Recursive Partitioning of Odd Integers into Primes and Semiprimes: A Novel Framework Toward Validating Lemoine’s Conjecture

The subject which I am going to discuss in this lecture excels in the richness of its ramifications and in the diversity of its relations to other mathematical topics. I think therefore that it will serve our present purpose better not to at tempt a systematic treatment, beginning with definitions and proceeding to lemmas, theorems, and proofs, but rather to look around and to envisage some outstanding marks scattered in various directions. I hope tha t the intrinsic relationships connecting the problems and theorems which I shall mention will nevertheless remain quite visible. A good deal of the investigations about which I shall report can be subsumed under the heading of analytic number theory, and, more specifically, analytic additive number theory. It would, however, be a misplacement of emphasis if we were to look upon analysis, which here means function theory, only as a tool applied to the investigation of number theory. I t is more the inner harmony of a system which I wish to depict, properties of functions revealing the nature of certain arithmetical facts, and properties of numbers having a bearing on the character of analytic functions. Whereas the multiplicative number theory, which deals with questions of factorization, divisibility, prime numbers, and so on, goes back more than 2000 years to Euclid, the history of additive number theory, in any noteworthy sense, begins with Euler less than 200 years ago. In his famous treatise, Introductio in Analysin Infinitorum (1748), Euler devotes the sixteenth chapter, De partitione numerorum, to problems of additive number theory. A partition is, after Euler, a decomposition of a natural number into summands which are natural numbers, for example, 6 = 1 + 1 + 4 . We can impose various restrictions on the summands ; they may belong to a specified class of numbers, let us say odd numbers, or squares, or cubes, or primes; it may be required that they be all different; or their number may be preassigned. I wish to speak here only about unrestricted partitions. By the way, only the parts are essential, not their arrangement, so that we do not count two decompositions as different if they differ only in the order of the summands; we can therefore take the summands ordered according to their size.

Number theory (or the theory of numbers [1]) starts from elementary (not necessarily simple and easy, and in fact, it is often very hard) number theory and grows up with analytic number theory (including additive and multiplicative number theory), algebraic number theory, geometric number theory (including arithmetic algebraic geometry), combinatorial number theory, computational number theory (including algorithmic number theory), to name just a few. Applied number theory, on the other hand, is involved in the application of various branches of number theory to a wide range of areas including, e.g., physics, chemistry, biology, graphics, arts, music, and particularly computing and digital communications [2]. Number theory was once viewed as the purest of the pure mathematics, with little application to other areas. However, with the advent of modern computers and digital communications, number theory becomes increasingly important and applicable to many areas ranging from natural sciences, engineering to social sciences.

For a number of years, French mathematicians have run regular number theory conferences to which they have invited number theorists from many countries. To repay their hospitality, the London Mathematical Society arranged for the 1980 'Journees Arithmeiques' to be held in Exeter. The papers published here are either based on the main invited lectures or on selected research talks given at the conference. They cover all branches of the subject: combinatorial and elementary methods; analytic number theory; transcendence theory; Galois module theory and algebraic number theory in general; elliptic curves and modular functions; local fields; additive number theory; Diophantine geometry, and uniform distribution. It will be necessary reading for all those undertaking research in number theory.

Heini Halberstam, 1926-2014

We exhibit a deterministic algorithm that, for some effectively computable real number c , decides whether a given integer n > 1 is prime within time \mathrm {log} n)^6\cdot(2+\mathrm {log\log} n)^c . The same result, with 21/2 in place of 6 , was proved by Agrawal, Kayal, and Saxena. Our algorithm follows the same pattern as theirs, performing computations in an auxiliary ring extension of \mathbb Z/n\mathbb Z . We allow our rings to be generated by Gaussian periods rather than by roots of unity, which leaves us greater freedom in the selection of the auxiliary parameters and enables us to obtain a better run time estimate. The proof depends on results in analytic number theory and on the following theorem from additive number theory, which was provided by D. Bleichenbacher: if t is a real number with 0 < t \le1 , and S is an open subset of the interval (0,t) with \int_S\mathrm d x/x > t , then each real number greater than or equal to 1 is in the additive semigroup generated by S . A byproduct of our main result is an improved algorithm for constructing finite fields of given characteristic and approximately given degree.

Recent Advances in Analysis and features selected contributions from the AMS conference which took place at Georgia Southern University, Statesboro in 2011 in honor of Professor Konstantin Oskolkov's 65th birthday. The contributions are based on two special sessions, namely Harmonic Analysis and Applications and Sparse Data Representations and Applications. Topics covered range from Banach space geometry to classical harmonic analysis and partial differential equations.Survey and expository articles by leading experts in their corresponding fields are included, and the volume also features selected high quality papers exploring new results and trends in Muckenhoupt-Sawyer theory, orthogonal polynomials, trigonometric series, approximation theory, Bellman functions and applications in differential equations. Graduate students and researchers in analysis will be particularly interested in the articles which emphasize remarkable connections between analysis and analytic number theory. The readers will learn about recent mathematical developments and directions for future work in the unexpected and surprising interaction between abstract problems in additive number theory and experimentally discovered optical phenomena in physics. This book will be useful for number theorists, harmonic analysts, algorithmists in multi-dimensional signal processing and experts in physics and partial differential equations.

