A Neighbourhood Of Infinity

Frustenberg showed that for multiplicative independent integers we have http://math.stanford.edu/~akshay/research/blmv.pdf

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Lectures on Algebraic Geometry from ELGA 2011: Lecture Videos Intersection Theory by Joe Harris: Cycles, Intersection Product, Chow Ring, Grassmanian, Lines on Surfaces, Chern Classes, 5-conic problem, Parameter spaces, Excess intersection. Lectures on Deformation Theory by Robin Hartshorne Lectures on Tropical Geometry by Diane Maclagan Lectures on Algebraic Codes by Peter Beelen

We want to count solutions to the equation over a finite field. Detecting non-zero powers of x in terms of characters, we get the following expression for point count in terms of the characters and Jacobi sums. We can write Jacobi sums in terms of Gauss sums by expanding in terms of additive characters. Now […]

Imagine we have a corpus representing a text in English language and we use it for training data. And we would like to assign probabilities to a given document of how likely is it to be a valid english text. A Language model assigns a probability to the document by assuming a probability distribution on […]

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.416.5018&rep=rep1&type=pdf

Zeroes of L-functions are conjectured to have spectral interpretation and the symmetry of a family of L-function linked to the distribution of zeros hence assumed to come from some suitable actions. In the following letter, Sarnak defines a symplectic pairing on a space of distributions (due to Connes) which annihilate some theta functions corresponding to […]

Consider finite abelian groups. We know their character theory which allows us to do harmonic analysis on these groups. The characters allow us to capture all the information about the group. In fact, we explicitly have that natural the character-group element pairing gives an isomorphism . There are several ways to read this isomorphism. For […]

From a set , assume that we want to pick elements that are not divisible by any primes in a set . Let be the set of element of divisible by If we write the inclusion-exclusion to sift out the element divisible by some prime in , we get Even if we know very accurately, […]

Our natural way to think of rationals is that they are ratios (pairs up to equivalence) of integers. So rationals are built out of integers. But what if we are given rationals abstractly, and we want to decide if a given rational is an integer? That is how do we define integers if we are […]

Given a finite extension of fields , how do we determine the elements of norm ? For instance, elements of the form , for in the Galois group, have norm because the norm is equal to Are there any other examples? Example: Consider the field extension . The automorphisms are identity and the conjugata map […]

I am almost always surprised and excited to find functorial changes and dualities turning the problems that look hard in one category into something manageable. Here is a puzzle that I came across recently. It’s called the rectangle tiling theorem and says that any rectangle that is tiled with rectangles with at least one integer […]

We shall start looking at some algebraic K theory. There are several perspectives on this topic. The theory is motivated by ideas in several areas like homological algebra, the theory of vector bundles, homotopy theory, etc. There are some constructions like group completions (symmetric monoids), plus constructions ( rings), Q-constructions (exact categories), Waldhausen constructions (categories […]

For a prime and an integer the Fermat quotient is defined as Let be the smallest valueof for which We want to understand how looks. For instance, Granville conjectured Lenstra conjectures Lenstra proved Granville showed that for . How do we get upper bounds on ? Main ideas: (Smooth numbers+ Equidistribution of multiplicative subgroups) If […]

Legendre transform is a duality concept that comes up in building the dual theory of Hamiltonian mechanics from Lagrangian mechanics.  Legendre transform of a function on a space gives a function on the dual space. In mechanics, for instance, it gives the hamiltonian of the cotangent phase space from the lagrangian of the configuration space( […]

Using Stepanov’s method with auxilliary polynomials inspired from Transcedence theory Heath-Brown get cancellation in Heilbronn’s exponential sum uniformly for

We want to understand statistics of “random” integer polynomials. How do we choose the polynomials? There are several ways to order polynomials — by degree, discriminants, heights of the coefficients etc etc. Here we fix the degree and focus on the ordering by the coefficient heights. If we take , where are chosen uniformly, how […]

Euclidean Algorithm: If given any two elements of a ring , we have quotients and remainders such that , where is strictly “smaller’” than , we say that the ring is Euclidean. For instance, for integers the measure of smallness is the size (absolute value as real number) and we have . How do we […]

