# Second Chebyshev function

From Number

This article is about a function defined on positive reals (and in particular, natural numbers) obtained as the summatory function of an arithmetic function, namely von Mangoldt 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

- Prime-counting function
- First Chebyshev function: This simply adds the logarithms of all the primes up to the point.

### 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 .