Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "Riemann Hypothesis"
m
 
(6 intermediate revisions by 5 users not shown)
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 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].
  
 
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>.
 
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>.
Line 10: Line 10:
 
* 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>
−
* The zeta function satisfies the inequality: <math> |\zeta(s)|>|\zeta(s)|</math> where <math>s=\sigma+it</math> and <math>0<\sigma<\frac{1}{2}</math> and <math>|t|\ge 6.5</math>
+
 
−
   
+
* <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.
−
{{wikify}}
 
  
 
==Links==
 
==Links==
  
 
[http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf Purported Disproof of the Mertens Conjecture]
 
[http://www.dtc.umn.edu/~odlyzko/doc/arch/mertens.disproof.pdf Purported Disproof of the Mertens Conjecture]
−
 
  
 
[[Category:Number theory]]
 
[[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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Riemann_Hypothesis&oldid=221269"