Problem 45988. Evaluate the zeta function for real arguments > 1

Created by ChrisR

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{"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>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Riemann_zeta_function\"><w:r><w:t>Riemann zeta function</w:t></w:r></w:hyperlink><w:r><w:t> is important in number theory. In particular, the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Riemann_hypothesis\"><w:r><w:t>Riemann hypothesis</w:t></w:r></w:hyperlink><w:r><w:t>, one of the seven</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Millennium_Prize_Problems\"><w:r><w:t>Millenium Prize Problems</w:t></w:r></w:hyperlink><w:r><w:t>, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.</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>This problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument </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=\"x\"/></w:customXmlPr><w:r><w:t>x,</w:t></w:r></w:customXml><w:r><w:t> the zeta function is the sum of the reciprocals of integers raised to the power 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=\"x\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t>. Euler showed that when </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=\"x\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t> is an even integer, the value of the zeta function is proportional to </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=\"pi^x\"/></w:customXmlPr><w:r><w:t>\\pi^x</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function\"><w:r><w:t>Cody Problem 45939</w:t></w:r></w:hyperlink><w:r><w:t> uses this fact to estimate </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:t>. Less is known about the zeta function for odd integer arguments, but Apery proved that </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(3)\"/></w:customXmlPr><w:r><w:t>\\zeta(3)</w:t></w:r></w:customXml><w:r><w:t>, now known as</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Apéry's_constant\"><w:r><w:t>Apery's constant</w:t></w:r></w:hyperlink><w:r><w:t>, is irrational.</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 zeta function for real arguments greater than 1.</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: 207px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 103.5px; transform-origin: 407px 103.5px; 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: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; 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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a><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: 145.733px 7.8px; transform-origin: 145.733px 7.8px; unicode-bidi: normal; "> is important in number theory. In particular, 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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Riemann_hypothesis">Riemann hypothesis</a><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: 55.6167px 7.8px; transform-origin: 55.6167px 7.8px; unicode-bidi: normal; ">, one of the seven<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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Millennium_Prize_Problems">Millenium Prize Problems</a><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: 335.65px 7.8px; transform-origin: 335.65px 7.8px; unicode-bidi: normal; ">, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.</div><div style="block-size: 105px; 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 52.5px; text-align: left; transform-origin: 384px 52.5px; 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: 378.467px 7.8px; transform-origin: 378.467px 7.8px; unicode-bidi: normal; ">This problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAkCAYAAACaJFpUAAAA60lEQVRIie2U0Q2DIBBA3w5s4AIu4ASdgA3cwA1cgRkcwR26gjOwgv3gLhKi0kT0o+UlJuIFH3B3QKVSqVR+kgawwCDvaWyQ+GVewATMwCrPFMVt9H3dWcwlVOoBA3TAIotyyUKK0LPtRGVdaUlMGwk9hXKWw4tweUIGIU8r8H5CZkR0S0Xu4RLhrTm0hLwZtjy6u2StyFoZax61cPSom2jsCH07yvgUIz+dCL3nCc2txP1oRWYP4qtIT+mSCX0SbzLx9LrzOaFOGji+SbpMXOfPPNRCysgXR1oKLbZs0ZSU3X45KJaHdlb5Ez4sHUr70Yy5uAAAAABJRU5ErkJggg==" alt="x" style="width: 14px; height: 18px;" width="14" height="18"><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: 248.15px 7.8px; transform-origin: 248.15px 7.8px; unicode-bidi: normal; "> the zeta function is the sum of the reciprocals of integers raised to the power of x<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: 80.5167px 7.8px; transform-origin: 80.5167px 7.8px; unicode-bidi: normal; ">. Euler showed that when x<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: 35.7833px 7.8px; transform-origin: 35.7833px 7.8px; unicode-bidi: normal; "> is an even integer, the value of the zeta function is proportional to <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAAmCAYAAACh1knUAAABLklEQVRYhe2WXa3EIBBGjwccYAADq6AK6qAO6mAtVMNKqIdauBqwwH1gJkAvm93tpjQ34SQ80D8+5psZCp1Op9P5CAfcsrmRuWklwAITsAFBFnfAj8znVkKUSRa+A6sIvAQnQgIwXCVCCURLLsWQ8uIrWywx0V6NZ4s8SAk7HhEwknbyznCVbwwi5CbPLMQIvV0xywcC9v5bYnXMxEgYGQHw8u28rzzlLruwMvYLjaRyrGHk/YXSrokUnZcYeUGZ+dt8VlKDaoan9F8j5FuKUAtyW7RLLi2FqAV5LmgVHSrDI6gFeS7krbrZyanl6yvXtuzawImHmOZBoKwg7Y4qxHFS0lrKTuopLcgbmObPKbmSR6J2ZG+7+48zREDcvXbV2n+Dk3srDaum0+l0/hW/ktdyQqlGSvEAAAAASUVORK5CYII=" alt="pi^x" style="width: 17px; height: 19px;" width="17" height="19"><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: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function">Cody Problem 45939</a><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: 80.8833px 7.8px; transform-origin: 80.8833px 7.8px; unicode-bidi: normal; "> uses this fact to estimate π<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: 27.6167px 7.8px; transform-origin: 27.6167px 7.8px; unicode-bidi: normal; ">. Less is known about the zeta function for odd integer arguments, but Apery proved that <img src="data:image/png;base64,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" alt="zeta(3)" style="width: 30.5px; height: 18.5px;" width="30.5" 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: 48.2333px 7.8px; transform-origin: 48.2333px 7.8px; unicode-bidi: normal; ">, now known as<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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Ap%C3%A9ry's_constant">Apery's constant</a><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: 10.8833px 7.8px; transform-origin: 10.8833px 7.8px; unicode-bidi: normal; ">, is irrational.</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: 186.7px 7.8px; transform-origin: 186.7px 7.8px; unicode-bidi: normal; ">Evaluate the zeta function for real arguments greater than 1.</div></div></div>

The <https://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function> is important in number theory. In particular, the <https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann hypothesis>, one of the seven <https://en.wikipedia.org/wiki/Millennium_Prize_Problems Millenium Prize Problems>, states that the non-trivial zeros of the zeta function all have real part equal to 1/2. The truth of the Riemann hypothesis has consequences for the distribution of prime numbers.

This problem deals only with values of the zeta function for real arguments greater than 1. For a positive integer argument x, the zeta function is the sum of the reciprocals of integers raised to the power of x. Euler showed that when x is an even integer, the value of the zeta function is proportional to pi^x, and <https://www.mathworks.com/matlabcentral/cody/problems/45939-estimate-pi-from-certain-values-of-the-zeta-function Cody Problem 45939> uses this fact to estimate pi. Less is known about the zeta function for odd integer arguments, but Apery provided that zeta(3), now known as <https://en.wikipedia.org/wiki/Apéry's_constant Apery's constant>, is irrational.

Evaluate the zeta function for real arguments greater than 1.

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Problem 45988. Evaluate the zeta function for real arguments > 1

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