Analytic number theory is a branch of number theory which inherits methods from mathematical analysis in order to solve difficult problems about the integers [...]

Klaus Friedrich Roth, 1925-2015

The authors set themselves two main objectives: to characterize the main stages of the life of the outstanding Russian mathematician, Honored Scientist of Russia, Professor of Moscow State University named after M. V. Lomonosov, head of the department of number theory Steklov Institute, Doctor of Physical and Sciences Anatolii Alexeevich Karatsuba and give a brief analysis of his scientific work, which has had a significant impact on the development of analytic number theory. Sufficiently detailed description of the research of Professor A. A. Karatsuba and his disciples on the analytic theory of numbers, which are allocated following the Karatsuba, three main areas: 1) trigonometric sums and trigonometric integrals; 2) the Riemann zeta function; 3) Dirichlet characters. A. A. Karatsuba, being a disciple of Professor N. M. Korobov, led the scientific schools and seminars on analytic number theory at Moscow State University named after M. V. Lomonosov. Among his many students defended their dissertations, with seven of them later became Doctor of Physical and Sciences. Anatoly Alekseevich has published 158 scientific papers, including 4 monographs and a classic textbook on analytic number theory, was the translator of a number of fundamental scientific monographs. He was a member of the editorial board of the journal Mathematical notesand a member of the program committees of several international conferences on algebra and number theory.

Since the deep paper by Bohr and Kalckar in 1938, it has been known that the Ramanujan formula in number theory is related to statistical physics and nuclear theory. From the early 1970s, there have been attempts to generalize number theory from the space of integers to the space of rational numbers, i.e., to construct a so-called analytic number theory. In statistical physics, we consider parameters such as the volume V, temperature T, and chemical potential μ, which are not integers and are consequently related to analytic number theory. This relation to physical concepts leads us to seek new relations in analytic number theory, and these relations turn out to be useful in statistical physics.

Elementary Methods in Number Theory begins with first course in number for students with no previous knowledge of the subject. The main topics are divisibility, prime numbers, and congruences. There is also an introduction to Fourier analysis on finite abelian groups, and a discussion on the abc conjecture and its consequences in elementary number theory. In the second and third parts of the book, deep results in number theory are proved using only elementary methods. Part II is about multiplicative number theory, and includes two of the most famous results in mathematics: the Erdos-Selberg elementary proof of the prime number theorem, and Dirichlets theorem on primes in arithmetic progressions. Part III is an introduction to three classical topics in additive number theory: Warings problems for polynomials, Liouvilles method to determine the number of representations of an integer as the sum of an even number of squares, and the asymptotics of partition functions. Melvyn B. Nathanson is Professor of Mathematics at the City University of New York (Lehman College and the Graduate Center). He is the author of the two other graduate texts: Additive Number Theory: The Classical Bases and Additive Number Theory: Inverse Problems and the Geometry of Sumsets.

Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics

Providing an algebraic proof that φ is indeed a bijection is an instructive exercise. By exploiting the multiplicative structure of the codomain, we can construct a map ψ : Z+ × Z+ → Z+ which is immediately recognized as a bijection. (No need to resort to algebraic calculation or a pictorial argument with diagonals.) For each pair of positive integers m and n, let ψ(m, n) = 2m−1(2n − 1). Bijectivity of ψ is equivalent to the fact that every positive integer has a unique representation as the product of an odd positive integer and a non-negative integer power of 2. As one referee noted, this fact is also key to Glaisher’s bijection between partitions of a positive integer into odd parts and partitions with distinct parts [1, Ex. 2.2.6; 4, p. 12].

We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number theory. On the elementary side, summation by parts, Euler Maclaurin summation, and Mobius inversion play a prominent role. In the slightly less elementary sphere, we find tools from analysis, such as Poisson summation, generating function methods, Cauchy's residue theorem, asymptotic methods, and the fast Fourier transform. We then describe conjectures and experiments that connect number theory and random matrix theory.

The aim of the series of Oberwolfach meetings on ‘Explicit methods in number theory’ is to bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and L -functions and their special values, modular forms and functions. The meeting provides a forum for presenting new methods and results on concrete aspects of number theory. Considerable attention is paid to computational issues, but the emphasis is on aspects that are of interest to the pure mathematician. In this respect the meetings differ from virtually all other computationally oriented meetings in number theory (most notably the ANTS series), which have a tendency to place actual implementations, numerical results, and cryptographic implications in the foreground. The 2018 meeting featured a mini-course on nonabelian Chabauty theory, so several of the talks were on this topic; the other talks covered a broad range of topics in number theory.