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As I was studying the Möbius $\mu$ function and Gram series, I got myself some pretty nice books:

Ribenboim - The New Book of Prime Number Records

Apostol - Introduction to Analytic Number Theory

Niven, Zuckerman, Montgomery - An Introduction to the Theory of Numbers

Iwaniec and Kowalski - Analytic Number theory

All of them deal with the Möbius $\mu$ function. But none of them dealt with the subject in details other than giving a few theorems and problems...

So I would like to know, If you guys know of some good books that deal exclusively with arithmetic functions?

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    $\begingroup$ Inwaniec ==> Iwaniec and Kowalski. $\endgroup$ – Anweshi Jan 27 '10 at 11:28
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    $\begingroup$ While it doesn't deal exclusively with arithmetic functions, Hardy and Wright's 'An introduction to the theory of numbers' has quite a bit on arithmetic functions and is a bit of a classic within the field. You can check out the table of contents here: books.google.com/… $\endgroup$ – user1073 Jan 27 '10 at 12:59
  • $\begingroup$ "But none of them dealt with the subject in details other than giving a few theorems and problems." I have to say, I'm completely mystified by this remark. Iwaniec and Kowalski is one of the most complete books on analytic number theory you'll find anywhere. $\endgroup$ – David Hansen Jul 22 '10 at 3:10
  • $\begingroup$ I'm sorry, David, but the OP is right. And to give you just one example of what is not covered in I&K, well, it's Probabilistic Number Theory. They do mention the Erdos-Kac but that's it. Otherwise the subject is very wast and deserves at least 2 books the size of I&K (for the moment we have Elliott's two-volume treatise). Also the flavor of analytic number theory, as done by Luca, Erdos, etc. is not covered in I&K. I think this is what the OP is looking for. $\endgroup$ – anon Sep 14 '10 at 3:29
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A MathSciNet search set to Books and with "arithmetic functions" entered into the "Anywhere" field yields 148 matches. Some of the more promising ones:

The theory of arithmetic functions. Proceedings of the Conference at Western Michigan University, Kalamazoo, Mich., April 29--May 1, 1971. Edited by Anthony A. Gioia and Donald L. Goldsmith. Lecture Notes in Mathematics, Vol. 251. Springer-Verlag, Berlin-New York, 1972. v+287 pp.

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Narkiewicz, Wƚadysƚaw Elementary and analytic theory of algebraic numbers. Monografie Matematyczne, Tom 57. PWN---Polish Scientific Publishers, Warsaw, 1974. 630 pp. (errata insert).

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Babu, Gutti Jogesh Probabilistic methods in the theory of arithmetic functions. Macmillan Lectures in Mathematics, 2. Macmillan Co. of India, Ltd., New Delhi, 1978.

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Elliott, P. D. T. A. Arithmetic functions and integer products. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 272. Springer-Verlag, New York, 1985.

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Sivaramakrishnan, R. Classical theory of arithmetic functions. Monographs and Textbooks in Pure and Applied Mathematics, 126. Marcel Dekker, Inc., New York, 1989.

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Schwarz, Wolfgang; Spilker, Jürgen Arithmetical functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic properties. London Mathematical Society Lecture Note Series, 184. Cambridge University Press, Cambridge, 1994.

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Tenenbaum, Gérald Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995.

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    $\begingroup$ See also Chandrasekharan, Intro. to Analytic Number Theory, ETH lectures. $\endgroup$ – Anweshi Jan 27 '10 at 12:45
  • $\begingroup$ Actually, Narkiewicz wrote a book called "Number theory". It is quite comprehensive, but there is a treatment of arithmetic functions. $\endgroup$ – anon Sep 14 '10 at 3:31
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Isn't it weird that nobody's mentioned so far K. Chandrasekharan's treatise on arithmetic functions? OK, they've already mentioned his book on the principles of analytic number theory, yet the book I'm now referring to is

K. Chandrasekharan. Arithmetical functions. Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 167, Springer-Verlag.

You should definitely take a look at it, esteemed rpg16! I consider that none of the texts listed above is more to the point than this one.

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There's also this one: McCarthy, Paul J. Introduction to arithmetical functions. Springer, 1986.

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(As I can't comment)

If you understand German, Springer published last year a translated version of Paul McCarthy's book - "Arithmetische Funktionen" (ISBN: 978-3-662-53731-2).

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