Problem 46111. Evaluate the prime zeta function

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>In its original definition, the zeta function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"zeta(s)\"/></w:customXmlPr><w:r><w:t>\\zeta(s)</w:t></w:r></w:customXml><w:r><w:t>, which is important in number theory, is the sum of the reciprocals of the positive integers raised to the power </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"s\"/></w:customXmlPr><w:r><w:t>s</w:t></w:r></w:customXml><w:r><w:t>. The </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>prime zeta function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"P(s)\"/></w:customXmlPr><w:r><w:t>P(s)</w:t></w:r></w:customXml><w:r><w:t> is the sum of the reciprocals of the </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>primes</w:t></w:r><w:r><w:t> raised to the power </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"s\"/></w:customXmlPr><w:r><w:t>s</w:t></w:r></w:customXml><w:r><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Evaluate the prime zeta function for the specified values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"s\"/></w:customXmlPr><w:r><w:t>s</w:t></w:r></w:customXml><w:r><w:t>, which are positive and greater than one.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 93px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 46.5px; transform-origin: 407px 46.5px; vertical-align: baseline; "><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 126.417px 7.79167px; transform-origin: 126.417px 7.79167px; unicode-bidi: normal; ">In its original definition, the zeta function <img src="data:image/png;base64,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" alt="zeta(s)" style="width: 29px; height: 18.5px;" width="29" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 226.233px 7.79167px; transform-origin: 226.233px 7.79167px; unicode-bidi: normal; ">, which is important in number theory, is the sum of the reciprocals of the positive integers raised to the power s<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8833px 7.79167px; transform-origin: 17.8833px 7.79167px; unicode-bidi: normal; ">. The <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.0667px 7.79167px; transform-origin: 61.0667px 7.79167px; unicode-bidi: normal; ">prime zeta function <img src="data:image/png;base64,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" alt="P(s)" style="width: 29.5px; height: 18.5px;" width="29.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 7.79167px; transform-origin: 110.45px 7.79167px; unicode-bidi: normal; "> is the sum of the reciprocals of the <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 7.79167px; transform-origin: 21.0083px 7.79167px; unicode-bidi: normal; ">primes<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.725px 7.79167px; transform-origin: 30.725px 7.79167px; unicode-bidi: normal; "> raised to the power s<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; ">. </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 183.2px 7.79167px; transform-origin: 183.2px 7.79167px; unicode-bidi: normal; ">Evaluate the prime zeta function for the specified values of s<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 128.367px 7.79167px; transform-origin: 128.367px 7.79167px; unicode-bidi: normal; ">, which are positive and greater than one.</div></div></div>

Solve

Solution Stats

33 Solutions
13 Solvers

Last Solution submitted on Jun 02, 2022

Last 200 Solutions

Problem Comments

Solution Comments

Solution 2808031

1 Comment

1 Comment

ChrisR on 7 Aug 2020

I loosened the tolerance on the test for s = 2. Increasing the number of primes seems to degrade the accuracy after a certain point, and the result becomes farther from the value listed in sequence A085548 in the Online Encyclopedia of Integer Sequences. This solution should pass now.

Problem Recent Solvers13

Problem Tags

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 46111. Evaluate the prime zeta function

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers13

Suggested Problems

More from this Author162

Problem Tags

Community Treasure Hunt

Problem 46111. Evaluate the prime zeta function

Solution Stats

Last 200 Solutions

Problem Comments

Solution Comments

Problem Recent Solvers13

Suggested Problems

More from this Author162

Problem Tags

Community Treasure Hunt

Players