The mysteries of real prime (Q2755074)

It has long been a goal of many number theorists to work with the real numbers and the \(p\)-adic numbers in a uniform way. The tendency has been to use ideas that were originally developed for the reals and carry them over to the \(p\)-adics. The present book aims to reverse this situation by taking a \(p\)-adic approach to the reals. Of course, one problem is that there is no ring \(\mathbb Z_{\eta}\) for the real place \(\eta\) that is the analogue of the ring \(\mathbb Z_p\) of \(p\)-adic integers. However, it is still possible to use \(\mathbb Z_{\eta}\) as a motivating influence. For example, the author takes the Gaussian \(e^{-\pi x^2}\) as the characteristic function of \(\mathbb Z_{\eta}\). This fits well with the definition of the local factors \(\pi^{-s/2}\Gamma(s/2)\) for \(\eta\) and \((1-p^{-s})^{-1}\) for \(p\) in the zeta function for \(\mathbb Q\). NEWLINENEWLINENEWLINEA major theme in the book is the relations between oriented graphs and Markov chains. When the graph is a tree, there is a bijection between chains and probability measures on the boundary. The sets \(\mathbb P^1(\mathbb Q_p)\), \(\mathbb Z_p\), and \(\mathbb Z_p^{*}\) can be expressed as inverse limits of corresponding objects for \(\mathbb Z/p^N\mathbb Z\). One goal of the book is to find real analogues of these limits. This leads to the theory of orthogonal polynomials of Jacobi, Laguerre, and Hermite, which are developed via certain Heisenberg relations. The set \(\mathbb P^1(\mathbb Q_p)\) can be regarded as the boundary of a tree. The set \(\mathbb P^1(\mathbb R)\) is viewed as the boundary of a graph, with Heisenberg relations compensating for the lack of a tree structure in the graph. Moreover, these ideas lead back to \(p\)-adic analogues of the classical orthogonal polynomials. NEWLINENEWLINENEWLINEThe \(q\)-zeta function \(\zeta_{(q)}(\beta)=\prod_{n\geq 0} (1-q^{\beta + n})^{-1}\) is introduced to interpolate between the real and \(p\)-adic local factors of the zeta function. Also, \(q\)-analogues of several functions appear, and certain \(q\)-chains are constructed. The \(q\) versions of various concepts are used to obtain real and \(p\)-adic phenomena in a uniform manner. NEWLINENEWLINENEWLINEThe Heisenberg group and its fundamental representation are studied over the reals and the \(p\)-adics. This leads to a study of the zeta function, where the \(q\) world is used to treat the real and \(p\)-adic contributions equally. Finally, a heuristic ``proof'' of the Riemann Hypothesis is given. NEWLINENEWLINENEWLINEThe book has many interesting ideas, only a few of which are mentioned above. It is not a traditional ``theorem-proof'' book. In fact, no theorems are stated (though a few formulas are proved). Rather, the book develops a dictionary for moving between the \(p\)-adic and real worlds.