What are Hecke Operators? These are some operators acting on automorphic forms on some arithmetic groups. Not all discrete subgroups have these operators and existence of these operators is a reason for why the spectrum of these group is different from a generic spectrum and has different statistics compared to the universal distributions expected. It’s […]

http://www.ma.huji.ac.il/~elon/Publications/Montreal.pdf

Computing zeta functions of nondegenerate hypersurfaces with few monomials (Steven Sperber and John Voight): https://math.dartmouth.edu/~jvoight/articles/sparse-dwork-092813.pdf Notices of the AMS- Bernard Dwork : http://youngp.people.cofc.edu/mem-dwork.pdf

The equation is called the Pell’s equation. There are trivial integer solutions for every . For and , a square, these are the only solutions. But for any non-square , there are infinitely many solutions and all the solution are generated by a fundamental solution. That is there exists a least solution , with such […]

Consider an elliptic curve over in Weierstrass form given by The discriminant is defined as which is essentially the discriminant of the polynomial . Also define the invariant (coming from the coefficients of Weierstrass function) The curve is non-singular precisely when . Let be the number of solutions to mod . We define the quantities […]

Siegel Modular Forms: Consider the Siegel upper-half space defined by where are symmetric real matrices, is a positive definite matrix. Convex combinations of psd matrices are positive define, hence this is a convex subset of the subspace of symmetric matrices. For example gives the usual upper-halfplane . We have the action of on by Mobius […]

Gorodnik’s lecture notes on Dynamical systems in Number theory:https://www.math.uzh.ch/gorodnik/dyn_num/index.html Dave Morris- Ratner’s Book: http://people.uleth.ca/~dave.morris/books/Ratner.pdf Lindenstrauss- Dynamics and Number Theory: http://www.ma.huji.ac.il/~elon/Publications/ICM2010.pdf Akshay Venkatesh- Littlewood Conjecturehttps://www.math.ias.edu/~akshay/research/eklexp.pdf Dmitry Kleinbock- Application of Homogenous dynamicshttps://arxiv.org/pdf/math/0210301.pdf ADDED LATER: MSRI Workshop https://www.msri.org/workshops/742/schedules/19308 Dave Morris Slides https://docs.google.com/viewer?url=https://www.msri.org/workshops/742/schedules/19303/documents/2445/assets/23238

Following is the survey of Katz :https://web.math.princeton.edu/~nmk/old/DeligneRHOverview.pdf

Consider the mapping from the coefficients to the roots. Consider loops in the coefficient space. As you move around in loop and come to same point, the set of roots will come back to the same set but the individual roots could be permuted.To show that the roots cannot be written in terms of nested […]

The equation has infinitely many solution for any coprime integers and . And every solution is given by the translate of a fixed solution by . One can find the solution by the gcd Euclidean algorithm applied to and . (Thus a solution can be obtained by looking at the last convergent in the continued […]

Mordell: https://www.maths.ed.ac.uk/~v1ranick/papers/mordell4.pdf Elkies: http://people.math.harvard.edu/~elkies/Misc/E8.pdf, Characterisation of lattice: https://arxiv.org/pdf/math/9906019.pdf

Theorem: On every smooth cubic surface in , there are exactly 27 lines. How do we prove such a result? Proof Sketch/Ideas: Find a nice cubic surface where you can exactly compute all the lines on the surface. Show that this count doesn’t change under deformations. And that the space of smooth cubic surface is […]

We want to understand how many discriminants D have solutions to negative Pell’s equation. First of all we have some local restrictions on the the prime factors of D if there is a solution. There cannot be prime factors 3 modulo 4. If we restrict to these prime factors for D, (special discriminants) how likely […]

What is largest size of set such that there is no square in the difference set?That is we don’t want solutions to Lower bounds: (Rusza) There exists a set of size contained in Proof: Rusza uses the following property: If there is a set such that its difference set modulo some number doesn’t have squares, […]

Look at the beautiful book The Theory of Irrationalities of Third Degree by Delone and Faddeev. The correspondence is between cubic rings and integral binary cubic forms up to equivalences. This correspondence is crucial to understand statistics of cubic rings (and fields). For instance, how do we count them by discriminants, how are the lattices […]

