What is the Riemann hypothesis?

The Riemann hypothesis is a conjecture about the solutions to the Riemann zeta function . This function is crucial in number theory for understanding the distribution of prime numbers . The hypothesis states that all "nontrivial" zeros of the Riemann zeta function lie on a specific line in the complex plane, known as the critical line (where the real part of the complex number is 1/2). German mathematician Bernhard Riemann proposed this hypothesis in 1859. While many zeros have been found on this critical line, a definitive proof remains elusive, making it one of the most significant unsolved problems in mathematics. A proof would have major implications for number theory and fields like cryptography .

Who proposed the Riemann hypothesis and when?

The Riemann hypothesis was proposed by German mathematician Bernhard Riemann in a paper published in November 1859. This hypothesis concerns the location of solutions to the Riemann zeta function and has significant implications for the distribution of prime numbers . Riemann extended the study of the zeta function to complex numbers and conjectured that all nontrivial zeros lie on a specific line, known as the critical line. This conjecture remains one of the most important unsolved problems in mathematics.

What is the Riemann zeta function?

The Riemann zeta function, denoted as (\zeta(s)), is an infinite series defined as (\zeta(s) = \sum_{n=1}^{\infty} n^{-s}). This function is crucial in number theory for understanding the distribution of prime numbers. While initially studied by Leonhard Euler, Bernhard Riemann significantly extended its analysis to complex numbers. Riemann discovered that the zeta function has "trivial zeros" at negative even integers and an infinite number of other zeros within a specific region of the complex plane called the critical strip. He conjectured that all these non-trivial zeros lie on a particular line within this strip, known as the critical line (where the real part of (s) is 1/2). This conjecture is famously known as the Riemann hypothesis and remains one of the most important unsolved problems in mathematics.

What does the Riemann hypothesis say about the zeros of the zeta function?

The Riemann hypothesis states that all of the non-trivial zeros of the Riemann zeta function lie on the critical line with a real part of 1/2. The zeta function is defined by the infinite series (\zeta(s) = 1 + 2^{-s} + 3^{-s} + 4^{-s} + \dots). While the function has "trivial zeros" at negative even integers ((-2, -4, -6, \dots)), the hypothesis concerns the location of its other zeros within the complex plane. Bernhard Riemann conjectured this in 1859, and it remains one of the most important unsolved problems in mathematics. Proving it would have significant implications for the distribution of prime numbers.

Why is the Riemann hypothesis important in mathematics, especially number theory?

The Riemann hypothesis is crucial in mathematics, particularly in number theory, because it directly impacts our understanding of the distribution of prime numbers . German mathematician Bernhard Riemann proposed this hypothesis in 1859, concerning the solutions to the Riemann zeta function . The hypothesis suggests that all non-trivial zeros of the zeta function lie on a specific line in the complex plane. If proven true, it would have significant implications for number theory and the security of cryptography . It is considered one of the most important unsolved problems in mathematics and was designated a Millennium Problem .

Has anyone proven or disproven the Riemann hypothesis, and what would happen if it were solved?

The Riemann hypothesis, a conjecture about the location of solutions to the Riemann zeta function, remains unproven. While mathematicians like Godfrey Harold Hardy proved an infinite number of solutions lie on the critical line, and computers have verified trillions of solutions on this line, a definitive proof for all nontrivial zeros has not been established. If proven true, the Riemann hypothesis would have significant implications for number theory, particularly concerning the distribution of prime numbers. This, in turn, could impact fields like cryptography. It is considered one of the most important unsolved problems in mathematics and was designated a Millennium Problem, with a $1 million prize offered for its solution.

Riemann hypothesis

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A graph illustrating the Riemann hypothesis, with a horizontal real axis and a vertical imaginary axis. The critical strip, where the real part of S is between 0 and 1, is shaded, and the critical line at real part 1/2 is marked. Trivial zeros are shown at negative even integers on the real axis.

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Key People:: Bernhard Riemann

Related Topics:: number theory; prime number theorem; Riemann zeta function; Millennium Problem; Bohr–Landau theorem

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Riemann hypothesis

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What is the Riemann hypothesis?
The Riemann hypothesis is a conjecture about the solutions to the Riemann zeta function. This function is crucial in number theory for understanding the distribution of prime numbers. The hypothesis states that all "nontrivial" zeros of the Riemann zeta function lie on a specific line in the complex plane, known as the critical line (where the real part of the complex number is 1/2).

German mathematician Bernhard Riemann proposed this hypothesis in 1859. While many zeros have been found on this critical line, a definitive proof remains elusive, making it one of the most significant unsolved problems in mathematics. A proof would have major implications for number theory and fields like cryptography.
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Who proposed the Riemann hypothesis and when?
The Riemann hypothesis was proposed by German mathematician Bernhard Riemann in a paper published in November 1859. This hypothesis concerns the location of solutions to the Riemann zeta function and has significant implications for the distribution of prime numbers. Riemann extended the study of the zeta function to complex numbers and conjectured that all nontrivial zeros lie on a specific line, known as the critical line. This conjecture remains one of the most important unsolved problems in mathematics.
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What is the Riemann zeta function?
The Riemann zeta function, denoted as (\zeta(s)), is an infinite series defined as (\zeta(s) = \sum_{n=1}^{\infty} n^{-s}). This function is crucial in number theory for understanding the distribution of prime numbers. While initially studied by Leonhard Euler, Bernhard Riemann significantly extended its analysis to complex numbers.

