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Difference between revisions of "Riemann Hypothesis"
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The '''Riemann Hypothesis''' is a well-known [[conjecture]] in [[analytic number theory]] that states that all nontrivial [[zero]]s of the [[Riemann zeta function]] have real part <math>1/2</math>. From the [[functional equation]] for the zeta function, it is easy to see that <math>\zeta(s)=0</math> when <math>s=-2,-4,-6,\ldots</math> These are called the trivial zeros. This hypothesis is one of the seven [http://www.claymath.org/millennium/ millenium questions].
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The '''Riemann Hypothesis''' is a famous [[conjecture]] in [[analytic number theory]] that states that all nontrivial [[root |zero]]s of the [[Riemann zeta function]] have [[real part]] <math>1/2</math>. From the [[functional equation]] for the zeta function, it is easy to see that <math>\zeta(s)=0</math> when <math>s=-2,-4,-6,\ldots</math>.  These are called the trivial zeros. This hypothesis is one of the seven [http://www.claymath.org/millennium-problems/millennium-prize-problems millenium questions].
  
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The Riemann Hypothesis is an important problem in the study of prime numbers. Let <math>\pi(x)</math> denote the number of primes less than or equal to ''x'', and let <math>\mathrm{Li}(x)=\int_2^x \frac{1}{\ln t}\; dt</math>. Then an equivalent statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>.
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The Riemann Hypothesis is an important problem in the study of [[prime number]]s. Let <math>\pi(x)</math> denote the number of primes less than or equal to ''x'', and let <math>\mathrm{Li}(x)=\int_2^x \frac{1}{\ln t}\; dt</math>. Then an equivalent statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</math>.
  
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One fairly obvious try to prove the Riemann Hypothesis (which unfortunately doesn't work) is to consider the reciprocal of the zeta function, <math>\frac{1}{\zeta(s)}{=}\sum_{n=1}^\infty \frac{\mu(n)}{n^s}</math>, where <math>\mu(n)</math> refers to the [[Möbius function]]. Then one might try to show that <math>\frac{1}{\zeta(s)}</math> admits an [[analytic continuation]] to <math>\Re(s)>\frac{1}{2}</math>. Let <math>M(n)=\sum_{i=1}^n \mu(i)</math> be the [[Mertens function]]. It is easy to show that if <math>M(n)\le\sqrt(n)</math> for sufficiently large <math>n</math>, then the Riemann Hypothesis would hold. However, A. M. Odlyzko and H. J. J. te Riele showed that this conjecture is in fact false. The Riemann Hypothesis would also follow if <math>M(n)\le C\sqrt{n}</math> for any constant <math>C</math>; however, this is believed to be false as well.
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One fairly obvious method to prove the Riemann Hypothesis is to consider the [[reciprocal]] of the zeta function, <math>\frac{1}{\zeta(s)}{=}\sum_{n=1}^\infty \frac{\mu(n)}{n^s}</math>, where <math>\mu(n)</math> refers to the [[Möbius function]]. Then one might try to show that <math>\frac{1}{\zeta(s)}</math> admits an [[analytic continuation]] to <math>\Re(s)>\frac{1}{2}</math>. Let <math>M(n)=\sum_{i=1}^n \mu(i)</math> be the [[Mertens function]]. It is easy to show that if <math>M(n)\le\sqrt{n}</math> for sufficiently large <math>n</math>, then the Riemann Hypothesis would hold. The Riemann Hypothesis would also follow if <math>M(n)\le C\sqrt{n}</math> for any constant <math>C</math>.
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Some equivalent statements of the Riemann Hypothesis are
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* The zeta function has no zeros with real part between <math>\frac{1}{2}</math> and 1
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* <math>\zeta_a(s)</math> has all nontrivial zeros on the line <math>Re(s)=\frac{1}{2}</math>
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* All nontrivial zeros of all [[L-series]] have real part one half where an L-series is of the form <math>\sum_{n=1}^\infty \frac{a_n}{n^s}</math>.  This is the [[generalized]] Riemann Hypothesis because in the Riemann Hypothesis, <math>a_n</math> is 1 for all n
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* <math>|M(x)|\le cx^{1/2+\epsilon}</math> for a constant c and where <math>M(x)=\sum_{n\le x}\mu(n)</math>
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* <math>\sigma(n) < e^\gamma n\log_{\log_n}</math> for all <math>n > 5040</math>.  This being equivalent to the Riemann Hypothesis is Robin's Theorem, proved in 1984 by Guy Robin.
  
 
==Links==
 
==Links==
  
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[http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf Disproof of the Mertens Conjecture]
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[http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf Purported Disproof of the Mertens Conjecture]
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[[Category:Number theory]]

Latest revision as of 02:18, 29 June 2024

The Riemann Hypothesis is a famous conjecture in analytic number theory that states that all nontrivial zeros of the Riemann zeta function have real part $1/2$. From the functional equation for the zeta function, it is easy to see that $\zeta(s)=0$ when $s=-2,-4,-6,\ldots$. These are called the trivial zeros. This hypothesis is one of the seven millenium questions.

The Riemann Hypothesis is an important problem in the study of prime numbers. Let $\pi(x)$ denote the number of primes less than or equal to x, and let $\mathrm{Li}(x)=\int_2^x \frac{1}{\ln t}\; dt$. Then an equivalent statement of the Riemann hypothesis is that $\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))$.

One fairly obvious method to prove the Riemann Hypothesis is to consider the reciprocal of the zeta function, $\frac{1}{\zeta(s)}{=}\sum_{n=1}^\infty \frac{\mu(n)}{n^s}$, where $\mu(n)$ refers to the Möbius function. Then one might try to show that $\frac{1}{\zeta(s)}$ admits an analytic continuation to $\Re(s)>\frac{1}{2}$. Let $M(n)=\sum_{i=1}^n \mu(i)$ be the Mertens function. It is easy to show that if $M(n)\le\sqrt{n}$ for sufficiently large $n$, then the Riemann Hypothesis would hold. The Riemann Hypothesis would also follow if $M(n)\le C\sqrt{n}$ for any constant $C$.

Some equivalent statements of the Riemann Hypothesis are

  • The zeta function has no zeros with real part between $\frac{1}{2}$ and 1
  • $\zeta_a(s)$ has all nontrivial zeros on the line $Re(s)=\frac{1}{2}$
  • All nontrivial zeros of all L-series have real part one half where an L-series is of the form $\sum_{n=1}^\infty \frac{a_n}{n^s}$. This is the generalized Riemann Hypothesis because in the Riemann Hypothesis, $a_n$ is 1 for all n
  • $|M(x)|\le cx^{1/2+\epsilon}$ for a constant c and where $M(x)=\sum_{n\le x}\mu(n)$
  • $\sigma(n) < e^\gamma n\log_{\log_n}$ for all $n > 5040$. This being equivalent to the Riemann Hypothesis is Robin's Theorem, proved in 1984 by Guy Robin.

Links

Purported Disproof of the Mertens Conjecture

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