A positive integer is squarefull if each prime in the factorization occurs at least twice. We want to count the number of squarefull integers less than for a large . In general, numbers with each prime occurring at least times are called powerfull numbers. The indicator functions of powerfull numbers are multiplicative functions. Erdos Szekeres […]

Large Sieve Inequalities are very powerful tools in analytic number theory to show cancellation in sums of harmonics twisted with coefficients that we know little about or hard to deal with (imagine non-smooth arithmetic coefficients). By regarding coefficients as general unknown coefficients, we can still obtain cancellation when we average over a set of “almost” […]

Dirichlet made deep contributions to mathematics, especially in number theory by his work on primes in arithmetic progressions, introducing Characters and L-functions, Class number formula, Units in number fields, Divisor problem, Diophantine approximations, reciprocity laws, special cases of Fermat’s last theorem. In analysis he introduced important ideas like Dirichlet Kernel (Fourier convergence), Dirichlet boundary conditions, […]

where Poisson summation is one of the most beautiful and powerful identities in mathematics. It’s a trace-formula that relates a sum over integer lattice to it’s sum over the dual lattice of integers. In general, it relates a sum over a lattice to a sum over the dual lattice. is periodic function , hence defined […]

Consider the function We have This identity is equivalent to some non-trivial polynomials relations in the coefficients of the function. For example Proof: Apply Hecke operator on to get Let us say is the Fourier expansion for We can compute and see that So we can observe the only negative coefficients that appear in is […]

Say we want to count rational points on the unit sphere of a bounded height and study their distribution. Counting: The number of rational points on the sphere of height bounded by is given by where The rational points of height correspond to primitive integer points on the sphere of radius by clearing the denominators. […]

In this post I shall give a proof of functional equation of Riemann zeta function by poisson summation formula and then a proof to a converse theorem. The Riemann zeta function , defined for   by &fg=000000 and extended meromorphically to other values of by analytic continuation, obeys the functional equation where , and the […]

We want to prove that equation has finitely many solution in integers for any homogeneous polynomial of degree at least One standard way to prove the result to use diophantine approximation and polynomial method. We can create very good rational approximations to the roots of the polynomial if there are solutions to the equation, and […]

Consider holomorphic modular forms and take their derivatives. The resulting functions are not modular. In fact, let Take any weight modular form , that is a holomorphic function satisfying and is holomorphic at infinity, hence looks like We have the following formula for it’s derivatives So instead of just the term , we have extra […]

In  this post I will describe certain applications of Euler’s formula for planar graphs. Euler’s formula states that for a planar graph. Thus acts a invariant for planar graph. Infact, more generally euler’s characteristic is a topological invariant of certain objects like simplical complexes. So a direct application euler’s formula shows that graphs and are […]

Chernoff bounds allow us to get the concentration of sum of independent binary random variables. (We can relax the independence to negative correlation/association etc). The standard proof goes by considering the idea of Bernstein to use moment generating function which allows us to decouple the variables. That is using the fact that exponentials behave nicely […]

Elkies found a solution giving a counter examples to Euler’s conjecture that there are no solutions in positive integers to , the generalizations of Fermat’s equations to multiple variables . In the case , a counterexample was found by computer search, but was much harder, and only after Elkies found the above solution, Frye found […]

To prove Dirichlet’s theorem for primes in arithmetic progressions consider the weighted sums over primes in an AP and sums over primes twisted by characters. and We have the following result which relates the L-values to primes. Proof: Use and evaluate Note that are bounded quantities, hence is bounded provided In the above calculation, if […]

How do the torsion elements of an elliptic curve look? The following theorem says that rational torsion should integral and say much more about the coordinates of elements of finite order. Theorem: Let be an elliptic curve over integers. Then a rational point is a torsion point, that is under the addition of the elliptic […]

The natural mod- projection map is surjective. Proof: We want to show that for any integer matrix with , there is an matrix such that 1) If , we can choose integers such that are coprime. WLOG assume that . Now if divides , then it cannot divide both and ; Because otherwise the determinant […]

