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Clearly (1.5) reduces to (1.3) in case k=l. Van der Corput proved a result equivalent to Theorem B in the special case in which g(n) is assumed multiplicative [2, Proposition 1 ]. However, van der Corput did not make explicit use of the multiple arithmetical integral, with the consequence that the essential simplicity of his argument is somewhat obscured. Moreover, the superfluous multiplicativity assumption leads to additional complications in his method. We shall need two simple lemmas.

Arithmetical notes. I. On a theorem of van der Corput