That's a neat idea, because it takes number of citizens into account. But if you want to get an idea of the wealth of a nation you may prefer to use a measure like total income or the GDP, rather. Possibly, dividing bribes by population size serves only to get a sense of the relative size of bribery that is going on, so from an ethical perspective you may run into trouble here. Bribery can be judged using absolute values, too. What do you think?

If you mean will the new variable be statistically independent of population, the answer is clearly no, because it will be highly correlated with population. But if you are not planning on using population size anywhere else in the analysis (e.g., as an independent variable), then this would be one way to "control" for that factor.

Beyond that, I think the main thing you might gain is interpretability, in terms of the substantive meaning of the variable you would create.

The new variable would thus be "per capita bribes." You are correct; it is now logically independent of  the population. However, it's possible that large or small countries might either be more corrupt or have better ways of estimating total bribes paid. If either of those cases are true, then per capita bribes will be correlated with population.

When you divide the number of Euros paid as bribe by the population you would get the "per capita bribes". i.e. the money value of the bribes paid per capita in a country. This variable will then be dependent on population statistically, but independent objectively. Also, Euros is a nominal variable and population is a real variable, hence one needs to consider inflation and value of money.

If you divide the variable by population this one way of normalizing the variable. To eliminate the effect of population you may consider diving by GDP.  What are your other independent and the dependent variable?

From the point of view of measurement both 'indicators' are valid, but of course they tell you different things. For instance, in your case, the absolute figure would give you how much the economy (and the society) spends on bribes without taking into account of how size (be it population or economy) is related to it.   However, as bribe can be influenced (among other things) by population size or size of an economy to compare likes with likes you would have to have a relative indicator that allows you to measure the "risk" of bribe across different population and economic settings. So, what do you achieve when you divide the amount of bribe by total number of people in a country or given area; well you get the “density of bribe” in that population. So, just for the sake of example if that turns out to be  0.50 Euro in country x in year y, what that means is that on average every person in that country (or area) could theoretically expect to pay 0.50 Euro as a bribe in a given period. But of course some people may face higher "risks" while others (such as children, for example) can't even do it. This is where you may want to refine your denominator such as by restricting it to the population that is potential exposed to the risk of giving or providing bribe. For example, instead of the total population you could just use the working age population, or you could focus on those employed in the public sector or even make it sector specific, if that is where the "risks" are found. On the other hand, if your interest is to adjust for the size of economy, then you would have to divide the amount by GDP or some equivalent measure. In this case of course the resulting quantity will have a different interpretation. So, I think the most important point to consider is the purpose for which the indictor is needed for.

The other alternative, which I don't think anyone has mentioned, is to include population size as an independent variable in the regression. In particular, you might begin by controlling for things like population size and GDP, and then look at the additional explanatory power that you get from more substantive variables.

Number of bribes is essentially a count variable. You could model it using a Poisson or Negative Binomial, using the population as the exposure variable.

I believe another construct would help in the analysis. For example level of poverty or education level etc. 

Considering health and epidemiology, when you divide the number of people with a particular disease by the total population, you get the prevalence for that disease. 

 When you divide the Euros by population you end up with what would essentially be a "Euro rate."   What would that mean?  If you're interested in the causal effects of population bribery size I would probably look at the correlation between them -- either Pearsons's or a non-parametric, or regress "number of Euros" on population. 

Good luck with your analysis

--Bill F