Primes in Arithmetic Progressions to Large Moduli. https://projecteuclid.org/journals/acta-mathematica/volume-156/issue-none/Primes-in-arithmetic-progressions-to-large-moduli/10.1007/BF02399204.full Primes in Arithmetic Progressions to Large Moduli. II. https://eudml.org/doc/164255Primes in Arithmetic Progressions to Large Moduli. III https://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0976723-6/ Dipsersion, sums of Kloosterman sums
Category: Mathematics
Geometric Langlands for GL_1
At the forefront of contemporary mathematics, the Langlands Program—initiated by Robert Langlands in the late 1960s—seeks to relate two seemingly distant worlds. On one side are Galois representations, which encode the symmetries of algebraic equations through continuous homomorphisms from absolute Galois groups into matrix groups. On the other side are automorphic forms, highly symmetric analytic […]
Diophantine Sets
Diophantine sets are subsets defined by polynomial equations over integers. More precisely We obtain them basically by projecting integers solutions of some polynomial equations. Observation 1: Using the statement iff we see that there is no difference between one polynomial equation and a system of polynomial equations. Example: Even numbers are Diophantine: Odd numbers: Non-negative […]
Weyl Quantization
The transition from classical to quantum mechanics represents one of the most profound shifts in the history of science. It is not merely a refinement of existing laws but a wholesale reconstruction of the conceptual and mathematical framework used to describe physical reality. At the heart of this transition lies the procedure of quantization, a […]
Computation of Riemann Zeta function
Euler-Maclaurin type formula with Bernoulli numbers One can accelerate the convergence of the Dirichlet series by acceleration methods to compute zeta function for small imaginary parts. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.9455&rep=rep1&type=pdf Poisson summation to get Riemann Siegel formula. Another way to see this is approximate functional equation. So now we are reduced to computing exponential sums . Using Taylor […]
Toda Lattice, Lax Pairs
In the vast landscape of theoretical physics, certain models, despite their apparent simplicity, serve as profound gateways to entire universes of mathematical structure. The Toda lattice is a preeminent example of such a system. Introduced by Morikazu Toda in 1967, it was conceived as a simple, one-dimensional model of a crystal, describing a chain of […]
Lagrangian Mechanics for Infinite-Dimensional Systems
The elegance of the Lagrangian formalism (from previous post) extends seamlessly from discrete particle systems to continuous physical systems, such as fields and fluids, which possess an infinite number of degrees of freedom. Lagrangian Density For classical field theory, the concept of a single Lagrangian is replaced by a Lagrangian density, denoted by . A […]
Lagrangian Mechanics: Hamilton’s Action Principle
Newtonian formulation of classical mechanics in terms of forces, while a foundational pillar, often confronts significant analytical hurdles in systems with intricate constraints or non-Cartesian coordinates. Its vectorial foundation, though immensely successful in many domains, can lead to intractable equations when grappling with complex geometries or coupled motions. Analytical mechanics, specifically the Lagrangian formalism, represents […]
The Goemans-Williamson Algorithm
Consider the problem of finding the Max-Cut in a weighted graph It is known to be NP-complete to decide if the maximum cut is bigger than some parameter . It’s NP-hard to even approximate the Max-Cut to an approximate ratio of (unless ) If we pick the cut randomly, that is if we decide randomly […]
Poisson Brackets and Nambu Brackets
In classical mechanics, we often start with Newton’s laws (). But there’s a more profound and symmetrical formulation: Hamiltonian mechanics. Here, a system isn’t described by position and velocity, but by generalized coordinates and their conjugate momenta . The set of all possible pairs forms the phase space, a stage where the system’s entire history […]
Classical Mechanics–Observables, states
This post will be an introduction to a series of posts that I am going to write on Quantum Mechanics. In this post I will discuss classical mechanics from a perspective which differs from traditional presentation. I will start with a basic question of what we do in physics. We can say physics is about […]
Regular points, Fundamental Theorem of algebra
Let be a smooth map between two manifolds. We call a regular point provided the derivative is non singular, so that there exists a neighbourhood of such that it is diffeomorphically mapped onto its image.(See Inverse function theorem) A point is called a regular value if contains only regular points. Otherwise we call them crtitical […]
The Feynman path integral, The Trotter Product Formula
The Transition Amplitude as a Path Integral In quantum mechanics, the probability amplitude for a particle to propagate from an initial position at time to a final position at time is given by the matrix element of the time-evolution operator: where is the system’s Hamiltonian. The path integral is derived by evaluating this matrix element […]
Tangent space, Derivative
In the previous post we have seen some examples of manifolds and the most general ways in which they occur. By now we know what we mean by differentiable functions on the differentiable manifolds. But we have not defined what do we mean by derivative of a function. So the the natural way to do […]
Differentiable structures-II
In the previous post we have defined a differentiable structure.The definition allows us to talk of differentiable functions i.e., we gave meaning to the term ” f is differentiable function on the manifold”. So basically, differentiable structure is the topological manifold together with sets of differentiable functions where the sets satisfy some properties. For instance […]
