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[March for reflection:|:Maynard] varying of base

已有 241 次阅读 2022-7-23 23:46 |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.

Fourier, Fourier, Fourier... 

    ♘   7        5

 

    ♗   2        3

Story - General appeared on Saturday.

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

5. Fourier analysis on digit functions

---- When was Fourier analysis introduced to number theory?

---- Or, what was the original context to apply Fourier analysis in number theory?

---- Does this mean to treat numbers as a signal?

.

The proofs of Theorems 2.1-2.3 are Fourier-analytic in nature, and ultimately rely on the fact that many digit-related functions are very well controlled by their Fourier transform.

 ---- "Fourier -analytic"... Analytic number theory appears to use utilities and philosophy from analysis theories.

---- By "analysis", one divides or projects the object of study for a close look.

.

Given a function f: ℤ → ℂ, we define the Fourier transform  ^fx : ℝ/ℤ → ℂ of f restricted to [0, x] by ^fx(θ) := (n<x) f(n)e(nθ).

---- Fourier transform is active in analytic number theory.

---- There have been views elsewhere that Fourier transform are outdated.

---- Apparently not in Maynard's paper.

.

Here, and throughout the paper, e(t): = e^2πit is the complex exponential.

---- clear.

.

Our weak version of Theorem 2.1 is based on understanding ^gx when g(n) = e(αsb(n)) where α∈{0, 1/m, ..., (m - 1)/m} and sb(n) is the sum of digits in base b.

---- The form of g(n) = e(αsb(n)) appears abrupt.

---- How did it arise in the inventor's mind?

---- What does "m" refer to?

.

In particular, writing n = ∑(i)nib^i in its base b expansion, we find


^gb^k (θ) = ∑(n) e(nθ)e(αsb(n))

                 = ∑(ni)e(∑(i)ni(α+b^i·θ))

                 = П(i)((ni)e(ni(α+b^i·θ)))

                    П(i)[ e(bα + b^i+1·θ) - 1) / (e(α + b^i·θ) - 1) ].

.

Shorthand notation: (n) refers to n < b^k; 

                              (ni) refers to 0≤n0, ..., nk-1< b;

                              (i) refers to i = 0, ..., k - 1. 

.

Thus ^gb^k has a product structure, which will be very convenient to work with.

---- It appears unexpectedly simple (not necessarily easy).

---- I leave it for an off-line check.

.

For our weak version of Theorem 2.3, we work with the Fourier transform of the indicator function 1B of the set B of integers with no base b digit equal to a0.

---- The expression of 1B is not given explicitly.

---- Need a guess...

.

Similarly to the calculation above, we have

.

^1B,b^k(θ) = ∑(n) e(nθ)^1B(n)

                = ∑(ni)e(∑(i)nib^i·θ)

                = П(i)[ e(b^i+1·θ) - 1) / (e(b^i·θ) - 1) - e(b^i·a0θ) ].

.

Shorthand notation: (n) refers to n < b^k; 

                              (ni) refers to 0≤n0, ..., nk-1< b and ni ≠ a0;

                              (i) refers to i = 0, ..., k - 1. 

.

Again, we find ^1B,b^k has a nice product structure.

---- leave for an off-line check.

.

Comment: The key unitities are singled out, good signs for a full reading. As a weird thought, I consider numbers of variable base.

 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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