  This monograph elucidates and extends many theorems and conjectures in
analytic number theory and algebraic asymptotic analysis via the natural
notions of degree and logexponential degree. The Riemann hypothesis, for
example, is equivalent to the statement that the degree of the function
$\pi(x)- \operatorname{li}(x)$ is $1/2$, where $\pi(x)$ is the prime counting
function and $\operatorname{li}(x)$ is the logarithmic integral function. Part
1 of the text is a survey of analytic number theory, Part 2 introduces the
notion of logexponential degree and uses it to extend results in algebraic
asymptotic analysis, and Part 3 applies the results of Part 2 to the various
functions that figure most prominently in analytic number theory.
  Central to the notion of logexponential degree are Hardy's
logarithmico-exponential functions, which are real functions defined in a
neighborhood of $\infty$ that can be built from $\operatorname{id}$, $\exp$,
and $\log$ using the operations $+$, $\cdot$, $/$, and $\circ$. Such functions
are natural benchmarks for the orders of growth of functions in analytic number
theory. The main goal of Part 3 is to express the logexponential degree of
various functions in analytic number theory in terms of as few "logexponential
primitives" as possible. The logexponential degree of the function $e^\gamma
\prod_{p\leq x}(1-1/p) -1/\log x$, for example, can be expressed in terms of
that of $\pi(x)- \operatorname{li}(x)$ and vice versa (where $\gamma \approx
0.5772$ is the Euler-Mascheroni constant), despite the fact that very little is
known about the logexponential degree of either function separately, even on
condition of the Riemann hypothesis.