The class number formula shows that is roughly Thus studying the distribution of class numbers can be related to the distribution of as varies. https://arxiv.org/pdf/0707.0237.pdf – Soundararajan The average value of the class number The distribution of the L values are governed by the statistics of a random Euler product. https://arxiv.org/pdf/math/0206031.pdf – Granville, Soundararajan

By using harmonic analysis to detect intervals, equidistribution of a sequence modulo one can is equivalent to the following: Weyl’s criterion: A sequence of real numbers is uniformly distributed mod one if and only if for every integer we have That is this criterion gives equidistribution . What is the rate of convergence? Can we […]

Hensel’s lemma gives necessary conditions to lift a solution mod of a polynomial equation to a solution mod . Collecting all these solutions mod , we get a solution in the p-adic integers. Basic Version: If an integer polynomial satisfies then there is unique solution to . All these solutions can be put together to […]

Consider the ellipse with eccentricity given by is called the modulus. (We just assumed the length of the minor axis ) Define to be the “angular” arc-length parameter given by Here are the polar coordinates. We define elliptic functions by the formulae (These can be tough of deformations of and , in fact we get […]

Hamiltonian discovered the following Quaternion Algebra over reals. Hamilton’s quaternions is the four dimensional algebra with the relations Here and hence all reals are assumed to commute with every element. This algebra can also be seen as a two dimensional algebra over complex numbers where is satisfies But the complex numbers are not in the […]

Dirac operator is a differential operator that is formally a square root of a second order operator like the Laplacian. For example, consider the Laplacian on given by . The Dirac operator here is given by which satisfies . Consider the operator that acts pairs of functions on the Euclidean plane given by Using and […]

Given non-zero elements and in a field , the quaternion algebra over a field is the 4-dimensional vector space with the relations In characteristic , the quaternion algebra is defined by the relations Any algebra generated by non-zero elements , satisfying these relations with and non-zero has to be the quaternion algebra, that is it […]

Orthogonal matrices are matrices such that . Skew symmetric matrices correpond to infinitesimal orthogonal matrices (Lie algebra). That is If , . Question: How do we parametrize orthogonal matrices over rational numbers? The standard method over reals is to pick orthonormal basis of columns We have to just pick any unit vector, and then move […]

How the spaces of all quadrilaterals look? What if we fix the side lengths? The analogous question for triangle is not that interesting, if we fix the side length up to plane isometries there are only two triangle with given side lengths that are obtained by intersection of two circles drawn on a base side, […]

Artin primitive roots conjecture is about the number of primes which have a given fixed integer as a primitive root. In fact, if the integer is not a square and not equal to , we have the following conjecture proved conditionally under GRH (Hooley). (See Hooley’s Proof) If or a perfect square, then there is […]

Inequalities of the form and more generally, for , are called Hilbert’s inequalities (Hardy-Hilbert Inequalities). This can be viewed in terms of matrices/operators. Let be defined as We are looking for bounds of the form . By Cauchy-Schwarz, this is equivalent to having Thus we are trying show the operator is bounded and the best […]

Given a integer , consider the set . For how many values of prime , the set exhausts all the non-zero residues. That is for how many primes , the given integer primitive element modulo ? There are some obvious examples where there are a very few/no such primes. For instance take only generates one […]

Rado graph is a graph on countable number of vertices wtih an extension property. For any two disjoint finite subsets of vertices, there is a vertex connected to all the vertices in first subsets and not connected to any of the vertices in the second subset. It can be seen that any two graph with […]

Many results are asymptotic in the sense that certain bounds are true for sufficiently large values of the parameters involved. Sometimes, one can determine the explicit bounds for how large we need the quantities to be by carefully working through the estimates and some other times, we just know that there is a finite number […]

The Ramanujan Tau function is defined by The sequence looks like Ramanujan observed multiplicativity property And these both formulae together can be written as Standard way to prove these properties by the use of Hecke operators. By the Petersson trace formula applied to holomorphic cusp forms on we get where is the Petersson norm of […]

