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[March for reflection |Maynard] view of world

已有 356 次阅读 2022-7-12 18:06 |个人分类:牛津大学|系统分类:科研笔记

[注:下文是群邮件的内容,标题是原有的。内容是学习一篇数学文章的笔记。]

["Terms of awareness /use" folded below] On going is to read a paper of primes to increase generic understanding on mathematics.

In mathematics, one defines the "world" before formal discussions.

 ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁻⁰ 1

Review: the inclusion-exclusion formula may have a concise form of

                #A(ξ)p = #A(η)n - Σ(p)#A(p)n.

Note: Σ(p) denotes Σ(η<p<ξ). It appears a convention to set η = x^1/4ξ = x^1/2.

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Resume the text from the original form of the inclusion-exclusion formula ——

#{ ∈ Ap > x^1/2} = #{ n ∈ A: P⁻(n) > x^1/4} - Σ(p) #{ n ∈ A: P⁻(n) = }.

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The term on the left hand side is counting primes in A.

---- In the last note, it is defined A(x^1/2)p = { p ∈ Ap > x^1/2 }.

---- By using the inequality, one creates a "buffer" for the inclusion shown in the first term on the right hand side.

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The first term on the right hand side is a quantity that can be estimated by sieve methods, and one can obtain reasonable upper and lower bounds for such quantities if one can estimate A in arithmetic progressions.

---- In the last note, it is defined A(x^1/4)n = { n ∈ A: P⁻(n) > x^1/4 }.

---- The properly smaller parameter x^1/4 (< x^1/2) is used to exploit the buffer created by the term on the left hand side.

---- One can label the operation here as "buffer skill".

---- "sieve methods" appear to exclude composite numbers from A by using previously known primes to detect the possible prime factors.

---- It seems more than one sieve methods have been invented.

---- Arithmetic progressions are associated to the upper and lower bounds for #A(η)n.

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The final term on the right hand side is counting products of 2 or 3 primes which lie in A where all factors are reasonably large, and so can be viewed as a special case of bilinear sums of the form

Σ(n)Σ(m) αnβm,   N < n < 2N, M < m < 2M,  nm ∈ A

for some complex coefficients |αn|, |βm| ≤ 1.

---- In the last note, it is defined A(p)n = { n ∈ A: P⁻(n) = p }.

---- In the inclusion-exclusion formula, p ranges between x^1/4 and x^1/2.

---- Alternatively, one may define A(η, ξ)n = { n ∈ A: η < P⁻(n) ≤ ξ}.

---- As such, one has  #A(η, ξ)n = Σ(p)#A(p)n.

---- In this way, the formula has the simpler form of

                          #A(ξ)p = #A(η)n - #A(η, ξ)n. 

---- According to the text, #A(η, ξ)n = Σ(n)Σ(m) αnβm.

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Comments: it's not apparent why complex coefficients of unit norm are involved.(?)

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By iterating such identities, we can therefore obtain {combinatorial decompositions for the number of primes in Awhich can be estimated by using our knowledge of A in {arithmetic progressions} or {certain bilinear sums over moderately large variables whose product lies in A}.

---- Question: how to iterate such identities?

---- If one denotes "primes in A" as A(p), the sentence states that ——

#A(p) ~> combinatorial decompositions ~> estimation through: (1) arithmetic progressions; (2) bilinear sums (over...).

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Typically it is rather harder to estimate the bilinear sums associated to a set A than it is to estimate A in arithmetic progressions...

---- Estimation through "bilinear sums" is harder than "arithmetic progressions".

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...and particularly so if A is sparse in the sense that it contains O(x^1-ε) elements.

---- "bilinear sums" hard for sparse cases.

---- What is a "spase" case?

---- Take x=10,000 and ε=0.1, one has x^1-ε ≈ 3981, about 40% of elements in [1, x].

---- Big O? Most of the time, one can forget that.

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(We fail to prove the n^2+1 conjecture precisely because we have no way of estimating the bilinear terms- the sparsity of the sequence means that it is insufficient to consider bilinear sums with general coefficients in this case.)

---- The sparsity of n^2+1 overflows the ability of bilinear sums.

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Moreover, the condition nm ∈ A in the bilinear sums is difficult to handle unless A has some `multiplicative structure'.

---- nm ∈ A ~> multiplicative structure.

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This is why the only sparse polynomials known to take infinitely many prime values (such as the results of Friedlander-Iwaniec [?] on primes of the form X^2 + Y^4) are closely associated to norm forms which have such multiplicative structure.

---- sparse polynomials ~ infinitely many prime values ~ multiplicative structure.

---- example of sparse polynomials: X^2 + Y^4.

---- Question: what is a "norm form"?

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Comment: it appears A serves as the "world" in the realm of analytic number theory.

 ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁻⁰ 1

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