I know that the maximum entropy distribution with over the non-negative integers fixed mean is a geometric distributions.
However, I cannot find conclusive information about what are the maximum entropy distributions over the whole integers (negative + non-negatives).
As per the comment below, in this case, we need at least to specify both the mean and the standard deviation of the distribution for the "maximum entropy" criterion to make sense.
It is well known that if we consider the whole real line, the maximum entropy distribution with a given mean and variance is a Gaussian normal distribution.
My questions are therefore: 1- For a given mean and variance, is there a maximum entropy distribution over the integers? 2- If it exists, can it be expressed in a closed form, like gaussian or geometric distributions?
My guess is that the answer is "yes" for 1) and "no" for 2) (since I cannot find any mention of a closed form anywhere). But I would be very happy to see a confirmation of this.
EDIT: Since it is now clear that the variance needs to be specified for the "maximum entropy" criterion to make sense, I rewrote the question accordingly.