Delange theorem

In 1961 H. Delange (see [a1]) proved that a multiplicative arithmetic function $f : \mathbf{N} \rightarrow \mathbf{C}$ of modulus $|f| \le 1$ possesses a non-zero mean value $$ M(f) = \lim_{x\rightarrow\infty} \frac{1}{x} \sum_{n\le x} f(n) $$ if and only if:

i) the Delange series $$ S_1 = \sum_p \frac{1}{p}(f(p)-1) \,, $$ extended over the primes, is convergent; and

ii) all the factors $\sum_{k=0}^\infty f(p^k) p^{-ks}$ of the Euler product of $\sum_n f(n) n^{-s}$ are non-zero.

Since $|f| \le 1$, condition ii) is automatically true for every prime $p>2$. In [a2] this theorem was sharpened.

An elegant proof of the implication "i) and ii) $\Rightarrow$ $M(f)$ exists" , using the Turán–Kubilius inequality, is due to A. Rényi [a4].

Using the continuity theorem for characteristic functions, for a real-valued additive arithmetic function $f$ Delange's theorem permits one to deal with the problem of the existence of limit distributions $$ \Psi(x) = \lim_{N\rightarrow\infty} \frac{1}{N} \sharp\{ n \le N : f(n) \le x \} \ . $$

Important extensions of Delange's theorem are due to P.D.T.A. Elliott and H. Daboussi; these theorems give necessary and sufficient conditions for multiplicative functions $f$ with finite semi-norm $$ \Vert f \Vert_q = \left({ \limsup_{x\rightarrow\infty} \frac{1}{x} \sum_{n \le x} |f(n)|^q }\right)^{1/q} $$ to possess a non-zero mean value (respectively, at least one non-zero Fourier coefficient $M(n\mapsto f(n) \exp(2\pi i a/r n)$). See Elliott–Daboussi theorem. See also Wirsing theorems.

E.V. Novoselov's theory of integration for arithmetic functions (see [a3]) also leads to many results on mean values of arithmetic functions.

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