It seems clear that developing countries which open up their economies to competitive mobile operators get rapid growth in tele-density (phones per capita) and this leads to rapid economic growth (increased GDP per capita), but I've never seen a rigorous proof that this is true. Over the weekend, I spent some time surfing the web in search of research on this subject, either pro or con. I've printed out a wealth of papers which likely will take weeks to decipher, but let me share one very interesting item I stumbled upon.
In a 1999 World Bank policy research working paper entitled Infrastructure's Contribution to Aggregate Output, David Canning examines the contributions of different factors of production to aggregate output looking at 57 countries over the period 1960-1990. Quoting from his conclusions,
Our estimates suggest that the returns to physical capital as a whole, human capital, electricity generating capacity and transport routes are close to those found from microeconomic evidence based on the private returns to these factors or cost benefit analysis. However, we find a large productivity effect of telephone networks over and above the normal productivity of capital...
... we find a large, statistically significant, impact of telephones on output. This suggests that investment in telephones is more productive than investment on average, by a substantial margin.
Of course this is for a period before telecom really started to take off. My sense is that the advent of mobile phone competition and soaring adoption has only accelerated the trends identified by Canning but, as yet, I haven't found an analysis based on more recent data.