DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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Chebyshev's psi function

Definition

For every real number $x \geq 1$ , Chebyshev's psi function is

$ψ (x) : = \sum_{n \leq x} Λ (n),$

where $Λ$ is the von Mangoldt function of The von Mangoldt function.

Remarks

The sum is finite for every real $x$ .
The next lemma rewrites $ψ (x)$ as a sum over prime powers, which is the form used throughout this page.

Depends on

The von Mangoldt function

Used by

Right-continuous psi has the wrong Perron endpoint Counterexample
The half-weighted Chebyshev function Definition
A table of pi(x), theta(x), and psi(x) Example
Prime-power expansion of Chebyshev's psi function Lemma
Psi and theta differ by at most a square-root term Lemma
The von Mangoldt harmonic sum is log x plus O(1) Lemma
Mertens sum for primes in an arithmetic progression Theorem

Dependency tree · two levels

4 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

Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 2 (standard reference, not scraped)