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 [[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 | + | 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>. |
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>. | ||
Revision as of 14:39, 12 January 2011
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 
and 1
has all nontrivial zeros on the line 
. This is the 
is 1 for all n
for a constant c and where 
