Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Chebyshev's theta function
Definition
For every real number , Chebyshev's theta function is
the sum being taken over the primes .
Remarks
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For , the sum is empty, so .
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Since there are only finitely many integers at most , the displayed sum is finite for every real .
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Like , the function is right-continuous and jumps only at primes.
Depends on
Used by
- A table of pi(x), theta(x), and psi(x) Example
- Abel summation recovers the prime-counting function from theta Lemma
- Prime-power expansion of Chebyshev's psi function Lemma
- Psi and theta differ by at most a square-root term Lemma
- Chebyshev bounds for the prime-counting function Theorem
- Chebyshev's theta function has linear lower and upper bounds Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Karl-Dieter Crisman, Number Theory: In Context and Interactive (standard reference, not scraped)
- Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 2 (standard reference, not scraped)