Category:Definitions/Second Chebyshev Function

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This category contains definitions related to Second Chebyshev Function.
Related results can be found in Category:Second Chebyshev Function.


The second Chebyshev Function $\psi: \R \to \R$ is defined as follows:


Definition 1

$\ds \forall x \in \R: \map \psi x := \sum_{k \mathop \ge 1} \sum_{p^k \mathop \le x} \ln p$

where, for each $k$, the summation extends over all powers of prime numbers $p$ such that $p^k \le x$.


Definition 2

$\ds \forall x \in \R: \map \psi x := \sum_{1 \mathop \le n \mathop \le x} \map \Lambda n$

where $\Lambda$ is the von Mangoldt function.


Definition 3

$\ds \forall x \in \R: \map \psi x := \sum_{p \mathop \le x} \floor {\log_p x} \ln p$

where:

the summation extends over all prime numbers $p$ such that $p \le x$
$\floor {\, \cdot \,}$ denotes the floor function.

Pages in category "Definitions/Second Chebyshev Function"

The following 4 pages are in this category, out of 4 total.