 From the definition (or Kluyver's formula) it is straightforward to prove that, if "p" is a prime number,

and if "p""k" is a prime power where "k" > 1,

This result and the multiplicative property can be used to prove

This is called von Sterneck's arithmetic function.The equivalence of it and Ramanujan's sum is due to Hölder.For all positive integers "q",

For a fixed value of "q" the absolute value of the sequence formula_38 is bounded by φ("q"), and for a fixed value of "n" the absolute value of the sequence formula_39 is bounded by "n".