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:: ... I added three classical references giving this less restrictive definition, and modified the lead accordingly. [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 15:41, 8 June 2017 (UTC)
::: I am sort of unhappy with this change. The previous version suffered from being not a precise definition but still morally accurate: the point of the name "arithmetic function" is that it expresses something about arithmetic, of interest to number theorists. The new definition does actually define something, but it is morally wrong: no one is really interested in the collection of all sequences of complex numbers, the vast majority of which have no meaning whatsoever for number theory (or for anything else). {{u|Sapphorain}}, do the sources you've added have more discussion of this definition, along the lines of the quote in your first comment, that could be used to explain that the cultural/historical background is important in the use of the name (not just a formal definition)? --[[User:Joel B. Lewis|JBL]] ([[User_talk:Joel_B._Lewis|talk]]) 20:44, 8 June 2017 (UTC)
:::: With the more simple and precise definition the set of arithmetical function with the operations of addition and Dirichlet convolution is a unitary ring. This is proved in the Bateman-Diamond, Niven-Zuckerman, Tenenbaum books, and in most introductory courses in analytic number theory. On the other hand Hardy and Wright don't clearly mention this (although it can be inferred from their chapter 17). [[User:Sapphorain|Sapphorain]] ([[User talk:Sapphorain|talk]]) 21:33, 8 June 2017 (UTC)
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