mathematics

Jurkat–Richert theorem

Jurkat–Richert theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Jurkat–Richert theorem rather than just read about it. In short: The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.

Key takeaways

Jurkat–Richert theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Jurkat–Richert theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Jurkat–Richert theorem from memory before moving on to harder problems.

Reference excerpt

The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture. It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert.

Statement of the theorem This formulation is from Diamond & Halberstam. Other formulations are in Jurkat & Richert, Halberstam & Richert, and Nathanson. Suppose A is a finite sequence of integers and P is a set of primes. Write Ad for the number of items in A that are divisible by d, and write P(z) for the product of the elements in P that are less than z. Write ω(d) for a multiplicative function such that ω(p)/p is approximately the proportion of elements of A divisible by p, write X for any convenient approximation to |A|, and write the remainder as

r A ( d ) = | A d | − ω ( d ) d X . {\displaystyle r_{A}(d)=\left|A_{d}\right|-{\frac {\omega (d)}{d}}X.}

Write S(A,P,z) for the number of items in A that are relatively prime to P(z). Write

V ( z ) = ∏ p ∈ P , p < z ( 1 − ω ( p ) p ) . {\displaystyle V(z)=\prod _{p\in P,p<z}\left(1-{\frac {\omega (p)}{p}}\right).}

Write ν(m) for the number of distinct prime divisors of m. Write F1 and f1 for functions satisfying certain difference differential equations (see Diamond & Halberstam for the definition and properties). We assume the dimension (sifting density) is 1: that is, there is a constant C such that for 2 ≤ z < w we have

∏ z ≤ p < w ( 1 − ω ( p ) p ) − 1 ≤ ( log ⁡ w log ⁡ z ) ( 1 + C log ⁡ z ) . {\displaystyle \prod _{z\leq p<w}\left(1-{\frac {\omega (p)}{p}}\right)^{-1}\leq \left({\frac {\log w}{\log z}}\right)\left(1+{\frac {C}{\log z}}\right).}

(The book of Diamond & Halberstam extends the theorem to dimensions higher than 1.) Then the Jurkat–Richert theorem states that for any numbers y and z with 2 ≤ z ≤ y ≤ X we have

S ( A , P , z ) ≤ X V ( z ) ( F 1 ( log ⁡ y log ⁡ z ) + O ( ( log ⁡ log ⁡ y ) 3 / 4 ( log ⁡ y ) 1 / 4 ) ) + ∑ m | P ( z ) , m < y 4 ν ( m ) | r A ( m ) | {\displaystyle S(A,P,z)\leq XV(z)\left(F_{1}\left({\frac {\log y}{\log z}}\right)+O\left({\frac {(\log \log y)^{3/4}}{(\log y)^{1/4}}}\right)\right)+\sum _{m|P(z),m<y}4^{\nu (m)}\left|r_{A}(m)\right|}

and

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Jurkat–Richert theorem

Start with the simplest possible case. Write down what Jurkat–Richert theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Jurkat–Richert theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Jurkat–Richert theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Jurkat–Richert theorem

In research: Jurkat–Richert theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Jurkat–Richert theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Jurkat–Richert theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Sieve theory, Theorems in analytic number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Jurkat–Richert theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Jurkat–Richert theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Jurkat–Richert theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Jurkat–Richert theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jurkat–Richert theorem in simple terms?

The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture.

Why does Jurkat–Richert theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Jurkat–Richert theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Jurkat–Richert theorem.