Additive Number Theory: Morley Davidson (Kent State University), June 26-30
The question of representing a positive integer as a sum of a certain number of squares was historically a central problem in the development of modern number theory. So our starting point will be the classical problems of representability of a positive integer as the sum of two, three, or four squares (theorems of Fermat, Gauss, and Lagrange). While these problems are solved completely, they are only a small step away from unsolved, notoriously difficult problems involving sums of cubes and higher powers (Waring's problem). Using a strong form of Dirichlet's theorem for arithmetic progressions, we will give Watson's short proof of Linnik's result that every sufficiently large positive integer is the sum of at most seven cubes. Then we will finally meet the powerful Hardy-Littlewood method, through Hua's elegant proof that at most 2k+1 summands suffice to represent every sufficiently large positive integer as the sum of kth powers. We will conclude with a survey of the state of knowledge in Waring's problem, Goldbach's Conjecture (that every even number larger than 2 is the sum of two primes), and the theory of partitions.
Afternoon (break-out) sessions will include problems meant to fill in proofs from the lectures, as well as theorems and problems from the theory of partitions. For example, we might attempt to prove, in digestible steps, the Hardy-Ramanujan-Rademacher formula for the number p(n) of partitions of n.
Reading list: The first volume of Mel Nathansons's two Springer-Verlag books, "Additive Number Theory", subtitled "The Classical Bases", contains much of the material we wish to cover, and are easily obtained.
Books abound on partitions: Apostol's two Springer-Verlag books on Analytic Number Theory are both in print and contain both elementary and advanced theorems.
I recommend reading the original paper of Hardy and Ramanujan which was the genesis of the circle method: "Asymptotic formulae in combinatory analysis" Proc. London Math. Soc. (2) 17 (1918), 75-115.
Parts of this and other articles will be required reading during the week.
Analytic Number Theory: Harold Diamond (University of Illinois at Urbana-Champaign), July 3-7
The lectures will focus on multiplicative number theory. Topics to be selected from following list, depending on time and audience preparation and interest.
1. Elementary theory of multiplicative functions. Convolutions
2. Summatory function. Counting square free numbers and primes
3. Analytic theory. Dirichlet series, Euler products, applications
5. Mean values. Elementary theory, Halasz theorem
6. Numbers having only small prime factors
A prose description of the topics:
An arithmetic function f ≠ 0 is called multiplicative if it satisfies the relation f(mn) = f(m) f(n) for all relatively prime positive integers m,n. This modest requirement imposes significant structure on an arithmetic function, and many interesting functions are either multiplicative or are `nearly so.' We are going to examine several aspects of multiplicative functions, including the following.
Some famous number theoretic questions such as the prime number theorem, the Dirichlet divisor problem, and the distribution of square-free numbers will be considered in terms of multiplicative functions.
Multiplicative functions will be characterized in terms of convolutions and exponentials of arithmetic functions. In particular, these functions will be shown to form a group under convolution.
Associated with each arithmetic function is a Dirichlet series, which can provide useful analytic information about the function. For multiplicative functions the Dirichlet series admits a representation in factored form, the so-called Euler product. Several examples of this relation will be studied, including the world's most famous Dirichlet series, the Riemann zeta function.
The summatory function associated with a multiplicative function is often of greater interest than the function itself. Some elementary and analytic techniques will be presented for estimating such functions. In particular, we shall ask Which multiplicative functions have a mean value? Theorems of Delange and Halasz will be presented which give conditions for a mean value.
We shall consider the counting function of integers having no small prime factors. The Dickman function will be introduced and its properties examined.
Suggested reading: G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge studies in advanced math. 46, 1995.
Probabilistic Number Theory: Peter Elliot (University of Colorado at Boulder), July 10-14
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.
Literature: 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.
"Zeta Functions" and/or "Automorphic Functions:" Kannan Soundararajan (Institute for Advanced Study), July 17-21