In this post I will discuss some basic category theory. I have discussed what a category is, in a previous post. A category basically consists of some objects and maps between these objects are called morphisms. Morphisms are a way to tell which object in our category are related. Now we want the relationship between […]

Consider the sum We want to prove lower bound on Consider the sieve weights which have the property The contribution from prime powers is small and we get On the other hand opening up the sum over the divisors and switching sums we get, So we have The contribution from is , therefore and partial […]

We will see how to accelerate the convergence of an alternating series to prove the above formula for Start with and write it as by distributing half of each term to two adjacent terms. For example applying this transformation to we get Applying it again, we get After few more times we get, In general, […]

Given two simply connected proper open subsets of the complex plane, there is a conformal bijection between them. How do we prove such a result? The result in particular implies that any simply connected proper open set should be biholomorphic to the open unit disk in the complex plane. In fact if we see that […]

How many numbers can be represented as sums of two cubes? How many of them in exactly one way? How many of them in at least ways etc? Counting the the number of points we count the numbers with multiplicity and it turns out to be . (Each of the variables has around choices which […]

Consider the system How do we get information about the number of periodic orbits of this system? Can we show that there are no periodic orbit given some system? What are some sufficient conditions? Bendixson: On a simply connected domain , if the quantity doesn’t change sign, then the above system has not periodic orbits. […]

Suppose a group acts on a vector space and we want to classify the invariant polynomials on the space. 1. For example, consider the action of on the three dimensional space of binary quadratic forms . The invariant polynomials are generated by the discriminant , that is the ring of invariants in . 2. Similarly […]

One way to define the genus of quadratic forms is by the values represented by the form or by the complete characters( the values of genus characters). In terms of the class group, forms in the principal genus are exactly the square classes, form in a particular genus form a coset of the square classes. […]

Question: Which primes are represented by a given quadratic form of discriminant ? If , then we can see that the form is equivalent to a form of the form . To observe this, choose a matrix of determinant and apply the coordinate transformation to get a form with coefficient . We can always do […]

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.54.5332&rep=rep1&type=pdf

Using properties of factorials and the prime factorization of to get lower and upper bounds on the number of primes less than a number Sylvester improved the idea to get better bounds. But can we reach prime number theorem along these lines?

Reaction-Diffusion Model: Two morphogens – one activator and one inhibitor which has larger diffusion coefficients. Use Fourier transform on the space variables to separate and then we a linear ODE in two variables.

If we multiply two expressions of the form , we get an expression of the same form. That is where and A way to see this is by writing , as a product of conjugates elements (norm) in . So we have Note the forms we are multiplying have the same discriminant ( in our […]

How do we prove stability of a critical point?Idea: If we can find a function non-negative function which doesn’t increase with time and vanishes only at the critical point, then we see that the trajectories sufficiently close to the critical point will stay close to the critical point. For consider the system The linearized approximation […]

Arnold, V. I., Ordinary Differential Equations Differential Equations, Dynamical Systems, and an Introduction to Chaos by Morris W. Hirsch, Stephen Smale and Robert L. Devaney Fred Brauer, John A. Nohel – The Qualitative Theory of Ordinary Differential Equations Nonlinear ordinary differential equations by Dominic Jordan, Peter Smith Guckenheimer, J. C., and Holmes, P., Nonlinear Oscillations, […]

Local Global principle for quadratic forms over a field. Why is it true? How do the local solutions “force” Global solutions? Let stick to rationals. What happens for variables? This case basically corresponds to understanding the form Having a non-trivial rational solution corresponds to being a square. And this can be checked locally! (Every non-square […]

How do we find rational solutions to the equation ? http://people.math.harvard.edu/~elkies/euler_09m.pdf

How does the spectrum of the following operator look? https://arxiv.org/pdf/0908.1093.pdf

June 1, 1796 Jenenser Intelligenzblatt : “Every beginner in geometry knows that it is possible to construct different regular polygons [with compass and straightedge], for example triangles, pentagons, 15-gons, and those regular polygons that result from doubling the number of sides of these figures. One had already come this far in Euclid’s time, and it […]

