Lectures on mean values of the riemann zeta function. Based on work of alain connes, i have constructed a spectral interpretation for zeros of lfunctions. Spectral theory of the riemann zetafunction motohashi. Truncated in nitesimal shifts, spectral operators and. A nonlinear equation for a new statistic has appeared in phys. Wall 127 spectral theory of the riemann zetafunction downloaded from university publishing online. Truncated in nitesimal shifts, spectral operators and quantized universality of the riemann zeta function hafedh herichi and michell. We recall alain connes spectral interpretation of the critical zeros of the riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable hilbert space. Spectral operators and an asymmetric criterion for the riemann hypothesis michel l. V, the multiset of all eigenvalues of ain c is denoted by speca. September 30, 2015 abstract we initiate the study of spectral zeta functions.
A relation between nontrivial zeros of the zeta function and eigenvalues of the convolution operator is given. Quantum chaos, random matrix theory, and the riemann. A condensed version, gue eigenvalues and riemann zeta function zeros. Weils proof of the riemann hypothesis for function fields section 12. Spectral zeta functions of graphs and the riemann zeta function in. We recall alain connes spectral interpretation of the critical zeros of the riemann zeta function as eigenvalues of the absorption spectrum of an unbounded. They enduringly defy the request of solids, others, and tau years. The riemann hypothesis was posed in 1859 by bernhard riemann, a mathematician who was not a number theorist and wrote just one paper on number theory in his entire career.
Basic zeta functions and some applications in physics klaus kirsten 1. Y motohashi the riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. These ideas are then utilized to unveil a new image of the zetafunction, revealing it as the main gem of a. Spectral theory of the riemann zetafunction book, 1997. Spectral zeta functions of graphs and the riemann zeta function in the critical strip fabien friedli and anders karlsson. Universality of the riemann zeta function universality of the spectral operator. Notes on the riemann zeta function january 25, 2007 1 the zeta function 1.
David lyon the physics of the riemann zeta function. In a second way, the spectral zeta function of the graph z enjoys properties analogous to the riemann zeta function, notably the relation. Spectral theory of the riemann zetafunction by yoichi motohashi. Spectral analysis and the riemann hypothesis request pdf. In this book, based on his own research, professor motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function. While it was not created by riemann, it is named after him. Cambridge core number theory spectral theory of the riemann zetafunction by yoichi motohashi skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. This book, however, already requires a solid background in analysis. This groundbreaking work combines the classic the zetafunction with the modern the spectral theory to create a comprehensive but elementary treatment of spectral resolution. Pdf in this article, some new representations of the riemann zeta function.
We discuss a possible spectral realization of the riemann zeros based. In the first section of this chapter we will briefly recall, addressing the reader to roberts paper 63 see also aramaki 1, the construction of the spectral zeta function. Its a good book, gives really good motivation for the theory and goes through the history nicely. The only annoying thing is the use of the pi function instead of the gamma function. Riemanns theory about those nontrivial zeros is documented in the riemann hypothesis, but his biggest achievement was to see a connection between the prime numbers and the nontrivial zeros. It is an analogue of the selberg transform in selbergs trace formula. The strongest result fonctlon this kind one can hope for is the truth of the riemann hypothesis, which would have many profound consequences in the theory of numbers. Last theorem, but that the proof of the irrationality of 2.
Cambridge core number theory spectral theory of the riemann zeta function by yoichi motohashi. Naturally, this single paper would go on to become one of the most important papers in number theory history, a depressing, frustrating, and angering. Pdf notes on the riemann zeta function researchgate. Short proofs for nondivisibility of sparse polynomials under the extended riemann hypothesis, d. In noncommutative geometry and the stadard model of elementary. The main aim of this article is to develop, in a fully detailed fashion, a \bf unified theory of the spectral theory of mean values of individual automorphic lfunctions which is a natural extension of the fourth moment of the riemann zetafunction. Riemann zeta function nontrivial zeros because of the di. Introduction it is the aim of these lectures to introduce some basic zeta functions and their uses in the areas of the casimir effect and boseeinstein condensation. A spectral interpretation for the zeros of the riemann zeta function. The main aim of this article is to develop, in a fully detailed fashion, a \bf unified theory of the spectral theory of mean values of individual automorphic lfunctions which is a natural extension of the fourth moment of the riemann zetafunction but does not admit any analogous argument and requires a genuinely new method. Lapidus university of california department of mathematics 900 university ave. Riemann zeta function but also a conjecture about the nontrivial zeros of this function.
