Mertens function. The Mertens Hypothesis is stronger than the Riemann Hypothesis, but was refuted in 1985.

The probabilistic interpretation of the Riemann Hypothesis by Arnaud Denjoy is related to this . Let be a random sequence of values ​​(1, −1) (that is, they have the same probability), then for each for the sum (using the Landau symbols )

that is, the amount of the deviation from the mean value 0 increases asymptotically at most as much as . Substituting for the Möbius function, the Riemann hypothesis is equivalent to the statement that this asymptotic growth behavior also applies to their sum (the Mertens function) (Littlewood 1912). The Riemann hypothesis can then be interpreted as a statement that the distribution of the Möbius function (i.e. whether numbers without double prime factors have an even or an odd number of prime factors) is completely random.

As already mentioned, from the Riemann Hypothesis according to Helge von Koch there are bounds for the growth of the error term of the prime number theorem. But von Koch's result is also equivalent to the Riemann hypothesis. Out

follows the Riemann hypothesis.

In a similar form as an asymptotic error term to the prime number theorem, the Riemann conjecture can also be expressed with the help of the Mangoldt function or its sum :

where the prime number theorem is equivalent to

is. From this one can derive another conjecture equivalent to the Riemann conjecture: The following applies to all :

with the least common multiple .

The Lindelöf hypothesis on the organization of the zeta function along the critical line is weaker than the Riemann hypothesis, but still unproven.

In 1916, Marcel Riesz showed the equivalence to a conjecture about the asymptotic behavior of the Riesz function. Jérôme Franel proved in 1924 the equivalence to a statement about Farey series . This clearly states that the arrangement of the rational numbers in the interval (0.1) after decimal fractions in linear form and the arrangement in Farey sequences are as different as possible in a well-defined mathematical sense.

In 2002 Jeffrey Lagarias put forward a conjecture of elementary number theory that is equivalent to the Riemann conjecture:

for everyone . It is the sum of the divider of and the th harmonic number .

A conjecture refuted in 1958 about a series formed with the Liouville function would also have resulted in the Riemann conjecture.

Evidence ideas from physics

New ideas for proving the conjecture came from physics. David Hilbert and George Polya had already noticed that the Riemann hypothesis would follow if the zeros were eigenvalues ​​of an operator , where it is a Hermitian (i.e. self-adjoint) operator that has only real eigenvalues, similar to the Hamilton operators in quantum mechanics. During a conversation with Freeman Dyson in the 1970s, Hugh Montgomery found out that the distribution of the distances between consecutive zeros showed a distribution similar to the eigenvalues ​​of Hermitian random matrices ( Gaussian unitary ensemble , GUE), which Andrew Odlyzko confirmed by numerical calculations. In the 1990s, physicists like Michael Berry also began to look for such an underlying system, for example in the theory of quantum chaos . For more support these considerations in an analogy of "explicit formulas" in the theory of Riemann zeta function with the Selberg - trace formula that the eigenvalues of the Laplace-Beltrami operator on a Riemann surface with the lengths of closed geodesics is related, and the Gutzwiller - Trace formula in the quantum chaos theory. This connects the eigenvalues ​​(energies) of the quantum mechanical version of a chaotic classical system with the lengths of the periodic orbits in the classical case. All of these trace formulas are identities between the sums of the respective zeros, trajectory period lengths, eigenvalues, etc.

One of the Fields medal winners Alain Connes specified 1996 Operator fits "almost". So far, Connes has not been able to rule out the existence of further zeros outside the critical straight line.

Another idea from physics that was discussed in connection with the Riemann Hypothesis is the “Yang-Lee zeros” of the sum of states, which is analytically continued into the complex, in models of statistical mechanics . Using a result from George Polya from the theory of the zeta function, to which they pointed out to Mark Kac, Chen Ning Yang and Tsung-Dao Lee proved that in certain models the zeros were on a circle, in other models they are on a straight line. The position of the zeros determines the behavior in phase transitions similar to how the zeros on the critical straight line control the fine distribution of the prime numbers.

All of these ideas are based on an analogy which, in simplified form, can be described as follows: The prime numbers are "elementary particles" that interact via multiplication and thus build up the combined numbers. At the same time, the “particles” are arranged by the addition. In the zeta function, both aspects (additive / natural numbers and multiplicative / prime numbers) are combined with one another in the form of a sum or product formula.

In 2009, Freeman Dyson proposed a connection of the Riemann Hypothesis to one-dimensional quasicrystals .

See also

literature

  • Marcus du Sautoy : The music of the prime numbers. On the trail of the greatest puzzle in mathematics. dtv / CH Beck, Munich 2003 and 2004, ISBN 3-423-34299-4 (popular representation of the history of the presumption).
  • Barry Mazur , William Stein : Prime Numbers and the Riemann Hypothesis. Cambridge University Press, 2015, ISBN 978-1-107-49943-0 , (PDF; 7.6 MB). ( Memento from September 15, 2013 in the Internet Archive ).
  • John Derbyshire : Prime obsession - Bernhard Riemann and the greatest unsolved problem in Mathematics. Washington 2003, ISBN 0-309-08549-7 .
  • Andrew Granville : Refinements of Goldbach's Conjecture, and the generalized Riemann hypothesis . In: Functiones et Approximatio, Commentarii Mathematici . tape 37 , no. 1 . Faculty of Mathematics and Computer Science of Adam Mickiewicz University, Poznań 2007, p. 159–173 ( umontreal.ca [PDF; 184 kB ]).
  • Harold Edwards : Riemann's Zeta Function. New York 1974, Dover 1991, ISBN 0-486-41740-9 .
  • Karl Sabbagh: Dr. Riemann's zeros. Atlantic books, 2002.
  • Edward Charles Titchmarsh : The Theory of the Riemann Zeta-Function. Modifications made by Heath-Brown. Oxford 1987, ISBN 0-19-853369-1 .
  • P. Borwein , S. Choi, B. Rooney, A. Weirathmueller: The Riemann hypothesis. A resource for the affinity and virtuoso alike. (CMS Books in Mathematics 27) Canad. Math. Soc., Springer-Verlag, 2008, ISBN 978-0-387-72125-5 .
  • Julian Havil : Gamma - Euler's constant, prime number beaches and the Riemann hypothesis. Springer Verlag, 2007.
  • Jürg Kramer : The Riemann Hypothesis. In: Elements of Mathematics. Volume 57, 2002, pp. 90-95. hu-berlin.de. (PDF; 400 kB).
  • Dan Rockmore: Stalking the Riemann Hypothesis. Pantheon Books, 2005.
  • Kevin Broughan: Equivalents of the Riemann Hypothesis. 2 volumes, Cambridge University Press, 2017.