What happens to say a linear program if the constants in the constraints/objective functions are not known with full certainity and you have to assume some distribution for them to account for uncertainity? Lectures on Stochastic Programming Modeling and Theory http://a https://www2.mathematik.hu-berlin.de/~romisch/papers/Berg07.pdf

Quadratic reciprocity is one of many deep facts of arithmetic relating different local properties (modulo primes). In this post I try to explain it in terms I understand. I always wanted to write about it, but have been postponing. Theorem: Let be two odd primes.We have Here denotes the Legendre symbol which detects whether is […]

We want to prove Ramanujan’s congruence identities for the partition function: We prove them using identities involving series of the generating function: Pentagonal identity: (You can directly prove this or obtain it as consequence of Jacobi -Triple Product identity) Let be a fifth root of unity. Using the following Ramanujan’s identity (obtained from specializations of […]

Norbert Schappacher: http://irma.math.unistra.fr/~schappa/NSch/Publications_files/1998e1_Eisenstein.pdf M.Schmitz: https://www.massey.ac.nz/massey/fms/Colleges/College%20of%20Sciences/IIMS/RLIMS/Volume06/The_life_of_Gotthold_Ferdinand_Eisenstein.pdf Andre Weil: https://projecteuclid.org/download/pdf_1/euclid.aspm/1529259084 Cubic FormsQuadratic Reciprocity and Higher reciprocity LawsElliptic Functions

Consider the differential equation This equation can be solved in integration (quadrature) and every solution is obtained by shifting one particular solution.(Constants of integration)Is it possible to change coordinates in the first equation and reduce to the second case where we can solve equation explicitly.One way to view the reason for this solvability is the […]

http://irma.math.unistra.fr/~schappa/NSch/Publications.html

Let us assume we have How do we prove the following? Here is how a proof goes: Using the identity and comparing the Dirichlet coefficients, we get Now summing over , Using PNT for sufficiently large and the Chebyshev bound for smaller , we get Replacing by introduced an error of because therefore and cancelling […]

https://www.math.arizona.edu/~swc/notes/VideoAll.html

Goldston’s Notes: https://arxiv.org/pdf/math/0412313.pdfhttps://web.williams.edu/Mathematics/sjmiller/public_html/ntandrmt/talks/PrCrlTransp_Gonek3.pdf

If we have map between two Banach spaces which preserves distances, then it has to be a linear! (Affine) Let be the map. To prove that is affine it is sufficient to show that Consider the defect which is bounded by Construct another isometry where is a reflection in the Y space about the point […]

The goal is to partition a dense graph into few pseudorandom parts. Edge density -regularity: For a pair of vertex sets and is called -regular, if for all subsets satisfying we have Regular partition: A partition of into sets is called an -regular partition if A way to achieve such partitions is to take: with1. […]

Binary Quadratic Forms: A integer quadratic form in two variables is (properly) equivalent to another form is one is obtained from the other by integer change of coordinates (with determinant 1). We can to find some “small” representatives in each of the equivalence classes. So we uses some equivalences(change of coordinates) to reduce a given […]

Consider where Write Ramanujan conjectured: Mordell’s Proof of the multiplicativity (1917): Consider the operator: We prove that Proof: For we have the modularity relations: One way to prove the modularity of is to use Jacobi triple product type relations to relate products to theta functions and the modularity of theta functions comes from Poisson summation. […]

I have been interested in sorting algorithms and structures since a long time. The simple looking theory has lot of rich and beautiful structure. Let us see what are the most natural ideas for sort. First of all given a sequence of elements, we use some properties of these elements to order these elements. The […]

Extensions of what? How are we given the Galois groups? How do we choose to view a given group to construct these extensions? Let’s say rationals. And some finite group. Start with a basic group like a cyclic group. How do we construct cyclic extensions? How do they look like? We have to start with […]

Consider the rationals (fractions) between and with denominator bounded by . That is These fractions are called Farey fractions of level and we denote them by The number of fractions is Neighbours: Any two neighbours have the property that In fact, two fractions are adjacent to each other in some Farey sequence iff . In […]

http://arxiv.org/abs/math.HO/9404236

Three Squares Theorem: if is a positive integer not of the form , then is the sum of three squares. We present a proof using geometry of numbers due to Ankeny. We prove it for the case , and squarefree. Let Find a prime such that for all . Therefore we find solutions to implies […]

