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with suitable coefficients ĝ(n) for n ∈ N. Via Dirichlet series, Hardy [6] shortly afterwards found a different approach to these Ramanujan expansions, and in 1932 Carmichael [1] gave generalizations based on certain orthogonality relations for the Ramanujan sums cn(a). Later on, the existence of many expansions of the type (1) could be explained by putting them into the context of harmonic analysis by which the consideration was restricted to almost even arithmetical functions g having, at least in general, a nonzero asymptotic mean value (see, for example, Wintner [22], Cohen [2], Schwarz and Spilker [18], Knopfmacher [9]). The progress achieved in mean value theorems for multiplicative functions by Delange [3], Halasz [5], Elliott [4] led to certain classes of multiplicative functions g having a pointwise convergent Ramanujan expansion (1) (compare Schwarz [14]–[17], Tuttas [20], Warlimont [21]). From this point of view, for example the expansions (see Ramanujan [12])

Weighted relationship theorems and Ramanujan expansions