I agree this is nonsensical. Hardy & Wright's definition is opaque: they had something in mind, but it is not so clear what.
A more modern definition is given at https://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-spring-2012/lecture-notes/MIT18_781S12_lec13.pdf Best wishes, John ________________________________ From: Chat <[email protected]> on behalf of Raul Miller <[email protected]> Sent: Saturday, February 19, 2022 9:53 AM To: Chat forum <[email protected]> Subject: [Jchat] Arithmetic Functions? https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FArithmetic_function&data=04%7C01%7Crandall%40newark.rutgers.edu%7C76c373847f6541578a2108d9f3b7a8e0%7Cb92d2b234d35447093ff69aca6632ffe%7C1%7C0%7C637808792843645377%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=RllCugHPwgMOmmHKbpR1FH9auMAxAVplid885gaJ2X4%3D&reserved=0 I am confused by the current version of the "Arithmetic Function" wikipedia page, and am hoping someone here can point me in the right direction. The page currently says: "In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers." But then it goes on to say: "There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions." My problem is that a prime counting function's domain is the positive integers, and the result (which contributes to the range) is a count, and a count is non-negative integer, and integers are a subset of the complex numbers. So the wikipedia page in its current form seems to me to be expressing nonsense. Likely, something important was removed from the page, perhaps because it was confusing to a wikipedia editor. There's a hint in the current first paragraph, which says 'Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n".' But that does not carry much information. I suppose maybe I ought to just buy that Hardy & Wright book -- https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.amazon.com%2FRamanujan-Lectures-Subjects-Suggested-Publishing%2Fdp%2F0821820230&data=04%7C01%7Crandall%40newark.rutgers.edu%7C76c373847f6541578a2108d9f3b7a8e0%7Cb92d2b234d35447093ff69aca6632ffe%7C1%7C0%7C637808792843645377%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=iBHrrwB9FVOawijlQsonXY8yR%2F0xsaojhhu06%2B2arzE%3D&reserved=0 -- but I was hoping that this would be a more easily explained concept. Does anyone here know what's going on there? Thanks, -- Raul ---------------------------------------------------------------------- For information about J forums see https://nam02.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.jsoftware.com%2Fforums.htm&data=04%7C01%7Crandall%40newark.rutgers.edu%7C76c373847f6541578a2108d9f3b7a8e0%7Cb92d2b234d35447093ff69aca6632ffe%7C1%7C0%7C637808792843645377%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=4STG3bg9r%2Bl3ZsRZKuPpkoD6vAewex7aI0LRwjoP0s4%3D&reserved=0 ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