Riemann discovered that the zeta function has "trivial zeros" at negative even integers and an infinite number of other zeros within a specific region of the complex plane called the critical strip. He conjectured that all these non-trivial zeros lie on a particular line within this strip, known as the critical line (where the real part of (s) is 1/2). This conjecture is famously known as the Riemann hypothesis and remains one of the most important unsolved problems in mathematics.
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What does the Riemann hypothesis say about the zeros of the zeta function?
The Riemann hypothesis states that all of the non-trivial zeros of the Riemann zeta function lie on the critical line with a real part of 1/2. The zeta function is defined by the infinite series (\zeta(s) = 1 + 2^{-s} + 3^{-s} + 4^{-s} + \dots). While the function has "trivial zeros" at negative even integers ((-2, -4, -6, \dots)), the hypothesis concerns the location of its other zeros within the complex plane. Bernhard Riemann conjectured this in 1859, and it remains one of the most important unsolved problems in mathematics. Proving it would have significant implications for the distribution of prime numbers.
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Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Why is the Riemann hypothesis important in mathematics, especially number theory?
The Riemann hypothesis is crucial in mathematics, particularly in number theory, because it directly impacts our understanding of the distribution of prime numbers. German mathematician Bernhard Riemann proposed this hypothesis in 1859, concerning the solutions to the Riemann zeta function.

The hypothesis suggests that all non-trivial zeros of the zeta function lie on a specific line in the complex plane. If proven true, it would have significant implications for number theory and the security of cryptography. It is considered one of the most important unsolved problems in mathematics and was designated a Millennium Problem.
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1. Riemann hypothesis
2. Riemann zeta function
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Has anyone proven or disproven the Riemann hypothesis, and what would happen if it were solved?
The Riemann hypothesis, a conjecture about the location of solutions to the Riemann zeta function, remains unproven. While mathematicians like Godfrey Harold Hardy proved an infinite number of solutions lie on the critical line, and computers have verified trillions of solutions on this line, a definitive proof for all nontrivial zeros has not been established.

If proven true, the Riemann hypothesis would have significant implications for number theory, particularly concerning the distribution of prime numbers. This, in turn, could impact fields like cryptography. It is considered one of the most important unsolved problems in mathematics and was designated a Millennium Problem, with a $1 million prize offered for its solution.
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2. Riemann zeta function
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Riemann hypothesis, in number theory, hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function, which is connected to the prime number theorem and has important implications for the distribution of prime numbers. Riemann included the hypothesis in a paper, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (“On the Number of Prime Numbers Less Than a Given Quantity”), published in the November 1859 edition of Monatsberichte der Berliner Akademie (“Monthly Review of the Berlin Academy”).

The zeta function is defined as the infinite series ζ(s) = 1 + 2^−s + 3^−s + 4^−s + ⋯, or, in more compact notation, formula for the zeta function, Riemann hypothesis , where the summation (Σ) of terms for n runs from 1 to infinity through the positive integers and s is a fixed positive integer greater than 1. The zeta function was first studied by Swiss mathematician Leonhard Euler in the 18th century. (For this reason, it is sometimes called the Euler zeta function. For ζ(1), this series is simply the harmonic series, known since antiquity to increase without bound—i.e., its sum is infinite.) Euler achieved instant fame when he proved in 1735 that ζ(2) = π²/6, a problem that had eluded the greatest mathematicians of the era, including the Swiss Bernoulli family (Jakob, Johann, and Daniel). More generally, Euler discovered (1739) a relation between the value of the zeta function for even integers and the Bernoulli numbers, which are the coefficients in the Taylor series expansion of x/(e^x − 1). (See also exponential function.) Still more amazing, in 1737 Euler discovered a formula relating the zeta function, which involves summing an infinite sequence of terms containing the positive integers, and an infinite product that involves every prime number: formula for the Euler zeta prime function, Riemann hypothesis

Riemann extended the study of the zeta function to include the complex numbers x + iy, where i = Square root of√−1, except for the line x = 1 in the complex plane. Riemann knew that the zeta function equals zero for all negative even integers −2, −4, −6,… (so-called trivial zeros) and that it has an infinite number of zeros in the critical strip of complex numbers that fall strictly between the lines x = 0 and x = 1. He also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1/2. Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis.

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In 1914 English mathematician Godfrey Harold Hardy proved that an infinite number of solutions of ζ(s) = 0 exist on the critical line x = 1/2. Subsequently it was shown by various mathematicians that a large proportion of the solutions must lie on the critical line, though the frequent “proofs” that all the nontrivial solutions are on it have been flawed. Computers have also been used to test solutions, with the first 10 trillion nontrivial solutions shown to lie on the critical line.

A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography.

The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics. It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert at the Second International Congress of Mathematics in Paris on Aug. 8, 1900. In 2000 American mathematician Stephen Smale updated Hilbert’s idea with a list of important problems for the 21st century; the Riemann hypothesis was number one. In 2000 it was designated a Millennium Problem, one of seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Mass., U.S., for a special award. The solution for each Millennium Problem is worth $1 million. In 2008 the U.S. Defense Advanced Research Projects Agency (DARPA) listed it as one of the DARPA Mathematical Challenges, 23 mathematical problems for which it was soliciting research proposals for funding—“Mathematical Challenge Nineteen: Settle the Riemann Hypothesis. The Holy Grail of number theory.”

William L. Hosch

Riemann hypothesis

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