Advances in Number Theory

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

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.

As we begin a new year full of potential and opportunity, we wish each of you much joy and success. As we embark on this journey, we also celebrate a significant milestone -the fifteenth anniversary of the European Journal of Geography. Over the past decade and a half, our journal has become a beacon of excellence in the field of geography and the social sciences. This journey has been characterised by unwavering commitment and tireless dedication, a collective endeavour led by the dedicated members of our editorial team and the European Association of Geographers (EUROGEO). Their diligence and passion have been instrumental in making our journal the respected publication it is today. Looking back on our shared history, we are proud to have published over 310 articles dealing with key topics in geography, planning and development. These scholarly contributions have not only explored and analysed important topics, but have also introduced new ideas (Koutsopoulos, 2022; Manetos et al., 2022 ), methods (Cramer-Greenbaum, 2023; Krevs et al., 2023; Morawski & Wolff-Seidel, 2023) and data (Hojati & Mokarram, 2016) that will inspire future generations of geographers to transcend conventional disciplinary boundaries. The richness of our content encompasses numerous facets and includes the fields of geography education (

Summary. The computational solution of a certain class of differential-difference equations requires numerical procedures involving an extremely high degree of precision to obtain accurate results over a large range of the independent variable. One method of solution uses an iterative procedure which relates the differentialdifference equation over a large range to a system of ordinary differential equations over a limited range. When the characteristic roots of the related system indicate borderline stability, it is evident that small perturbations in obtaining successive initial values eventually grow out of control as the system increases. To investigate this phenomenon, we examine the equation u' (x) =-u (x -1) /x. arising in analytic number theory.

Combinatorial group theory has roots in Poincaré's work on the topology of manifolds, which in turn was based on problems in differential equations and analytic number theory. Thus the Fuchsian groups, which are the fundamental (first homotopy) groups of oriented negatively curved compact surfaces, served as important models in their day. In the last few years there have been advances in the understanding of the structure of fundamental groups of negatively curved manifolds, some of them based on examples from analytic number theory. Here I describe one of these developments and pose a few difficult combinatorial questions.

An interview with I.M. Kolthoff

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .

This paper introduces the Spectral Riemann Conjecture (SRC), a novel reformulation of the classical Riemann Hypothesis (RH) through the lens of spectral breaking theory. The Riemann zeta function ζ(s), central to analytic number theory, possesses trivial zeros at negative even integers and non-trivial zeros conjectured to lie on the critical line Re(s) = 1/2. Building on the verified simplicity of the first 109 non-trivial zeros, the SRC proposes a two-step transformation: decoherence (D), which projects all non-trivial zeros onto Re(s) = 1/2, creating a zero of infinite multiplicity; followed by renormalization (R), which regularizes this multiplicity to one. The resulting function, spectral(s), resides in the spectral space—a domain of functions with real and simple zeros. The paper rigorously defines the operators D and R, demonstrates their action on ζ(s), and proves the convergence of spectral(s) using zeta and Hurwitz regularization techniques. Numerical simulations validate the SRC by approximating ζ(s) with finite products and applying D and R to yield spectral(s), whose zeros are located at {-2, -4, ..., 1/2}. This transformation preserves the infinite zeros, maintaining the arithmetic essence of ζ(s) while simplifying its non-trivial structure. The SRC offers a fresh perspective on RH, aligning with physical regularization analogies and suggesting potential applications in prime number distribution. Future work may extend these simulations and explore deeper implications within analytic number theory and mathematical physics.

We obtain several new bounds on exponents of interest in analytic number theory, including four new exponent pairs, new zero density estimates for the Riemann zeta-function, and new estimates for the additive energy of zeroes of the Riemann zeta-function. These results were obtained by creating the Analytic Number Theory Exponent Database to collect results and relationships between these exponents, and then systematically optimising these relationships to obtain the new bounds. We welcome further contributions to the database, which aims to allow easy conversion of new bounds on these exponents into optimised bounds on other related exponents of interest.

Since 1893, the Royal Netherlands Academy of Arts and Sciences has awarded, once every 10 years, a golden Buys Ballot Medal to a scientist who has made outstanding contributions to the development of meteorology. The award was instituted in 1888 in honor of C. H. D. Buys Ballot (1817–90), upon his retirement as professor of physics at the University of Utrecht. Today, Buys Ballot is remembered best by Buys Ballot’s law, which gives the relation between wind and pressure. He derived this law from observations and published it in 1857.1 Buys Ballot was a pioneer both in weather forecasting and in recognizing the need for international cooperation in meteorology. He founded the Royal Netherlands Meteorological Institute (KNMI) in 1854 and served as its chief director until his death, and was first president (1873–79) of the International Meteorological Committee, a predecessor of the World Meteorological Organization (WMO).

