On generating functions in additive number theory, II: lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form $$\begin{aligned} \sum _{n=1}^P e\left( {\alpha }_k\,n^k\,+\,{\alpha }_1 n\right) , \end{aligned}$$ ∑ n = 1 P e α k n k + α 1 n , involving lower order main terms. As an application, we show that for almost all $${\alpha }_2 \in [0,1)$$ α 2 ∈ [ 0 , 1 ) one has $$\begin{aligned} \sup _{{\alpha }_{1} \in [0,1)} \Big | \sum _{1 \le n \le P} e\left( {\alpha }_{1}\left( n^{3}+n\right) + {\alpha }_{2} n^{3}\right) \Big | \ll P^{3/4 + \varepsilon }, \end{aligned}$$ sup α 1 ∈ [ 0 , 1 ) | ∑ 1 ≤ n ≤ P e α 1 n 3 + n + α 2 n 3 | ≪ P 3 / 4 + ε , and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.

Computers as Fresh Mathematical Reality. II. Waring Problem

In this part I discuss the role of computers in the current research on the additive number theory, in particular in the solution of the classical Waring problem. In its original XVIII century form this problem consisted in finding for each natural k the smallest such s=g(k) that all natural numbers n can be written as sums of s non-negative k-th powers, n=x_1^k+ldots+x_s^k. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that almost all n can be expressed in this form. In the XX century this problem was further specified, as for finding such G(k) and the precise list of exceptions. The XIX century problem is still unsolved even or cubes. However, even the solution of the original Waring problem was [almost] finalised only in 1984, with heavy use of computers. In the present paper we document the history of this classical problem itself and its solution, as also discuss possibilities of using this and surrounding material in education, and some further related aspects.

Additive Number Theory via Approximation by Regular Languages

We prove some new theorems in additive number theory, using novel techniques from automata theory and formal languages. As an example of our method, we prove that every natural number [Formula: see text] is the sum of at most three natural numbers whose base-[Formula: see text] representation has an equal number of [Formula: see text]’s and [Formula: see text]’s.

Computers as novel mathematical reality. III. Mersenne numbers and divisor sums

Nowhere in mathematics is the progress resulting from the advent of computers is as apparent, as in the additive number theory. In this part, we describe the role of computers in the investigation of the oldest function studied in mathematics, the divisor sum. The disciples of Pythagoras started to systematically explore its behaviour more that 2500 years ago. A description of the trajectories of this function — perfect numbers, amicable numbers, sociable numbers, and the like — constitute the contents of several problems stated over 2500 years ago, which still seem completely inaccessible. A theorem due to Euclid and Euler reduces classification of even perfect numbers to Mersenne primes. After 1914 not a single new Mersenne prime was ever produced manually, since 1952 all of them have been discovered by computers. Using computers, now we construct hundreds or thousands times more new amicable pairs daily, than what was constructed by humans over several millenia. At the end of the paper, we discuss yet another problem posed by Catalan and Dickson