Talk:Riemann zeta function: Difference between revisions
defence of gene's derivation as the best one imo so that it deserves inclusion on the main page, and perhaps some mathematical motivations/details that might not be obvious to the reader could be added to make it more accessible |
m clarify purpose of the post as the title and address the only other genuine issue that im aware of that i forgot to address (but which for reasons explained i dont think is actually much of an issue) |
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– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:01, 5 April 2025 (UTC) | – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:01, 5 April 2025 (UTC) | ||
== Recommendation to include Gene's derivation and explanation of its motivations == | |||
: Of all the derivations I've seen, I've found Gene's derivation to be the most intuitive, understandable and well-motivated. There's only really a small handful of minor issues, which except for one issue are not issues with the derivation but rather with how the derivation is likely to be perceived by a less mathematically-inclined audience: | : Of all the derivations I've seen, I've found Gene's derivation to be the most intuitive, understandable and well-motivated. There's only really a small handful of minor issues, which except for one issue are not issues with the derivation but rather with how the derivation is likely to be perceived by a less mathematically-inclined audience: | ||
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: --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 14:40, 8 April 2025 (UTC) | : --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 14:40, 8 April 2025 (UTC) | ||
: Oh I forgot there's one other unaddressed motivational issue which is the change of weighting to the reciprocal of a power of a prime; specifically, while the principle of addressing errors of prime powers is correct, this change definitely deserves some elucidation beyond "to make it converge". Luckily, I do have an explanation: if we count the number of times a given prime ''p'' occurs in prime factorisations of all harmonics in the harmonic series (meaning we are ''counting with repetition''), we find it's 1/(''p'' - 1), which surprisingly means the average number of 2's in a harmonic, ''when counting with repetition'', is exactly 1. This comes from half of all harmonics having at least one 2 (+1/2), a quarter having at least two 2's (+1/4), an eighth having at least three 2's (+1/8), etc. so that we have 1/2 + 1/4 + 1/8 + ... = 1, which can be verified empirically: the sum from 1 to 2<sup>n</sup> will always have 2<sup>n</sup> factors of 2 counted. Point being, if we fix {{nowrap| ''s'' = 1, }} the only change made by zeta is using 1/''p'' instead of 1/(''p'' - 1), so that it's slightly biased to larger primes but is still asymptotically correct, so a bonus if anything (given we want more high-limit behaviour captured and we know the behaviour at {{nowrap| ''s'' = 1 }} is fine because it can be reached by converging to {{nowrap| Re(''z'') = 1 }} from {{nowrap| ''s'' > 1}}). --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 15:11, 8 April 2025 (UTC) | |||