Roger Heath-Brown (37) is a Reader in Pure Mathematics at Oxford University, and a Tutorial Fellow at Magdalen College. He works in analytic number theory, with interests covering the distribution of prime numbers, Waring's problem and Fermat's Last Theorem This term I'm giving a first year lecture course—Groups, Rings and Fields. I'm not at my best at 9 am and despite my efforts the undergraduates find me rather dull. The weaker ones have to have everything spelled out in full, with the result that the interesting things are too sparsely scattered for the stronger students. I'm back in college by 10.30 am having collected a pile of mail. There's a request from the Mathematical Gazette for a “day in the life of” style article, and I toy with the idea of describing an occasion on which a journal asks me to describe a typical day's work …. I also receive a referee's report on a paper submitted to me as an editorial advisor to the London Mathematical Society. The report is not encouraging and I reject the paper—a frequent enough duty, but not one I relish. The associated paperwork fills the next three-quarters of an hour. This particular manuscript is rather too similar to something published in 1963. While double-checking this in “Reviews in Number Theory” I notice an abstract of a quite unrelated paper, by a Finnish mathematician.

Lectins: Carbohydrate-specific Reagents and Biological Recognition Molecules

Collections made in Kentucky Reservoir during 1958 indicate that the pre-impoundment assemblage of species, characterized by a preponderance of species belonging to the Unioninae, is doomed; only Quadrula quadrula has adapted to the altered ecological conditions. This species along with two species of Leptodea, one species of Carunculina, and three species of Anodonta comprise the dominant species assemblage in post-impoundment habitats. The impoundment of the waters of the Tennessee River by the TVA has produced a radically altered environment for the survival of the native mussel fauna. The singularly rich pre-impoundment mussel fauna of this area consisted almost exclusively of typical large river forms, many of which are of considerable commercial value; the pigtoe (Pleurobema cordatum) and the niggerhead (Fusconaia ebenus) are the two most abundant commercial species. Present concern for the future of this mussel fauna is evidenced by the 1956-57 study of this resource undertaken by the U. S. Fish and Wildlife Service (Scruggs, 1960). Scruggs' study adequately summarizes the present status of the native pre-impoundment fauna occurring above the area considered here but does not record any mussel invasion of the extensive mud shallows which have resulted from impoundment. Ecologically, the invasion of these virgin shallows is of prime interest and is the subject of this report. Grateful acknowledgment is due Dr. Henry van der Schalie, Museum of Zoology, The University of Michigan, and Dr. Ruth Patrick, Department of Limnology, The Academy of Natural Sciences of Philadelphia, for their generous help and criticism in the preparation of this paper. During October, 1958, sampling was completed over approximately sixteen river miles.2 On TVA Navigation Charts, this area extends from mile 88 to mile 105 and includes the entrance of Duck River. This region is from 4 miles above to 12 miles below Ithe U.S. Route 70 bridge at New Johnsonville, Tennessee. The deep water collections for adult mussels were made with a small crowfoot brail operated by a local clammer. A Peterson dredge was used to, obtain juveniles. In the shallows, hand picking, pollywogging, raking, and screening were employed. The Kentucky Reservoir in the area studied presents three main types of habitats suitable for mussel populations. The first is the old river channel with its firm sand and gravel substrate which gradually is being covered with a deposit of silt; the second is the extensive mud 1 Present address: 1900 Dexter Avenue, Ann Arbor, Michigan. 2 These collections were made under the auspices of the Limnology Department, The Academy of Natural Sciences of Philadelphia.

We begin with the duality between analytic number theory, combinatorial identities and q-series, to indicate the historical development of the allied disciplines. It is irrelevant what notation we use for the Γ-function, the essential part is that we keep this notation. Section 3.7 is devoted to this important function and the hypergeometric function. We use a vector notation for the Γ-function and introduce the concepts well-poised and balanced series. The binomial coefficients also play an important part since a finite hypergeometric series can always be expressed in two equivalent ways. The three Kummerian summation formulae (and their multiple q-analogues) will follow us in future chapters. We summarise the different schools for Theta functions and show that the elliptic function snu can be written as a balanced quotient of infinite q-shifted factorials. We conclude this chapter with definitions of the most important orthogonal polynomials; we keep Jacobi’s definition for the Jacobi polynomials.