On the additive theory of prime numbers II

The undecidability of the additive theory of primes (with identity) as well as the theory Th(N,+, n -> p\_n), where p\_n denotes the (n+1)-th prime, are open questions. As a possible approach, we extend the latter theory by adding some extra function. In this direction we show the undecidability of the existential part of the theory Th(N, +, n -> p\_n, n -> r\_n), where r\_n is the remainder of p\_n divided by n in the euclidian division

Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy

Number theory is considered, by proposing quantum mechanical models and string-like models at zero and finite temperatures, where the factorization of number into prime numbers is viewed as the decay of particle into elementary particles conserving energy. In these models, energy of a particle labeled by an integer $n$ is assumed or derived to being proportional to $\ln n$. The one-loop vacuum amplitudes, the free energies and the partition functions at finite temperature of the string-like models are estimated and compared with the zeta functions. The $SL(2, {\bf Z})$ modular symmetry, being manifest in the free energies is broken down to the additive symmetry of integers, ${\bf Z}_{+}$, after interactions are turned on. In the dynamical model existing behind the zeta function, prepared are the fields labeled by prime numbers. On the other hand the fields in our models are labeled, not by prime numbers but by integers. Nevertheless, we can understand whether a number is prime or not prime by the decay rate, namely by the corresponding particle can decay or can not decay through interactions conserving energy. Among the models proposed, the supersymmetric string-like model has the merit of that the zero point energies are cancelled and the energy levels may be stable against radiative corrections.Comment: 16 pages, no figure

Noncommutative solenoids and their projective modules

Let p be prime. A noncommutative p-solenoid is the C*-algebra of Z[1/p] x Z[1/p] twisted by a multiplier of that group, where Z[1/p] is the additive subgroup of the field Q of rational numbers whose denominators are powers of p. In this paper, we survey our classification of these C*-algebras up to *-isomorphism in terms of the multipliers on Z[1/p], using techniques from noncommutative topology. Our work relies in part on writing these C*-algebras as direct limits of rotation algebras, i.e. twisted group C*-algebras of the group Z^2 thereby providing a mean for computing the K-theory of the noncommutative solenoids, as well as the range of the trace on the K_0 groups. We also establish a necessary and sufficient condition for the simplicity of the noncommutative solenoids. Then, using the computation of the trace on K_0, we discuss two different ways of constructing projective modules over the noncommutative solenoids.Comment: To appear in the AMS Contemporary Mathematics volume entitled Commutative and Noncommutative Harmonic Analysis and Applications edited by Azita Mayeli, Alex Iosevich, Palle E. T. Jorgensen and Gestur Olafsson. 19 Page

The ternary Goldbach problem

The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth century. In 2013 -- following a line of research pioneered and developed by Hardy, Littlewood and Vinogradov, among others -- the author proved the conjecture. In this, as in many other additive problems, what is at issue is really the proper usage of the limited information we possess on the distribution of prime numbers. The problem serves as a test and whetting-stone for techniques in analysis and number theory -- and also as an incentive to think about the relations between existing techniques with greater clarity. We will go over the main ideas of the proof. The basic approach is based on the circle method, the large sieve and exponential sums. For the purposes of this overview, we will not need to work with explicit constants; however, we will discuss what makes certain strategies and procedures not just effective, but efficient, in the sense of leading to good constants. Still, our focus will be on qualitative improvements.Comment: 29 pages. To be submitted to the Proceedings of the ICM 201

Some topics in the analytic number theory of polynomials over a finite field

There are striking similarities between the ring of integers and the ring of polynomials in one variable over a finite field. This thesis explores some of these similarities from an analytic number theoretic perspective. It develops a polynomial analogue of techniques for extracting number theoretic information from analytic functions known as the Selberg--Delange method. A motivating problem for the original development of this theory was the problem of counting integers with a prescribed number of prime factors. After presenting the theory in the context of counting polynomials with a prescribed number of prime factors in arithmetic progressions and short intervals, a refined version of the method is presented to study some related quantities in more detail. This work has applications to the study of so-called prime number races questions for polynomials with a prescribed number of prime factors. As a prelude to this work on the Selberg--Delange method, an application from the integer version is given. It concerns the distribution of the values of $\omega(n)$, the number of prime divisors of $n$, in different residue classes. We also prove some results concerning the existence and number of prime polynomials whose coefficients satisfy certain conditions. These can be compared with results about the existence and number of prime numbers whose digits satisfy certain conditions. In particular, we study prime polynomials whose coefficients are restricted to a given subset of the underlying finite field and those whose coefficients satisfy a given linear equation. These results make use of additive characters and as prelude to them, a result concerning the correlation of the polynomial analogue of the exponential function with the multiplicative M\"{o}bius function is presented

Rough Numbers and Variations on the Erdős--Kac Theorem

The study of arithmetic functions, functions with domain N and codomain C, has been a central topic in number theory. This work is dedicated to the study of the distribution of arithmetic functions of great interest in analytic and probabilistic number theory. In the first part, we study the distribution of positive integers free of prime factors less than or equal to any given real number y\u3e=1. Denoting by Phi(x,y) the count of these numbers up to any given x\u3e=y, we show, by a combination of analytic methods and sieves, that Phi(x,y)\u3c0.6x/\log y holds uniformly for all 3\u3c=y\u3c=sqrt{x}, improving upon an earlier result of the author in the same range. We also prove numerically explicit estimates of the de Bruijn type for Phi(x,y) which are applicable in wide ranges. In the second part, we turn to the topic of weighted Erdős--Kac theorems for general additive functions. Our results concern the distribution of additive functions f(n) weighted by nonnegative multiplicative functions alpha(n) in a wide class. Building on the moment method of Granville, Soundararajan, Khan, Milinovich and Subedi, we establish uniform asymptotic formulas for the moments of f(n) with a suitable growth rate. Our method also enables us to prove a qualitative result on the moments which extends a theorem of Delange and Halberstam on the moments of additive functions. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem with simple and interesting applications to the Ramanujan tau function and Euler\u27s totient function, the latter of which generalizes an old result of Erdős and Pomerance which shows that as an arithmetic function, the total number of prime factors of values of Euler\u27s totient function satisfies a Gaussian law