The fact that this function is analytic in this region of the complex plane is a consequence of the following basic fact. The story starts with a basic but unabridged treatment of the spectral resolution of the noneuclidean laplacian and the trace formulas. Riemanns hypothesis predicts that all nontrivial zeros. Lectures on mean values of the riemann zeta function by a. This is as far as one gets when considering chaotic quantum oscillator given by hamiltonian h px so far. Spectral thry riemann zetafunction cambridge tracts in. Here tr denotes a functional trace the zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues of the operator by. As a direct consequence we obtain a short, motivated proof of riemanns functional equation for s. Pdf spectral zeta functions of graphs and the riemann. Professor motohashi shows that the riemann zeta function is closely bound with automorphic forms and that many results from there can be woven with techniques and. They have attractions and volunteers read in problems and researches. Every nontrivial zero of the riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on.
Spectral zeta functions of graphs and the riemann zeta. Hawaii conference in number theory honolulu, march 7, 2012. Basic zeta functions and some applications in physics. Here we specialise this construction to the riemann. The author achieves this by the use of standard tools from analysis rather than. The completed zeta function and the riemann hypothesis 35 more speci. One of the methods to do so is to make use of spectral zeta functions. This function appears in the theory of eisenstein series although presumably. Zs, and it appears incognito as fudge factor in a few instances in the classical theory, such as in the fourier development of eisenstein series. Elizalde, ten physical applications of spectral zeta functions. Importance of the zeta function the zeta function is a very important function in mathematics. The riemann zeta function is one of my favorite function and this text is filled with useful properties of the zeta function. The author achieves this by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory.
The specialized topic of mean values of the riemann zetafunction permitted me to go into considerable depth. The riemann zeta function or eulerriemann zeta function. Eisenstein series, continuous spectrum of surfaces, langlands program. Spectral analysis and the riemann hypothesis sciencedirect. An introduction to the theory of the riemann zetafunction.
On spectral theory of the riemann zeta function springerlink. The riemann zeros as spectrum and the riemann hypothesis. The second term is minus the average number of riemann zeros, which according to connes, become missing spectral lines in the continuum 17,20. The completed zeta function and the riemann hypothesis. Pdf spectral zeta functions of graphs and the riemann zeta. The main aim of this article is to develop, in a fully detailed fashion, a \bf unified theory of the spectral theory of mean values of individual automorphic lfunctions which is a natural. The riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. Hilbert and p olya put forward the idea that the zeros of the riemann zeta function may have a spectral origin. The aim of these lectures is to provide an intorduction to the theory of the riemann zetafunction for students who might later want to do research on the subject. Publication date 1930 topics natural sciences, mathematics, number theory publisher at the university press collection universallibrary contributor osmania university language english. A spectral interpretation for the zeros of the riemann. The riemann hypothesis is shown to be equivalent to an approximate functional equation.
The zeta function of a mathematical operator is a function defined as. We construct an operator on a nuclear frechet space. Every nontrivial zero of the riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a hilbertpolya space. The explicit formulas of riemann and guinandweil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of riemann. The riemann zeta function plays a pivotal role in analytic number theory and has applications in. Titchmarsh 21 is a true classic book on the riemann zeta function with excellent endofchapter notes by d. Show full abstract analytic number theory, riemann zeta function, and tauberian theorems. Spectral theory of the riemann zetafunction by yoichi. We construct an operator on a nuclear frechet space whose spectrum is the set of. While the evidence for the spectral nature of the zeros of any. Fractal strings, complex dimensions and the spectral. Edwards book riemanns zeta function 1 explains the historical context of riemanns paper, riemanns methods and results, and the subsequent work that has been done to verify and extend riemanns theory.