computer science

Odlyzko–Schönhage algorithm

Odlyzko–Schönhage algorithm is a computer science 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 Odlyzko–Schönhage algorithm rather than just read about it. In short: In mathematics, the Odlyzko–Schönhage algorithm is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by (Odlyzko & Schönhage 1988). The main point is the use of the fast Fourier transform to speed up the evaluation of a finite Dirichlet series of length N at O(N) equally spaced values from O(N2) to O(N1+ε) steps (at the cost of storing O(N1+ε) intermediate values).

Key takeaways

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

Reference excerpt

In mathematics, the Odlyzko–Schönhage algorithm is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by (Odlyzko & Schönhage 1988). The main point is the use of the fast Fourier transform to speed up the evaluation of a finite Dirichlet series of length N at O(N) equally spaced values from O(N2) to O(N1+ε) steps (at the cost of storing O(N1+ε) intermediate values). The Riemann–Siegel formula used for calculating the Riemann zeta function with imaginary part T uses a finite Dirichlet series with about N = T1/2 terms, so when finding about N values of the Riemann zeta function it is sped up by a factor of about T1/2. This reduces the time to find the zeros of the zeta function with imaginary part at most T from about T3/2+ε steps to about T1+ε steps. The algorithm can be used not just for the Riemann zeta function, but also for many other functions given by Dirichlet series. The algorithm was used by Gourdon (2004) to verify the Riemann hypothesis for the first 1013 zeros of the zeta function.

References Gourdon, X., Numerical evaluation of the Riemann Zeta-function Gourdon (2004), The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height Odlyzko, A. (1992), The 1020-th zero of the Riemann zeta function and 175 million of its neighbors This unpublished book describes the implementation of the algorithm and discusses the results in detail. Odlyzko, A. M.; Schönhage, A. (1988), "Fast algorithms for multiple evaluations of the Riemann zeta function", Trans. Amer. Math. Soc., 309 (2): 797–809, doi:10.2307/2000939, JSTOR 2000939, MR 0961614

Worked examples

Example 1 — a first encounter with Odlyzko–Schönhage algorithm

Start with the simplest possible case. Write down what Odlyzko–Schönhage algorithm claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, 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 Odlyzko–Schönhage algorithm 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 Odlyzko–Schönhage algorithm 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 Odlyzko–Schönhage algorithm

In research: Odlyzko–Schönhage algorithm appears in computer science 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 Odlyzko–Schönhage algorithm 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: Odlyzko–Schönhage algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algorithms and data structures stubs, Analytic number theory, Computational number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Odlyzko–Schönhage algorithm 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 Odlyzko–Schönhage algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Odlyzko–Schönhage algorithm 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 Odlyzko–Schönhage algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Odlyzko–Schönhage algorithm in simple terms?

Why does Odlyzko–Schönhage algorithm matter?

Because it connects several computer science 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 Odlyzko–Schönhage algorithm?

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 Odlyzko–Schönhage algorithm.