[注：下文内容是一篇完整的邮件笔记，照例是逐句"手读"论文；末尾有关于二次型起源的AI问答。原标题"spirit of detour"。]
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.
Seeing or showing the answer alone is not sufficient.
♖ ♘
♔ ♗
Story - Spirit is the first.
♙ 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 3 ——
The usual approach to proving Dirichlet's theorem involves several standard analytic techniques (See [2], for example);...
---- "several standard analytic techniques"?
---- How did they arise in author's mind?
---- Seeing or showing the answer alone is not sufficient.
---- An ideal proof is to show the path on its own.(TOM)
.
...the main non-formal step is showing that ∑χ(n)/n ≠ 0 where χ is real and non-principle (the result is also needed for non-real χ, but this is fairly easily handled).
---- Can one view ∑χ(n)/n as the resolvent for the problem (arithmetic progression of primes)?
.
Dirichlet's original non-vanishing proof involved a detour through the theory of binary quadratic forms.
---- What is the true spirit behind binary quadratic forms?
---- How did the binary quadratic form arise anyway?
.
Comments: after a query with AI, I realized the quadratic forms formed a chapter in a course titled "advanced algebra" in my college era. It was not addressed in a spirital manner, however, nor reavealing its significance by showing its most important applications, say, the Dirichlet theorem. Well, I'm very sure that, even if the Dirichlet theorem were mentioned in that souless textbook, it would not be addressed in a spirital manner. Why is that? Very simply, Eastern culture or their characters or their spirits are generally of inhibition.
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Modern proofs generally use ideal theory in quadratic number fields or some complex variable theory.
---- Seeking simpler or shorter proofs are totally justified.
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Elementary proofs are also known, but are more complicated than the one I'll now present.
---- I agree that some problems in primary school can be difficult to solve through the primary approach.
---- In many cases, primary approaches are highly skillful.
.
PS: Questions and Answers with AI.
Say something about "resolvent"?
The term "resolvent" has multiple meanings depending on the context in which it is used. In mathematics, particularly in linear algebra, the resolvent of a matrix or an operator is another matrix or operator that is defined as the inverse of the difference between the original matrix and a scalar multiple of the identity matrix.
The resolvent is commonly used in the study of differential equations, where it plays a crucial role in determining the spectral properties of the underlying operators. It also has important applications in signal processing, control theory, and quantum mechanics.
In general, the resolvent can be interpreted as a tool for solving equations by transforming them into more manageable forms. Its properties and behavior have been extensively studied by mathematicians and scientists from different fields, making it one of the most versatile concepts in modern mathematics.
Whould you view ∑χ(n)/n as the resolvent for the problem concerning arithmetic progression of primes?
The sum ∑χ(n)/n is closely related to the Dirichlet L-function attached to the non-principal character modulo q, where q is the modulus of the arithmetic progression in question. This function is commonly denoted as L(s,χ), where s is a complex number and χ is the non-principal character modulo q.
In particular, when q=1 (i.e., the usual primes), the non-principal character is trivial and the corresponding L-function reduces to the Riemann zeta function. So, in this case, the sum ∑χ(n)/n would not be considered as the resolvent for the problem concerning arithmetic progressions of primes, but rather as a special case of the more general theory of L-functions.
However, when q>1 (i.e., arithmetic progressions other than the usual primes), the non-principal character plays a crucial role in determining the distribution of primes in the corresponding arithmetic progression. In this context, the sum ∑χ(n)/n can be viewed as a resolvent or generating function for counting the number of primes in the arithmetic progression, and has been extensively studied by mathematicians in the field of analytic number theory.
Dirichlet's original non-vanishing proof involved a detour through the theory of binary quadratic forms.
Yes, that's correct. Dirichlet's original proof of the non-vanishing of L(s,χ) at s=1 used a detour through the theory of binary quadratic forms.
Dirichlet observed that the properties of the L-function attached to the non-principal character modulo q are closely related to the behavior of certain sums involving binary quadratic forms. In particular, he showed that the sum of values of χ(n) multiplied by a certain multiplicative arithmetic function over all positive integer solutions of a given binary quadratic form is related to the value of L(1,χ). This reduction allowed him to rephrase the problem of proving the non-vanishing of L(s,χ) at s=1 in terms of solving equations involving binary quadratic forms.
Dirichlet was able to use known results about the distribution of integral points on certain families of quadratic forms to establish a lower bound for the sum in question, which then implied the desired non-vanishing property of L(s,χ). This approach was highly innovative and influential, and provided a new connection between number theory and the theory of quadratic forms.
Later developments in the theory of modular forms and elliptic curves provided a more direct and algebraic proof of the non-vanishing of L(s,χ) at s=1, but Dirichlet's original approach remains an important historical milestone in the development of analytic number theory.
What is the true spirit behind binary quadratic forms?
The true spirit behind binary quadratic forms lies in the deep connections between the theory of quadratic forms and many areas of mathematics, including number theory, algebraic geometry, topology, and physics.
A binary quadratic form is a polynomial of degree two in two variables with integer coefficients. It can be represented as ax^2+bxy+cy^2, where a, b, and c are integers. The study of binary quadratic forms involves understanding their properties and behavior under various operations, such as composition, reduction, and equivalence.
One of the key features of binary quadratic forms is that they provide a natural language for expressing certain arithmetic and geometric questions. For example, problems involving the distribution of prime numbers or solutions to Diophantine equations can often be phrased in terms of finding solutions to particular quadratic forms. In this sense, quadratic forms serve as a bridge between algebra and number theory.
Binary quadratic forms also have important connections to modular forms, elliptic curves, algebraic geometry, and topology. These connections arise from the fact that certain arithmetic functions associated with quadratic forms (e.g., the number of integral points on a curve defined by a quadratic form) can be expressed in terms of Eisenstein series, theta functions, or other special functions. This opens up a rich interplay between different areas of mathematics and leads to new insights and discoveries.
Overall, the true spirit behind binary quadratic forms is the same as that behind much of mathematics: to uncover deep patterns and structures in the natural world and to use them to solve problems and make predictions.
How did the binary quadratic form arise anyway?
The study of binary quadratic forms can be traced back to ancient times, where they appeared in problems of geometry and mechanics. However, it was in the 18th and 19th centuries that the theory of quadratic forms was developed into a powerful tool for studying number theory and algebra.
One origin of binary quadratic forms arises from the problem of finding solutions to Diophantine equations of the form ax^2+bxy+cy^2=m, where a, b, c, and m are integers. Such equations arise naturally in the context of geometry, where they correspond to the question of finding integral points on conic sections (such as circles, ellipses, and hyperbolas).
Another source of binary quadratic forms is the problem of classifying lattices in Euclidean space, which arose in the study of crystal structures and solid-state physics. A lattice is a discrete set of points in n-dimensional space that is invariant under translations by elements of a fixed subgroup of all possible translations. The study of lattices led to the development of generalizations of binary quadratic forms, such as ternary quadratic forms, which involve polynomials of degree two in three variables.
In both cases, the study of quadratic forms involves understanding their properties under various operations, including composition, reduction, and equivalence. This leads to a rich theory with connections to many areas of mathematics, including number theory, algebraic geometry, topology, and physics. Today, the theory of quadratic forms continues to be an active area of research with important applications in cryptography, coding theory, and computer science.
♙ 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
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