# Second Chebyshev function

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## Definition

Let  be a positive real number. The second Chebyshev function of , denoted , is defined as the following sum:

.

Here,  is the von Mangoldt function.

This summation is taken over all the natural numbers less than or equal to ; however, a positive contribution comes only from prime powers, and the contribution of a prime power  is .

## Relation with other functions

### Exponential

The exponential of the second Chebyshev function gives the lcm of all numbers so far. In other words:



where  denotes the greatest integer function of , i.e., the largest integer less than or equal to .