[注：下文是一篇完整的邮件笔记，有关迪利克雷定理中 ∑χ(n)/n≠0 的证明。]

This is coming to you from Yiwei LI (PhD, Applied math), Taiyuan University of Science and Technology  (TYUST) Taiyuan, China

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

H onesty shows the way.

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Story - to Arts.

♙ ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ∉ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ χ ξ η π Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ⌈ ⌉ ∨∧∞Φ⁺⁻⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹

Notice: learning notes here are homemade and may permit guesses, mistakes or critical thinking. I may use chatGPT for asistance.

(P.M.) Para 7 ——

Now, χ is periodic of period D, and ∑(n)χ(n) = 0.

---- (n) ~ 1 ≤ n ≤ D.

---- This is abrupt that should be avoided. (I cannot see ∑(n)χ(n) = 0 with n = 1,...,D).

---- Recall, by definition, χ(a) = 0 if and only if (a, D) > 1.

---- So, among χ(1), χ(2), ..., χ(D), one only knows χ(1) = 1 and χ(D) = 0 and χ(D - 1) ≠ 0.

Remark: ∑(n)χ(n) = 0 appears not used in the proof.(?)

So the numbers χ(1), χ(1) + χ(2), χ(1) + χ(2) + χ(3), ... are bounded in absolute value by D.

---- This is to say | χ(1) + χ(2) + .... + χ(n) + ... | ≤ D. (for n -> ∞)

---- Well, ∑(n)χ(n) = 0  is used here (with n = 1,..., D).

Since bn↘0, the standard Abel rearrangement of the infinite sum ∑χ(n)bn shows that |∑χ(n)bn| ≤ Db1 = D, contradicting the unboundedness of f on [0, 1).

---- What is Abel rearrangement ?

General remarks: The proof has three four key steps ——

1. Introduction of cn, the sum of χ(d) for all d dividing n;

2. Introduction of T(n), to form f(t) = ∑χ(n)T(n) or ∑cnt^n, showing f unbounded in [0, 1).

3. Introduction of bn by attaching 0 = (1 - t)∑χ(n)/n to -f (= 0 - f), to form -f(t) = ∑χ(n)bn, showing f bounded in [0, 1).

bn     T(n)

.

χ(n)     cn

Another Royal story ——

His Majesty expressed wishes to seize territory of natural numbers (∑χ(n)/n≠0). Bishop suggested taking the way to the Arts (∑χ(d)). Army was formulated (t^n/(1 - t^n)) to form Royal force (∑χ(n)T(n)). "Let the Arts take effect". And the Arts take effect(∑cnt^n), showing the unboundedness of Royal force. Noble felt stressed. They expressed the opposite opinion (∑χ(n)/n=0). Now, Royal force was reformulated(∑χ(n)bn) and bounded. Historian had a comment: Royal court followed a null periodic table while Noble declined.