[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.

I've identified Maynard's major and latest work on primes related to "large moduli" (papers I, II, III) which appears rather complicated... 

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

I might need to look up the meaning of notations there. Instead of feeling the strength there, at the last moment, I decide to look into the shorter paper titled "digits of primes" (paper 0), more or less interested. While I feel proper to include Maynard's email address,  it appears wiser to favor number theorists' obsession to be a `hermit'. I do not have a plan to enter the field of number theory. I do not think that reading a paper of primes is a crime. I do not give primes a special rank. I do not... Well, it's ok.

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According to a post*, Maynard has proved in an earlier paper (2016?) that there are infinitly many members in M(d), the set of primes with the missing digit d's. If one uses P(2,3,5,7) to denote the set of primes composed of the digits 2, 3, 5, 7, (i.e. all or some of the four digits), one would have P(2,3,5,7) = M(0) ∩ M(1) ∩ M(4) ∩ M(6) ∩ M(8) ∩ M(9). I refer "pure primes" to the members in the set of P(2,3,5,7). All the subjects on the primes can be re-considered on M(d) or P(2,3,5,7). Say, consider the gaps among the members in M(d) or P(2,3,5,7) or M(0) ∩ M(1), etc. One might use P0 to replace P(2,3,5,7) for short. Here I temporarily use the homemade notations P(·) and M(·) or the term "pure primes" to get involved. Is P0 a trap?  Is it a concern of the paper? So, I'm motivated to see what has happend in paper 0 ——

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Abstract. We discuss some different results on the digits of prime numbers, giving a simplified proof of weak forms of a result of Maynard and Mauduit-Rivat.

---- Digits matter in primes.

---- There are more than one result.

---- Simplified proof is accepted in mathematics.

---- Weak forms can be considered.

---- Former results may not be forgotten.

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1. Introduction: Counting primes

---- Counting matters in primes.

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Many problems in prime number theory can be phrased as `given a set A of integers, how many primes are there in A?'.

---- A set of integers can be a concern in the context of prime theory.

---- A mode of problem is identified: counting primes in a set of integers.

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Two famous examples are whether there are infinitely many primes of the form n^2 +1, and where there is always a prime between two consecutive squares.

---- Two specific examples of prime number theory.

---- For the first one, 5=2^2+1 is a case; 17= 4^2 +1 is another case; 131=11^2+10 is not the case.

---- For the second example, in the case of 11^2=121 and 12^2=144, one has a prime number of 127.

---- Both of the examples provide ways to search primes in the form of counting.

---- Perhaps, the motivation behind is to search or locate primes in an efficient way.

---- For an instant application, one may ask —— whether there are infinitely many pure primes of the form n^2 + 1?

---- Or —— whether there is minimum k, such that there is always a pure primes between n^k and (n+1)^k ?

---- Similar questions can be asked for M(d).

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Here `how many' might be asking for at least one prime, whether there are finitely many or infinitely many primes, or an asymptotic estimate for the number of primes in A of size depending on some parameter x.

---- To summarize the three sub-modes of the problem mode in prime number theory.

---- A parameter x is attached to the size of the domain set A for the third sub-mode.

---- Such as, pi(A) with |A| = f(x).

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Typically in analytic approaches to such questions, one tries to count the number of primes in a set A of a given size.

---- analytic number theory is featured by a set (of integers) of a give size.

---- What does "analytic approaches" mean?

---- By "analytic solutions" one refers to the "solutions with exact expressions".

---- So, "analytic approaches" might be of exact expressions (on primes).

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Almost all our approaches rely on the following rough principle, originally due to Vinogradov but with important refinements due to many authors including Fouvry, Friedlander, Harman, Heath-Brown, Iwaniec, Linnik, Vaughan, as well as many others (see [9] for more details).

---- I do not see a Chinese author on this list of names concerning the "rough principle".

---- I'll check "many others" in [9] later, if available. 

---- When did this rough principle arise?

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Principle. Given a set of integers A ⊆ [1, x], you can count the number of primes in A if you are `good' at counting

* The number of elememts of A in arithmetic progressions to reasonably large modulus (at least on average).

* Certain bilinear sums associated with the set A.

---- prime counts in A is related to member counts in A.  

---- "arithmetric progressions" might refer to a arithmetric procedure.(?)

---- What is the form of "bilinear sums"?

---- I'd like to see a form of A·a + B·b + C·c + ... .

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Here we have been deliberately vague as to what we mean by `good', `reasonably large', or bilinear sums, since these can vary from application to application.

---- I expect to understand these terms by examples.

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Comments: I see the sign of a friendly paper and expect to finish this reading.

♙ 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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Terms of awareness/ use