Second Chebyshev function

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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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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


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 .