Hi,


Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}\_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not reduce modulo p, and for any $p$ not in $S$, let $\gamma\_p$ be the size of $\Gamma \mod p$. My question is what is known about the function



> 
> $f(x)= \sum\_{p\not\in S,\ p\leq x}\frac{\log p }{\gamma\_p}$
> 
> 
> 


In particular what is the asymptotic behavior of $f$? Is the corresponding infinite series convergent whenever $\Gamma$ is *not* contained in an algebraic subgroup of $\mathbb{G}\_m^2$? Do you know of any references that might be relevant to those questions?


Thanks in advance,



Estimating the size of reduction of rational points on $\mathbb{G}_m^2$
Presumably "exceptional" means primes where either one of the generators of $\Gamma$ is 0 or $\infty$ mod p, or where $\Gamma$ mod $p$ has rank smaller than $2$. The following reference is possibly relevant to your question, although we consider a somewhat different sum. We give an upper bound (that should be fairly sharp) for the sum
$$\sum\_{p} \frac{\log p}{p\cdot\gamma\_p^\epsilon}.$$
In particular, we prove that 
$$\limsup\_{\epsilon\to0} ~~\epsilon \cdot \sum\_{p} \frac{\log p}{p\cdot\gamma\_p^\epsilon}
\le 1+\frac{1}{\text{rank}~\Gamma}.$$
The article is


Murty, M. Ram and Rosen, Michael and Silverman, Joseph H., Variations on a theme of Romanoff, *Internat. J. Math.* **7** (1996), 373-391 (MR1395936).


