The Circle and Divisor Problem, Essay Example

Dirichlet and Gauss derived the asymptomatic formula for respectively, as x ?infinity. The asymptomatic formulas are evident if we let r₂ to denote the representations of the n integer and d(n) to represent the positive divisors of n. The two asymptomatic expansions have error terms of unknown magnitude. Gauss Circle Problem and Dirichlet’s Divisor Problem are the determined by the exact orders of the error terms, respectively. The exact error terms represent the two most difficult and famous unsolved problems in number theory.

r₂ (n) is an arithmetic function that denotes the integer n’s representation as the sum of two squares. The convention of the arithmetic mean is that the different orders and different signs result to distinct representations. The circle problem determines r₂ (n) and the error term. The d(n) is an arithmetic function that equals n, the value of positive divisors.  E(x) is an error term and The d(n) is an average order that can be determined by the divisor problem. When n  is represented as a sum of squares, it relates to lattice points in a plane. All the lattice points in a circle are related with a unit square.

The divisor problem (The Dirichlet divisor problem) is based on the estimation of n?x d(n). The divisor problem is primarily concerned with the estimation of the lattice points’ values under a particular hyperbola. Dirichlet’s estimation of errors is as follows.

For x > 0, set D(x) := X n?x 0 d(n) = x(log x + 2? ? 1) + 1 4 + ?(x),

On the left hand side, the summation sign’s prime shows that 1 2 d(x) can only be counted as integers. The Euler’s constant is denoted by ? and the error term is shown by ? (x).

It therefore follows that as x ? ?, ?(x) = O( ? x).

Variations of Lehmer’s Conjecture

Lehmer’s unending curiosity on the Ramanujan’s tau-function necessitated the question of whether the a fixed integer is the value of r(n) or it is a Fourier coefficient of a given newform. There are results which hold that the infinitely many spaces are presented for which the primes l ? 37 are not the absolute values of coefficients of any new forms with integer coefficients. In the case of Ramanujan’s tau-function, such results indicate that, for n? 1, _t (n) € (±l : l< 100 is odd prime). The minors of a given graph are the major concepts in graph theory. There are three operations on graphs that determine the concept of graph theory. These concepts are the edge contraction, edge selection and vertex selection.

Ramanujan introduced the Fourier coefficients ? (n) throughout the equation ?(z)₌ ? n=1 ? (n)qn := q ? n=1 (1?qn) 24 = q?24q2 +252q3 ?1472q4 +4830q5 ?·· The tau-function has been a perfect testing ground for the Ramanujan theory of modular forms. The multiplicative properties of the modular forms also foreshadow the Hecke’s operators’ theory. Lehmer had a conjecture that t(n) remains open and does not vanish. For odd ?, it can be proven that t(n) = ? for n that is sufficiently large. The gigantic bounds develops as a result of application of the theory of linear forms. The theory in logarithm is the technique for proving the Lehmer’s conjecture when n is not operated for any ? = ±1. There are natural variants in the Lehmer’s conjectures which are unsolved and that there t(n) never vanishes. For n>1, it can be proven that:

?(n)?{±1,±3,±5,±7,±691}.

Further Progress towards Hadwiger’s Conjecture

Hadwiger conjectured in 1943 that every graph without K_t  minor is (t-1)- colorable for every t? 1. Kostochka and Thomason independently showed in 1943 that every graph without K_t minor has an average degree O (t ? log t) and hence K_t minor is O (t? log t)^?)- colorable for every ?> ¼, making the first improvement on the order of the magnitude of the O(t(log t)^?)- colorable for every ?>0. More specifically, O(t(log log t)⁶)-colorable.

In vertex deletion G represents a graph while the vertex v € V (G) results to a new graph G\v represented by V (G\v) = V (G) \ {v} and E(G\ v) = {e ? E(G) : e. The value of the new graph is incident to v in G} if H € G.

In edge deletion, an edge is defined by e ? E(G). A deletion of the edge of e results to a new graph represented by G\e. The graph is defined by the equation V (G\e) = V (G) and E(G\e) = E(G) \ {e}.

In edge contraction, the edge is defined by e=uv. A contraction of e results to another graph defined by V (G/e) = V (G) \ {u,v} ? {x} and denoted by G/e. G/e denotes a new graph. The new graph will retain all the edges of G that are not incident to v or u thus creating a vertex x which becomes adjacent to NG(u) ? NG(v).

References

EIMI Number Theory Seminar. (2020). https://researchseminars.org/seminar/EIMINT

Extremal and probabilistic combinatorics webinar. (2020). https://researchseminars.org/seminar/EPC