{"id":484,"date":"2012-10-03T12:30:18","date_gmt":"2012-10-03T19:30:18","guid":{"rendered":"http:\/\/zackmdavis.net\/blog\/?p=484"},"modified":"2012-10-08T19:10:31","modified_gmt":"2012-10-09T02:10:31","slug":"introducing-the-fractional-arithmetic-derivative","status":"publish","type":"post","link":"http:\/\/zackmdavis.net\/blog\/2012\/10\/introducing-the-fractional-arithmetic-derivative\/","title":{"rendered":"<em><strong>[RETRACTED]<\/strong><\/em> Introducing the Fractional Arithmetic Derivative"},"content":{"rendered":"<p>[<strong>NOTICE<\/strong>: <em>The conclusion of this post is hereby <strong>retracted<\/strong><\/em> because it turns out that the proposed definition of a \"fractional arithmetic derivative\" doesn't actually make sense. It fails to meet the basic decideratum of corresponding with an iterated <a href=\"http:\/\/en.wikipedia.org\/wiki\/Arithmetic_derivative\">arithmetic derivative<\/a>. <em>E.g.<\/em>, consider that 225\u2033 = (225\u2032)\u2032 = ((3<sup>2<\/sup>\u00b75<sup>2<\/sup>)\u2032)\u2032 = (2\u00b73\u00b75<sup>2<\/sup> + 3<sup>2<\/sup>\u00b72\u00b75)\u2032 = (150 + 90)\u2032 = 240\u2032 = (2<sup>4<\/sup>\u00b73\u00b75)\u2032 = 4\u00b72<sup>3<\/sup>\u00b73\u00b75 + 2<sup>4<\/sup>\u00b75 + 2<sup>4<\/sup>\u00b73 = 480 + 80 + 48 = 608. Whereas, under the proposed definition we would <em>allegedly<\/em> equivalently have 225<sup>(2)<\/sup> = (2!\u00b73<sup>0<\/sup>\u00b75<sup>2<\/sup> + 3<sup>2<\/sup>\u00b72!\u00b75<sup>0<\/sup>) = 50 + 18 = 68. I apologize to anyone who read the original post (??) who was thereby misled. The original post follows (with the erroneous section struck through).]<\/p>\n<p><!--more--><\/p>\n<p><em>Wikipedia<\/em> <a href=\"http:\/\/en.wikipedia.org\/wiki\/Arithmetic_derivative\">informs us of<\/a> the idea of an <em>arithmetic derivative<\/em>\u2014personally, I think this is a terrible name because it doesn't seem to be a rate of change of anything, but the motivation is clear enough, so let's go with it. It's a function on the natural numbers which we'll denote with the prime mark (&quot; \u2032 &quot;\u2014the name of this symbol is not to be confused with the name for positive integers with exactly two divisors, which are also of\u2014forgive me\u2014&quot;prime&quot; importance in this discussion). It works like this: 0\u2032 is 0, 1\u2032 is 0, <em>p<\/em>\u2032 is 1 for any prime number <em>p<\/em>, and the composite numbers get filled in with the <em>product rule<\/em> (<em>ab<\/em>)\u2032 = <em>a<\/em>\u2032<em>b<\/em> + <em>ab<\/em>\u2032 (hence the &quot;derivative&quot; moniker).<\/p>\n<p>For prime powers, the product rule degenerates into a <em>power rule<\/em>:<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_1c3a3623079737d915dcb564491d66ee.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"(p^a)' = \\sum_{j:=1}^{a} 1 \\cdot p^{a-1} = ap^{a-1}\" \/><\/span><script type='math\/tex;  mode=display'>(p^a)' = \\sum_{j:=1}^{a} 1 \\cdot p^{a-1} = ap^{a-1}<\/script><\/p> <\/p>\n<p>And this in turn makes it easy to compute the arithmetic derivative in general. Say that <em>n<\/em>\u2208\u2115 has the prime factorization \u03a0<sub>i<\/sub> <em>p<\/em><sub><em>i<\/em><\/sub><sup><em>a<\/em><sub><em>i<\/em><\/sub><\/sup>. Then\u2014<\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_0fd42a9baeeaafee7bdf31e776360e15.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"n' = \\sum_{i} a_{i}p_{i}^{a_{i}-1} \\prod_{j:\\neq i} p_{j}^{a_{j}} = \\sum_{i} a_{i}\\frac{n}{p_{i}} = n \\sum_{i} \\frac{a_{i}}{p_{i}}\" \/><\/span><script type='math\/tex;  mode=display'>n' = \\sum_{i} a_{i}p_{i}^{a_{i}-1} \\prod_{j:\\neq i} p_{j}^{a_{j}} = \\sum_{i} a_{i}\\frac{n}{p_{i}} = n \\sum_{i} \\frac{a_{i}}{p_{i}}<\/script><\/p><\/p>\n<p>Arithmetic derivatives for small natural numbers are given as <a href=\"http:\/\/oeis.org\/A003415\">sequence A003415<\/a> in the <em>Online Encyclopedia of Integer Sequences<\/em>.