[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 a genuine progress of mathematics a great cancellation is inevitable.

♖    ♘   7        5

♔    ♗   2        3

Story - The sword was ready.

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

So comes the last step ——

П(i)(∑(nj)e(nj(α+bⁱ·θ))) ~> П(i)[ e(bα + bⁱ⁺1·θ) - 1 / e(α + bⁱ·θ) - 1 ] 

shorthands:

--- ( i ) ~ i = 0, ..., k - 1;

--- ( nj ) ~ 0 ≤ nj < b.

Note: here, the index of ni is replaced by nj for clarity. 

The actual target is ——

∑(nj)e(nj(α+bⁱ·θ)) ~> e(bα + bⁱ⁺1·θ) - 1 / e(α + bⁱ·θ) - 1

---- The denominator appears the handle...

---- First guess ——

∑(nj)e(nj(α+bⁱ·θ)) = e(α + bⁱ·θ) - 1·∑(nj)e(nj(α+bⁱ·θ)) / e(α + bⁱ·θ) - 1

---- Now, one focuses on ——

    e(α + bⁱ·θ) - 1·∑(nj)e(nj(α+bⁱ·θ))

= ∑(nj)e(α + bⁱ·θ)e(nj(α+bⁱ·θ)) - ∑(nj)e(nj(α+bⁱ·θ))  (#)

---- Now, one focuses on ——

   e(α + bⁱ·θ)e(nj(α+bⁱ·θ)) 

= e(α + bⁱ·θ + nj(α+bⁱ·θ))

= e((1+nj)(α+bⁱ·θ))

---- It appears to have a "right" look.

---- Now, return to (#) for a replacement ——

∑(nj)e((1+nj)(α+bⁱ·θ)) - ∑(nj)e(nj(α+bⁱ·θ))

= e((1+n0)(α+bⁱ·θ)) + e((1+n1)(α+bⁱ·θ)) + ... + e((1+nb-1)(α+bⁱ·θ))

- [ e(n0(α+bⁱ·θ)) + e(n1(α+bⁱ·θ)) + e(nj(α+bⁱ·θ)) + ... + e(nb-1(α+bⁱ·θ)) ]  ($)

---- We are ariving at the target ——

---- For the fixed i of (α + bⁱ·θ), the coefficient nj is nothing but the digital number 0, 1, ..., b -1 in order.

Note: One may need to check this point closely in the last note* (near the end).

---- So, 1 + nj is just 1, 2, ..., b, a shift of nj to the right-hand side by one step.

---- In a genuine progress of mathematics, a great cancellation is inevitable.(TOM) 

---- So the left terms of ($) are ——

   e((1+nb-1)(α+bⁱ·θ)) - e(n0(α+bⁱ·θ))

= e(bα + bⁱ⁺1·θ) - 1.

---- Take it to the top, one has ——

∑(nj)e(nj(α+bⁱ·θ)) = e(bα + bⁱ⁺1·θ) - 1 /  e(α + bⁱ·θ) - 1

.

Comment: in retrospect, it was a correct decision to read this paper.

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

.

Terms of awareness/ use