Difference between revisions of "Riemann Hypothesis"

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The '''Riemannn Hypothesis''' is a yet unproven conjecture stating that all interesting solutions to <math>\zeta (x) = 0</math> lie on the vertical line denoting <math>\frac{1}{2}</math> in the four-dimensional graph of the [[harmonic series|zeta function]] with a [[complex number]] as input.  This hypothesis is one of the [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 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 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.
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==Links==
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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 12:01, 20 October 2016

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