I intend to give a broad sweep of the methods and results of Probabilistic Number Theory insofar as they apply to Arithmetic Functions. Topics will be developed not only to illustrate the key innovations but also to place them in an historical perspective.
I shall start with first steps in the nineteen twenties and continue until the developments of recent times. Since I believe the mathematician to be more important than the method, I shall include an appreciation of the founders of the field, with many of whom I have been personally acquainted, and of their contributions.
As far as time allows , I shall give complete proofs. Otherwise I shall give references. I shall also indicate other parts of mathematics where the ideas considered play a role. The investigation of such topics will depend upon audience response.
There are few books that explicitly consider the field.
The first book to consider multiplication in terms of the notion
of independence was probably
Statistical Independence in probability, analysis and number theory, M. Kac, Wiley, N.Y.,1959.
I should mention
Probabilistic Methods In the Theory of Numbers, J. Kubilius, Amer. Math. Soc. Translations of Math. Monographs 11. R.I., 1964.
Probabilistic Number Theory I: Mean-Value Theorems,II: Central Limit Theorems, P.D.T.A. Elliott, Grund. der math. Wiss. 239,240, Springer Verlag,Berlin, New York, 1979,1980.
On the Correlation of Multiplicative and the sum of Additive Arithmetic Functions, P.D.T.A. Elliott, Amer. Math. Soc. Memoirs 538, R.I., 1994.
Introduction to analytic and probabilistic number theory, (inFrench) G.Tenenbaum, Inst. Elie Cartan 13, 1990; translated into English in an edition published by Cambridge University Press.
For those interested in topics of a wider nature
Arithmetic Functions and Integer Products, P.D.T.A. Elliott, Grund. der math. Wiss. 272, Springer Verlag, Berlin, New York, 1985.
Duality in Analytic Number Theory, P.D.T.A.Elliott Cambridge Tracts in Math. 122,Cambridge University press, 1997.
There are very many papers in Probabilistic Number theory, published in English, French, German, Hungarian and Russian, according to the school to which the author belongs.
A little familiarity with the basic notions of probability , complex variables would be helpful .