<\/p>\n<p>Some generalizations of this arithmetic derivative idea are discussed online (<em>e.g.<\/em>, they say you can extend it to rational numbers using the familiar <em>quotient rule<\/em>), but (to my surprise) I didn't see any mentions of the obviously compelling idea of a <em>fractional arithmetic derivative<\/em> (in analogy to the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Fractional_calculus\">fractional calculus<\/a>). <del datetime=\"2012-10-09T01:11:01+00:00\">Repeated applications of the power rule for a prime power give us<\/del><\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_13ecb2691b8c3ffaa854af65b4ddb25a.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"(p^{a})^{(k)} = \\frac{a!}{(a-k)!}p^{a-k}\" \/><\/span><script type='math\/tex;  mode=display'>(p^{a})^{(k)} = \\frac{a!}{(a-k)!}p^{a-k}<\/script><\/p><\/p>\n<p><del datetime=\"2012-10-09T01:11:01+00:00\">where the superscript parenthetical <em>k<\/em> indicates the <em>k<\/em>th arithmetic derivative for natural number <em>k<\/em>. But then if we just swap out those factorials for their gamma-function equivalents, we should have a <em>q<\/em>th power rule for real <em>q<\/em>\u2014<\/del><\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_a54798b31370c6c93f881c170d042301.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"(p^{a})^{(q)} = \\frac{\\Gamma(a+1)}{\\Gamma(a-q+1)}p^{a-q}\" \/><\/span><script type='math\/tex;  mode=display'>(p^{a})^{(q)} = \\frac{\\Gamma(a+1)}{\\Gamma(a-q+1)}p^{a-q}<\/script><\/p><\/p>\n<p><del datetime=\"2012-10-09T01:11:01+00:00\">which in turn should give us a \"fractional\" <em>q<\/em>th arithmetic derivative for natural numbers:<\/del><\/p>\n<p><p style='text-align:center;'><span class='MathJax_Preview'><img src='http:\/\/zackmdavis.net\/blog\/wp-content\/plugins\/latex\/cache\/tex_c0c73e2e49c491e1bcb9156fa4f058a8.gif' style='vertical-align: middle; border: none;' class='tex' alt=\"n^{(q)} = \\sum_{i} \\frac{\\Gamma(a_{i}+1)}{\\Gamma(a_i-q+1)} \\frac{n}{p_{i}^{q}}\" \/><\/span><script type='math\/tex;  mode=display'>n^{(q)} = \\sum_{i} \\frac{\\Gamma(a_{i}+1)}{\\Gamma(a_i-q+1)} \\frac{n}{p_{i}^{q}}<\/script><\/p><\/p>\n<p><del datetime=\"2012-10-09T01:11:01+00:00\">So this is a cute definition that seems to work, but what can we <em>do<\/em> with it? At time of writing I can only demur that further research is needed.<\/del><\/p>\n","protected":false},"excerpt":{"rendered":"<p>[NOTICE: The conclusion of this post is hereby retracted because it turns out that the proposed definition of a \"fractional arithmetic derivative\" doesn't actually make sense. It fails to meet the basic decideratum of corresponding with an iterated arithmetic derivative. &hellip; <a href=\"http:\/\/zackmdavis.net\/blog\/2012\/10\/introducing-the-fractional-arithmetic-derivative\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[],"_links":{"self":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/484"}],"collection":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/comments?post=484"}],"version-history":[{"count":14,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/484\/revisions"}],"predecessor-version":[{"id":509,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/posts\/484\/revisions\/509"}],"wp:attachment":[{"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/media?parent=484"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/categories?post=484"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/zackmdavis.net\/blog\/wp-json\/wp\/v2\/tags?post=484"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}