Talk:Mertens' theorems

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 Field:  Number theory

Lindqvist[edit]

I have erased the link to Peter Lindqvist and Jaak Peetre’ s paper, as they don’t give any reference for the result they mention is « the best we know about », and as it it obviously not the best known today. Moreover they mention in their note 3 that another error term is « very difficult to estimate », which is definitely not the case once one has at his disposition the prime number theorem. So I think they didn’t really spend much time on this particular problem: the best known estimate I gave is really of the level of a textbook exercise, and belongs to the « folklore » (facts that everybody in the field knows but nobody cares to publish). Sapphorain (talk) 17:02, 10 November 2016 (UTC)

Mertens' second theorem and the prime number theorem, revisited[edit]

I have erased the link to Peter Lindqvist and Jaak Peetre’ s paper, as they don’t give any reference for the result they mention is « the best we know about », and as it it obviously not the best known today. Moreover they mention in their note 3 that another error term is « very difficult to estimate », which is definitely not the case once one has at his disposition the prime number theorem. So I think they didn’t really spend much time on this particular problem: the best known estimate I gave is really of the level of a textbook exercise, and belongs to the « folklore » (facts that everybody in the field knows but nobody cares to publish). Sapphorain (talk) 17:02, 10 November 2016 (UTC)

The logs in this section are written as "log()" whereas in the first section they are given as natural logs "ln()". I assume that natural logs are intended, and that therefore these formula should be changed to use the "ln()" form for consistency and to be more explicit.80.189.15.164 (talk) 10:55, 26 November 2018 (UTC)

I would rather do the opposite: replace the "ln" by "log", as "log" is used by the classical books on elementary and analytic number theory. (Landau (Handbuch, paragraphs 26 and 28); Hardy and Wright (The theory of numbers, 22.7); Niven and Zuckermann don't need the full force of Merten's theorem (Introduction to the theory of numbers, chapter 8), but they nevertheless use the size order of the sum which they denote by loglog x). Sapphorain (talk) 12:41, 26 November 2018 (UTC)