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Riemann hypothesis
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Unsolved problem in mathematics: Do all non-trivial zeroes of the Riemann zeta function have a real part of one half? <a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">(more unsolved problems in mathematics)</a> <p>In mathematics, the Riemann hypothesis is the <a href="/facts/Conjecture/oPWUb7SF">conjecture</a> that the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> has its <a href="/facts/Root_of_a_function/mlK3dhoT">zeros</a> only at the negative <a href="/facts/Even_integer/7zq2UM4V">even integers</a> and <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> with <a href="/facts/Real_part/2w7mqI7e">real part</a> 1/2. Many consider it to be the most important <a href="/facts/List_of_unsolved_problems_in_mathematics/RroxGPpP">unsolved problem</a> in <a href="/facts/Pure_mathematics/V29nQC1C">pure mathematics</a>. It is of great interest in <a href="/facts/Number_theory/zhoa7zrc">number theory</a> because it implies results about the distribution of <a href="/facts/Prime_numbers/L20FLkgn">prime numbers</a>. It was proposed by <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a> (1859), after whom it is named. </p><p>The Riemann hypothesis and some of its generalizations, along with <a href="/facts/Goldbach%27s_conjecture/967UhtUU">Goldbach's conjecture</a> and the <a href="/facts/Twin_prime_conjecture/272VWQbc">twin prime conjecture</a>, make up <a href="/facts/Hilbert%27s_eighth_problem/qbldYArn">Hilbert's eighth problem</a> in <a href="/facts/David_Hilbert/1vhlHn4b">David Hilbert</a>'s list of <a href="/facts/Hilbert%27s_problems/HaQWIxvV">twenty-three unsolved problems</a>; it is also one of the <a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a> of the <a href="/facts/Clay_Mathematics_Institute/9sCIUpD7">Clay Mathematics Institute</a>, which offers <a href="/facts/US$/naEEDUhC">US$</a>1 million for a solution to any of them. The name is also used for some closely related analogues, such as the <a href="/facts/Riemann_hypothesis_for_curves_over_finite_fields/jIu1cg89">Riemann hypothesis for curves over finite fields</a>. </p><p>The Riemann zeta function <i>ζ</i>(<i>s</i>) is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> whose <a href="/facts/Argument_of_a_function/QkuPc28I">argument</a> <i>s</i> may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, <i>ζ</i>(<i>s</i>) = 0 when <i>s</i> is one of −2, −4, −6, .... These are called its <i>trivial zeros</i>. The zeta function is also zero for other values of <i>s</i>, which are called <i>nontrivial zeros</i>. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: </p> <p>The real part of every nontrivial zero of the Riemann zeta function is 1/2.</p> <p>Thus, if the hypothesis is correct, all the nontrivial zeros lie on the <i>critical line</i> consisting of the complex numbers 1/2 + <i>i t</i>, where <i>t</i> is a <a href="/facts/Real_number/R02gw5Pb">real number</a> and <i>i</i> is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>. </p>