{"id":511,"date":"2019-06-29T22:05:34","date_gmt":"2019-06-29T22:05:34","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=511"},"modified":"2019-10-18T21:46:21","modified_gmt":"2019-10-18T21:46:21","slug":"eulers-product-formula","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/eulers-product-formula\/","title":{"rendered":"Euler&#8217;s Product Formula"},"content":{"rendered":"\n<p>Here is an amazing formula due to\u00a0Euler:<br>SUM<sub>n=1 to infinity<\/sub>\u00a0n<sup>-s<\/sup>\u00a0= PROD<sub>p prime<\/sub>\u00a0(1 &#8211; p<sup>-s<\/sup>)<sup>-1<\/sup>\u00a0.<br>What&#8217;s interesting about this formula is that it relates an expression involving all the positive integers to one involving just primes!<\/p>\n\n\n\n<p>And you can use it to prove there must be\u00a0infinitely many primes. For, if there were only finitely many primes, then the right side of the expression is a finite product, and in particular for s=1. But for s=1, the left side of the equation is the\u00a0harmonic series\u00a0which we know must diverge! This is a\u00a0contradiction, so there must be infinitely many primes.<\/p>\n\n\n\n<p><strong>Presentation&nbsp;Suggestions:<\/strong><br>Interested students may wish to take a few terms on the right hand side, use a power expansion, and multiply them out&#8230; to get an idea of why the equality holds.<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>The left hand side, when s is viewed as a complex variable, is also known as the\u00a0Riemann zeta function. Because of the above relationship, the study of zeta functions is closely related to the study of the\u00a0distribution of primes!<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Euler&#8217;s Product Formula.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:<\/strong>   <br>Lesley Ward <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is an amazing formula due to\u00a0Euler:SUMn=1 to infinity\u00a0n-s\u00a0= PRODp prime\u00a0(1 &#8211; p-s)-1\u00a0.What&#8217;s interesting about this formula is that it&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,20,7,100,130,41,10,42],"class_list":["post-511","page","type-page","status-publish","hentry","tag-advanced","tag-analysis","tag-calculus","tag-distribution-of-primes","tag-functions","tag-infinite-series","tag-numtheory","tag-riemann-zeta-function"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/511","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=511"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/511\/revisions"}],"predecessor-version":[{"id":1319,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/511\/revisions\/1319"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=511"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}