Differentiable structures
I am taking a course on differential topology this semester and will be writing a series of posts on this topic. In this post, we will look at the notion of differentiable structures on a topological manifold. The foundational concepts of calculus, including differentiation, tangent vectors, and integration, are robustly defined within the familiar framework […]
Baker-Campbell-Hausdorff formula
The Baker-Campbell-Hausdorff (BCH) formula addresses the fundamental question of how to combine exponentials of non-commuting operators. Given two operators (or matrices), and , the goal is to find an operator such that If and commute (i.e., ), the solution is simply . However, when they do not commute, is given by a more complex infinite […]
Construction of measures using random variables
Often we come across a problem to construct measures having certain properties. For instance we may want to construct one with its restriction to a class of subsets to be a desired function. We expect such a thing to exist may be because some of previous insights in, say, the finitary aspects of subject or […]
Mordell’s Equation
The equation is called Mordell’s equation. For , it is non-singular and defines an elliptic curve (genus 1, group structure). Rational Points: We first describe the torsion part of the rational solutions. By Nagell-Lutz, these solutions should have integer coordinates. Torsion: Assume that is free of sixth powers. If not we can use the change […]
Euclid’s Elements, Pasch’s Axiom
For centuries, the gold standard for mathematical reasoning wasn’t just inspired by Euclid’s Elements – it was Euclid’s Elements. Compiled around 300 BCE, this monumental 13-book collection systematically derived a vast body of geometry and number theory from a small set of explicit starting points. It begins with fundamental plane geometry (Book I covers basic […]
Sophie Germain’s Theorem
As early attempts to the proof of Fermat’s last theorem, many mathematicians solved the problem for small exponents. While these special cases are being studied, Sophie Germain, a French mathematician, came up with the following interesting result. (Look at https://www.agnesscott.edu/lriddle/women/germain.htm for her fascinating and revolutionary story) Theorem 1: For any odd prime such that is […]
Topological Spaces and Continuity
Our intuitive understanding of continuity is that, a function is continuous at when points close to are mapped to points close to When we have a notion of distance, for points which get closer and closer to x, the corresponding values of a function should get closer and closer to . Thus, we can formulate […]
A Topological Proof of the Infinitude of Primes
The following is a topological proof of the infinitude of primes due to Furstenberg. Consider a topology on generated by family of sets . It is easy to verify that this collection forms a basis for the topology. Now every non empty open set in is infinite so that no finite set is open. Also […]
Clairut’s Relation: Geodesics on Surfaces of Revolution
One of the recurring themes in mathematics is how symmetry simplifies problems. In differential geometry, surfaces of revolution – shapes like spheres, cylinders, cones, or donuts, formed by spinning a curve around an axis – possess a fundamental rotational symmetry. It turns out this symmetry provides a powerful shortcut for understanding the “straightest paths,” or […]
Kürschâk and Nagel’s theorems (Erdos 1932)
None of the above quantities are integers. Proof: For the first expression, look at the largest prime- when we clear denominators, the denominator is divisible by this prime and numerator is not.For the second expression, if the is smaller then then the quantity is less than one, otherwise there will be a prime between $m$ […]
Betrand Postulate : Erdos( 1932)
The theorem, known as Bertrand’s Postulate, asserts that for any integer , there is always a prime number satisfying . Erdős’s proof (1932) is centered on a careful analysis of the prime factors of the central binomial coefficient, . Contradiction Hypothesis: Assume there exists an integer for which no prime p exists in the interval […]
.Morley’s Trisector Theorem
Morley’s Trisector Theorem states that for any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle. Its beauty lies in the surprising emergence of a perfectly regular shape from an arbitrary starting triangle Let’s first look at direct proof by trigonometric computations. For a with angles , we have […]
Bernoulli numbers, Faulhaber’s formula, Umbral Calculus, Volkenborn integrals
Within the study of special functions and number theory, certain notational conventions occasionally arise that are so elegant and effective they appear to be a kind of magic. One of the most beautiful examples of this is the umbral calculus developed for Bernoulli numbers, where the indexed numbers are formally treated as powers of a […]
Zeta(2)
Basel Problem 1644 asks to find the value of Euler (1735) showed that The result, , is a cornerstone of analysis, and there are diverse methods of proof from function theory, harmonic analysis, and geometry. All of them essentially uncover how integers sit inside real numbers. Euler’s Proof: Euler’s original method relies on two representations […]
Eisenstein’s Lattice Point Proof of Quadratic Reciprocity
The Law of Quadratic Reciprocity is a cornerstone of classical number theory—Gauss himself called it the “Theorema Aureum” (Golden Theorem). Although Gauss provided multiple proofs, Eisenstein’s geometric argument simplifies Gauss’s third proof by employing a lattice‐point counting method. Statement of the Law of Quadratic Reciprocity Let and be distinct odd primes. The Law of Quadratic […]
A Neighbourhood Of Infinity 