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
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 […]
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 […]
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 […]
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 […]
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. Differentiable structure on a manifold is a basic setting in which we can talk of differentiability of functions […]
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 […]
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 […]
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- Tchebyshev estimates – Erdos( 1932 -1)
Basic idea to approximately count primes is to look at and look at it prime factorization. To get access to primes between and , we need to look at the binomial coefficients Another way to see it: Use and sum over . Use the resulting expression to estimate the number of primes. (This is just […]
Bernoulli numbers, Faulhaber’s formula, Umbral Calculus, Volkenborn integrals
Consider the following defining formula for Bernoulli numbers in terms of sums of powers of integers: If we formally think of as this formula becomes The formal replacement of with has many nice properties. For instance, the generating series of Bernoulli numbers can be thought of as The Bernoulli polynomial can seen as Consider the […]
Zeta(2)
Basel Problem 1644: Find the value of Euler (1735) showed that Euler’s Proof: But Comparing the coefficients of we get, Fourier Proof: Consider on as a periodic function. Parseval gives Apostol’s Proof: To compute the integral using the change of coordinates to get Using we get The integral are computed below. Hence
Eisenstein Lattice Proof of Quadratic Reciprocity:
Let be distinct odd primes. Start with the following expression for Legendre symbol. which detects if is square modulo or not. By writing , and multiplying the quantities , it is easy to see that where the sum is over even integers . We used that is just a permutation of Therefore we get Now […]
A Neighbourhood Of Infinity 