Problem 46111. Evaluate the prime zeta function

Created by ChrisR

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<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>

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Last Solution submitted on Jun 02, 2022

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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.

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Problem 46111. Evaluate the prime zeta function

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