Statistical Independence in Probability, Analysis, and Number Theory, by Mark Kac -- an amazingly potent piece of mathematical writing given its rather minuscule size. While now slightly out of date in terms of best possible bounds, he fairly seamlessly collects some of the most important results in analytic number theory. This was the first book (for me) that calculated not only expected values for the standard arithmetic functions (various divisor-counting functions, sum-of-divisors function,etc.), but also the expected deviation, in terms of a probability distribution function, from this value. These types of results don't get emphasized enough -- almost as important as knowing asymptotic values for a quantity in question is knowing how frequently the value is far from that asymptote.