mathematics

Lowell Schoenfeld

Lowell Schoenfeld 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 Lowell Schoenfeld rather than just read about it. In short: Lowell Schoenfeld (April 1, 1920 – February 6, 2002) was an American mathematician known for his work in analytic number theory. Career Schoenfeld received his Ph.D. in 1944 from University of Pennsylvania under the direction of Hans Rademacher.

Key takeaways

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

Reference excerpt

Lowell Schoenfeld (April 1, 1920 – February 6, 2002) was an American mathematician known for his work in analytic number theory.

Career Schoenfeld received his Ph.D. in 1944 from University of Pennsylvania under the direction of Hans Rademacher. In 1953, as an assistant professor at the University of Illinois Urbana-Champaign, he married (as his second wife) associate professor Josephine M. Mitchell, causing the university to fire her from her tenured position under its anti-nepotism rules while allowing him to keep his more junior tenure-track job. They both resigned in protest, and after several short-term positions they were both able to obtain faculty positions at Pennsylvania State University in 1958. They were both promoted to full professor in 1961, and moved to the University at Buffalo in 1968.

Contributions Schoenfeld is known for obtaining the following results in 1976, assuming the Riemann hypothesis:

| π ( x ) − l i ( x ) | ≤ x ln ⁡ x 8 π {\displaystyle |\pi (x)-{\rm {li}}(x)|\leq {\frac {{\sqrt {x}}\,\ln x}{8\pi }}}

for all x ≥ 2657, based on the prime-counting function π(x) and the logarithmic integral function li(x), and

| ψ ( x ) − x | ≤ x ln 2 ⁡ x 8 π {\displaystyle |\psi (x)-x|\leq {\frac {{\sqrt {x}}\,\ln ^{2}x}{8\pi }}}

for all x ≥ 73.2, based on the second Chebyshev function ψ(x).

References

External links Lowell Schoenfeld at the Mathematics Genealogy Project

Illustrations

Worked examples

Example 1 — a first encounter with Lowell Schoenfeld

Start with the simplest possible case. Write down what Lowell Schoenfeld 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 Lowell Schoenfeld 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 Lowell Schoenfeld 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 Lowell Schoenfeld

In research: Lowell Schoenfeld 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 Lowell Schoenfeld 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: Lowell Schoenfeld is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1920 births, 2002 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Lowell Schoenfeld 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 Lowell Schoenfeld in 20 minutes

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

Frequently asked questions

What is Lowell Schoenfeld in simple terms?

Lowell Schoenfeld (April 1, 1920 – February 6, 2002) was an American mathematician known for his work in analytic number theory. Career Schoenfeld received his Ph.D. in 1944 from University of Pennsylvania under the direction of Hans Rademacher.

Why does Lowell Schoenfeld 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 Lowell Schoenfeld?

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 Lowell Schoenfeld.