[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...
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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?
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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.
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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---- There have been views elsewhere that Fourier transform are outdated.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.
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Here, and throughout the paper, e(t): = e^2πit is the complex exponential.
---- clear.
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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?
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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) ].
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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.
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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...
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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) ].
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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.
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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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