Problem: Show that the cyclotomic polynomial is irreducible. The standard presentation of irreducibility is by Eisenstein’s criterion: Consider the shift Now observing that every term is and the last term is , we are done by Eisenstein’s criterion. We give an alternate proof by Schonemann. Schonemann’s proof of irreducibility of For a prime consider the […]

A natural number can be written can sum of three squares, if and only if is not of the form for nonnegative integers and Modulo every square has to be , or and hence cannot be Also if divides then all have to be even. These two facts prove the only if part. Now to […]

In this post, we will show that choosing one of the vectors from a epsilon-biased set (that is pseudorandom for for statistics like bias of parity functions), the linearity test works. Constructing small epsilon biased sets reduces the size of our sample space for the randomness used in the BLR test. A set is called […]

Elementary Prime number Theorem1. Selberg symmetry identity : Proof: 2. Tauberian argument Iterative arguments- Using Brun-Titchmarsh Relation to the analytic proof and non-vanishing of zeta(s) on Where are the zeroes?Smoothing of mobius- to get these higher Von Mangoldt functions.Relation to Chebyshev’s estimates? Approximations to mobius.. Error terms in this elementary proof- use generalized Selberg identities […]

A function is linear if either of the following conditions hold: for some (Global description)For all (Local description) What can we say about function which satisfies many of these local requirements? How close is it to a linear function? How does the distance to linear functions related to the fraction of local requirements satisfied? BLR […]

Following is a historical survey by Dorian Goldfeld about the elementary proof of prime number theory and the Erdos-Selberg dispute.

Minimum length of the curve bounding an area A? Maximum area bound by a closed curve of length L? Optimal case: Circle. Steiner Proof: Consider a curve with given perimeter 1. If the region bound by the curve is not convex – there is a chord joining two point on the curve which lies outside […]

A way to think of groups is that they correspond to symmetries. Symmetries of an object satisfy some properties which exactly correspond to the group axioms. An equivalence between an object and its other transformed form is what we mean by symmetry. Act of doing nothing to the object does not transform the object. It […]

The peg solitaire (Hi-Q) is a game defined by moves where one can move a peg orthogonally over an adjacent peg to an empty peg while removing the peg that’s jumped over. One can ask many question about the game, about the possible states that can be reached from a given initial state or about […]

For an odd prime , the Legendre symbol is defined as the quadratic residue symbol, This is a character modulo and is helpful as a “harmonic” in contrast to the Gauss’s notation and which serve as indicators for being a quadratic residue and non-residue. We also distinguish and rest of the quadratic residues. Some properties […]

http://www.crm.umontreal.ca/anatomie06/programme.htmlhttp://www.crm.umontreal.ca/anatomie06/pdf/GoldstonMontrealTalk1.PDF

Weierstrass: Factorials (Shift Identity): Proof: Integration by parts! Relection Formula: (Euler) Proof 1: Start with any product expansion fo But we have Note by differentiating twice, this is equivalent to showing and this can be established by noting that the difference of both sides is a bounded analytic function. Therefore Proof 2: Beta function: Proof: […]

Look at the Gauss Sum for a multiplicative character It’s easy to see by executing the double sum (with a change of variables) that (determining sign is more harder- you can apply poisson summation or Gauss’s -binomial identities etc to determine the sign) Consider the Jacobi Sums. Gauss Sums are the analogues of the Gamma […]

Given a prime we want to find solutions to Method 1: Find the solution of where Expand into a simple continued fraction to the point where the denominators of its convergents satisfy the inequality Then Proof: For any , we have which is true for any continued fraction expansion. Let Take We have Therefore Method […]

Quadratic irrational are solutions to quadratic equations with integer coefficients. Continued fractions: Expansion of the form If then Relation to GL_2 (invertible integers matrices): Quadratic Irrationals and periodic continued fractions: Theorem: An infinite integral continued fraction is periodic if and only if it represents a quadratic irrational. Proof: (Easy) Multiplying it out, we see satisfies […]