Web links

References and comments

  1. ^ Carl Friedrich Gauss Works , Volume Two , published by the Royal Society of Sciences in Göttingen, 1863, (letter) , pp. 444–447.
  2. ↑ The following applies to the definition of the Bernoulli numbers used here :
  3. a b Bernhard Riemann: About the number of prime numbers under a given size . October 19, 1859. In: Monthly reports of the Royal Prussian Academy of Sciences in Berlin. 1860, pp. 671-680.
  4. ^ For example, Terry Tao's blog: Complex analytic multiplicative number theory.
  5. With
    follows
    .
    Now is
    .
    Since is, the
    power series of the two remaining logarithms are needed, so
    .
    The sum of the two power series combined results in only even-numbered powers in , so that applies
    and finally you get the desired expression:
  6. ^ Helge von Koch: Sur la distribution des nombres premiers. Acta Mathematica, Vol. 24, 1901, pp. 159-182.
  7. ↑ Can be derived from Koch's result, but not the other way around.
  8. ^ Siegel: About Riemann's papers on analytic number theory. In: Studies on the History of Math. Astron. and phys. Dept. B: Studies, Volume 2, 1932, pp. 45-80.
    Siegel: Collected Treatises. Volume 1, Springer Verlag, 1966.
  9. ^ Laugwitz: Bernhard Riemann. 1996, p. 178.
  10. ^ HM Edwards: Riemann's Zeta Function. Dover, ISBN 978-0-486-41740-0 , pp. 164-166.
  11. Gram: Sur les zéros de la fonction de Riemann. In: Acta Mathematica. Volume 27, 1903, pp. 289-304.
  12. ^ Calculations relating to the zeros. Chapter 15. In: Titchmarsh: The Theory of the Riemann Zeta function.
  13. ^ Jacques Hadamard: Sur la distribution des zéros de la fonction ζ (s) et ses conséquences arithmétiques. In: Bulletin de la Société Mathématique de France. 24, 1896, pp. 199-220. (PDF; 1.3 MB), there p. 199 ff.
  14. There was no evidence of this evidence in Stieltjes' estate. Derbyshire: Prime Obsession. P. 160 f. The Mertens presumption has now been refuted.
  15. 143 year old problem still has mathematicians guessing. In: New York Times . The anecdote can also be found in Constance Reid's Hilbert biography.
  16. On the other hand, Hilbert is attributed the perhaps apocryphal statement that if he woke up after 1000 years of sleep, his first question would be whether the Riemann hypothesis would be solved. Borwein et al: The Riemann Hypothesis. P. 58 (without citing the source).
  17. Du Sautoy: The music of the prime numbers. P. 147.
  18. The history of his evidence is given by Karl Sabbagh in Dr. Riemann's Zeros is shown.
  19. ^ A b Granville: Refinements of Goldbach's Conjecture. See bibliography .
  20. Weitz / HAW Hamburg: Mathematics is more than arithmetic - example: Mertens' hypothesis on YouTube , accessed on March 22, 2020.
  21. AM Odlyzko, HJJ te Riele: Disproof of the Mertens conjecture. In: J. pure angew. Math. Volume 357, 1985, pp. 138-160, Andrew Odlyzko: Papers on Zeros of the Riemann Zeta Function and Related Topics.
  22. Denjoy: L'hypothesis de Riemann sur la distribution of zéros de , reliée à la théorie the probabilités. In: Comptes Rendus Acad. Sc. Volume 192, 1931, pp. 656-658. Edwards: Riemann's Zeta Function. 1974, p. 268. Edwards comments on this interpretation as follows: “… though it is quite absurd when considered carefully, gives a fleeting glimmer of plausibility to the Riemann hypothesis”.
  23. Littlewood: Quelques conséquences de l'hypothèse que la fonction n'a pas de zéros dans le demi-plan In: Comptes Rendus. Volume 154, 1912, pp. 263-266. Edwards, loc. cit. P. 261. Littlewood proved more precisely that the Riemann hypothesis is equivalent to the following statement: For each converges to zero for against .
  24. ^ Edwards: Riemann's Zeta function. Chapter 5.
  25. Eric W. Weisstein : Mangoldt function . In: MathWorld (English).
  26. ^ Andrew Granville in Princeton Companion to Mathematics. Chapter IV.2.
  27. Lagarias: An elementary problem equivalent to the Riemann hypothesis. In: American Mathematical Monthly. Volume 109, 2002, pp. 534-543.
  28. Alain Connes: Trace formula in non commutative geometry and the zeros of the Riemann zeta function. November 10, 1998.
  29. Freeman Dyson: Birds and Frogs. In: Notices AMS. 2009. (PDF; 800 kB).