Difference between revisions of "Riemann Hypothesis"
Billybob42 (talk | contribs) m |
|||
Line 1: | Line 1: | ||
The '''Riemann Hypothesis''' is a well-known [[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/ millenium questions]. | The '''Riemann Hypothesis''' is a well-known [[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/ millenium questions]. | ||
− | 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>. | + | 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>. |
− | 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>. | + | 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>. |
Some equivalent statements of the Riemann Hypothesis are | Some equivalent statements of the Riemann Hypothesis are | ||
* The zeta function has no zeros with real part between <math>\frac{1}{2}</math> and 1 | * The zeta function has no zeros with real part between <math>\frac{1}{2}</math> and 1 | ||
* <math>\zeta_a(s)</math> has all nontrivial zeros on the line <math>Re(s)=\frac{1}{2}</math> | * <math>\zeta_a(s)</math> has all nontrivial zeros on the line <math>Re(s)=\frac{1}{2}</math> | ||
− | * 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 | + | * 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 |
* <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> | * <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> | ||
Revision as of 13:09, 10 April 2009
The Riemann Hypothesis is a well-known conjecture in analytic number theory that states that all nontrivial zeros of the Riemann zeta function have real part . From the functional equation for the zeta function, it is easy to see that
when
. 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 denote the number of primes less than or equal to x, and let
. Then an equivalent statement of the Riemann hypothesis is that
.
One fairly obvious method to prove the Riemann Hypothesis is to consider the reciprocal of the zeta function, , where
refers to the Möbius function. Then one might try to show that
admits an analytic continuation to
. Let
be the Mertens function. It is easy to show that if
for sufficiently large
, then the Riemann Hypothesis would hold. The Riemann Hypothesis would also follow if
for any constant
.
Some equivalent statements of the Riemann Hypothesis are
- The zeta function has no zeros with real part between
and 1
has all nontrivial zeros on the line
- All nontrivial zeros of all L-series have real part one half where an L-series is of the form
. This is the generalized Riemann Hypothesis because in the Riemann Hypothesis,
is 1 for all n
for a constant c and where