Quantum Simulation of the Factorization Problem.

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.

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

Complex analysis is found in many areas of applied mathematics, from fluid mechanics, thermodynamics, signal processing, control theory, mechanical and electrical engineering to quantum mechanics, among others. And of course, it is a fundamental branch of pure mathematics. The coverage in this text includes advanced topics that are not always considered in more elementary texts. These topics include, a detailed treatment of univalent functions, harmonic functions, subharmonic and superharmonic functions, Nevanlinna theory, normal families, hyperbolic geometry, iteration of rational functions, and analytic number theory. As well, the text includes in depth discussions of the Dirichlet Problem, Green’s function, Riemann Hypothesis, and the Laplace transform. Some beautiful color illustrations supplement the text of this most elegant subject.

This thesis consists of three topics. The first one is on quadratic Waring-Goldbach problems. The second topic is about some additive problems involving fourth powers. The last topic is to consider an average result for the divisor problem in arithmetic progressions.\n\n\n\nChapter 1 is an introduction. In Chapter 2 the Lagrange equation with almost prime variables is studied, based on a method of Heath-Brown and Tolev. The techniques involve the circle method and the sieve method in analytic number theory. It is established that every sufficiently large integer, congruent to 4 modulo 24, can be represented as the sum of four squares of integers, each of which has at most four prime factors. In order to derive the desired result, it is important to study the asymptotic formula for a smooth counting function involving the sum of one square of a prime number and three squares of integers in arithmetic progressions. For this purpose, the square sieve of Heath-Brown and the Kloosterman refinement are employed. This is the main task of Chapter 3.\n\n\n\nChapter 4 is devoted to the investigation on the sum of four squares of primes and K powers of two. Inspired by the work of Wooley, this problem is dealt with by applying the linear sieve instead of the four dimensional vector sieve.\n\n\n\nThe theme of Chapter 5 is the three squares theorem with almost prime variables, firstly investigated by Blomer and Br?udern. A sharper result is obtained, in this chapter, by developing the three dimensional weighted sieve.\n\n\n\nIn Chapter 6, the main concern is the exceptional set for the sum of fourth powers. By developing Vaughan's p-adic iteration method, some new estimations are established. The new estimations improve upon the earlier result obtained by Kawada and Wooley. As a consequence of the new estimates, it is established that every sufficiently large integer under a natural congruence condition can be expressed as a sum of six fourth powers of primes and six fourth powers of integers. Moreover, a pair of diagonal quartic forms is considered. In Chapter 7, it is shown that a pair of diagonal quartic equations with 23 variables has nonzero integral solutions under certain reasonable conditions.\n\n\n\nThe purpose of Chapter 8 is to consider a Waring-Goldbach type problem involving twelve fourth powers of primes and one fourth power of a positive integer. The idea of proof is to combine Chen's switching principle and Iwaniec's linear sieve. A new result is obtained, which refines that of Ren and Tsang on this topic.\n\n\n\nIn the last chapter, a distribution problem for the divisor function on arithmetic progressions is investigated. Denote by ?q,b(X) the error term in the divisor problem in the arithmetic progression b(mod q). New asymptotic formulae for the variance A(q, X) =?qb=1|?q,b(x)|2 for X1=4+E ?· q?·X1=2-E and X1=2+E ? q ·?X1-E are derived. The distinct behaviors of A(q, X) in these two ranges are unveiled.

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.

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.

Power spectrum of the fluctuation of Chebyshev's prime counting function

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.

This paper presents an exploratory analytic framework for examining the distribution of prime numbers through complex exponential embeddings and their associated Residue Recurrence Classes (RRCs). Each prime p is mapped to a unit-magnitude complex oscillator ω( p )= e ip , allowing the study of angular dynamics modulo 2π within a multiplicative and rotational setting. This representation generalises classical modular analyses and reveals local angular recurrences and clustering within the prime sequence. Residue Recurrence Classes are defined as collections of primes whose angular residues approximate those generated by the integer or fractional roots of smaller primes. Empirical computations suggest that such classes contain numerous primes and exhibit coherent alignments on the unit circle, indicating possible small-scale regularities within the global modular uniformity of primes. We further introduce a quasi-harmonic prime function, formulated as a root-weighted cosine sum over prime angular residues, and analyse the cumulative complex prime centre summation. The latter exhibits an approximate power-law relation between its real and imaginary components, consistent with self-similar or fractal-like scaling. These observations are empirical in nature and do not contradict the asymptotic equidistribution of primes modulo 2π . Rather, they suggest that within finite intervals, the prime sequence may display local harmonic organisation and interference effects that warrant further formal investigation within analytic number theory.

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.

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by \(\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}\) , we show that for any given α > 0 the interval \((X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]\) contains an integer having no prime factor exceeding Xα for all X sufficiently large.

A number of quantum algorithms have been performed on small quantum computers; these include Shor's prime factorization algorithm, error correction, Grover's search algorithm and a number of analog and digital quantum simulations. Because of the number of gates and qubits necessary, however, digital quantum particle simulations remain untested. A contributing factor to the system size required is the number of ancillary qubits needed to implement matrix exponentials of the potential operator. Here, we show that a set of tunneling problems may be investigated with no ancillary qubits and a cost of one single-qubit operator per time step for the potential evolution, eliminating at least half of the quantum gates required for the algorithm and more than that in the general case. Such simulations are within reach of current quantum computer architectures.

Teaching Quantum Computing with the QuIDE Simulator

The key feature at this conference was the 33 invited papers from the world's leading number theorists. Talks were in three sessions: analytical number theory; arithmetical algebraic geometry; and diophantive approximation. Speakers included: F.Beukers (University of Utrecht); R. Heath-Brown (Oxford); H.L. Montgomery (Ann Arbor, Michigan); T. Nakahara (Saga University, Japan); Y. Zarhin (Academy of Science, USSR).