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What can be said about the partial sums of a complex-valued completely multiplicative function, let's say bounded by 1 in absolute value, if its Dirichlet series has an essential singularity?

As a concrete example, consider the completely multiplicative function defined by $f(p)=i$ for all primes $p$. The Dirichlet series of this function has the Euler product $$L(s,f)=\prod_p \left(1-\frac{i}{p^s}\right)^{-1},$$ and by taking logs, we see that $$\log L(s,f) = -\sum_p \log\left(1-\frac{i}{p^s}\right) = i \sum_p \frac{1}{p^s} + O(1) = i\log\left(\frac{1}{s-1}\right)+\theta(s),$$ where $\theta(s)$ is analytic if $\Re(s)\geq 1$. By taking the limit as $s\to 1$ in this right half-plane, we see that the argument of $L(s,f)$ goes to infinity while its absolute value converges to something non-zero (at least along the real axis), whence $L(s,f)$ has an essential singularity at $s=1$. Standard Dirichlet series techniques (Perron's formula, for example) let us say that the partial sums of $f$, $$S_f(x):=\sum_{n\leq x} f(n),$$ are not $O(x^{1-\epsilon})$ for any $\epsilon>0$, but to my knowledge, this is the best these techniques can achieve. By using a quantitative version of Halasz's theorem, I imagine it should be possible to show that $$S_f(x) \ll x \frac{\log\log^A x}{\log x}$$ for some $A\geq 1$. This bound is pretty good, in that it achieves quantitative savings over the trivial bound $O(x)$, but I have no idea if it is the truth in some sense, and it's also easy to imagine situations in which using Halasz's theorem is not feasible (essentially, $f(p)=i$ is a nice, consistent choice).

My question is this: What can be said about $S_f(x)$? Are there good lower bounds for it? Is there a way, even heuristically, to determine its order of magnitude, or even just to get better upper bounds? More generally, are there techniques other than the standard Dirichlet series approach and Halasz's theorem to get good upper bounds on partial sums of such functions?

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Hi Robert. For your particular $f$, you'll want to look at this article:

A Note on the Compositeness of Numbers
A. W. Addison
Proceedings of the American Mathematical Society
Vol. 8, No. 1 (Feb., 1957), pp. 151-154
Article Stable URL: http://www.jstor.org/stable/2032831

Addison gives the details for the case when $f(p)=\omega$, where $\omega$ is a primitive $3$rd root of unity, but remarks that the the case of $f(p)=i$ (or in general, any $\zeta_n$) is similar.

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As for your general question, it appears that certain sorts of essential singularities (e.g., the sort that appear in this example) can be handled by the Selberg--Delange method, which is described in Chapter II.5 of Tenenbaum's book. –  so-called friend Don Jul 25 '11 at 14:03
    
This seems great. I was interested in that specific example as well as the general case, so this was definitely helpful. Suppose I choose f(p) such that the sum of f(p)/p doesn't converge but also doesn't diverge to infinity in any direction. It seems like this shouldn't be amenable to study using the Selberg-Delange method, at least as presented in Tenenbaum. Are you aware of any techniques which might apply? –  rlo Jul 25 '11 at 15:02
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