# First 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.

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## Definition

Let be a positive real number. The **first Chebyshev function** of , denoted or , is defined as:

,

where the sum is only over the prime numbers less than or equal to .

## Relation with other counting functions

- Prime-counting function: Denoted , this simply counts the number of primes less than or equal to .
- Second Chebyshev function: A similar summation, but this time of the von Mangoldt function over all prime powers less than or equal to .