Abstract: The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. 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 itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Abstract: The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. 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 itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the development of number theory, namely the fusion of two main fields in mathematics that were previously studied separately.

Abstract: The form factor, k(t), is the spectral statistic which best displays nonuniversal quasiclassical deviations from random matrix theory. Recent estimations of k(t) for a single spectrum found interesting new effects of this type. It was supposed that k(t) is {ital self-averaging} and thus did not require an ensemble average. We here argue that this supposition sometimes fails and that for many important systems an ensemble average is essential to see detailed properties of k(t). In other systems, notably the nontrivial zeros of Riemann zeta function, it will be possible to see the nonuniversal properties by an analysis of a single spectrum. {copyright} {ital 1997} {ital The American Physical Society}

Abstract: We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x ) (log log x ) is not asymptotic as x →+∞ to a composition of semialgebraic functions, log and exp.

Abstract: The prime number theorem, that the number of primes < x is asymptotic to x/log x, was proved (independently) by Hadamard and de la Vallee Poussin in 1896. Their proof had two elements: showing that Riemann's zeta function ;(s) has no zeros with Sc(s) = 1, and deducing the prime number theorem from this. An ingenious short proof of the first assertion was found soon afterwards by the same authors and by Mertens and is reproduced here, but the deduction of the prime number theorem continued to involve difficult analysis. A proof that was elementaty in a technical sense-it avoided the use of complex analysis-was found in 1949 by Selberg and Erdos, but this proof is very intricate and much less clearly motivated than the analytic one. A few years ago, however, D. J. Newman found a very simple version of the Tauberian argument needed for an analytic proof of the prime number theorem. We describe the resulting proof, which has a beautifully simple structure and uses hardly anything beyond Cauchy's theorem. Recall that the notation f(x) g(x) ("f and g are asymptotically equal")

Abstract: After recalling the precise existence conditions of the zeta function of a pseudodifferential operator, and the concept of reflection formula, an exponentially convergent expression for the analytic continuation of a multidimensional inhomogeneous Epstein-type zeta function of the general form \zeta_{A,\vec{b},q} (s) = \sum_{\vec{n}\in Z^p (\vec{n}^T A \vec{n} +\vec{b}^T \vec{n}+q)^{-s}, with $A$ the $p\times p$ matrix of a quadratic form, $\vec{b}$ a $p$ vector and $q$ a constant, is obtained. It is valid on the whole complex $s$-plane, is exponentially convergent and provides the residua at the poles explicitly. It reduces to the famous formula of Chowla and Selberg in the particular case $p=2$, $\vec{b}= \vec{0}$, $q=0$. Some variations of the formula and physical applications are considered.

Abstract: The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial $f$ and an arithmetical invariant associated to a polynomial $f$ over a $p$-adic field. When $f$ is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of $f^{-1} \{ 0 \}$ in $\Bbb A^2$. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘$q$-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity. 1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)

Abstract: Somerapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for ζ(4n+3) that generalizes Apervs seriesfor ζ(3), and appears to give the best possible series relations of this type, at least for n < 12. The formula reduces to a finite but apparently nontrivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a formula that provides strange evaluations of a large new class of nonterminating hypergeometric series. [Editor's Note: The beautiful formulas in this paper are no longer conjectural. Seenote on page 194.]

Abstract: Next I considered the case where k is half of any positive integer and proved (1) (however with C1 depending possibly on k). Next D. R. Heath-Brown [1] considered the case H = T and k any positive rational number and proved (1) (however with C1 depending possibly on k). Next M. Jutila [4] considered the case H = T and k = q−1 and proved (1) with C1 independent of k. For all these references see also my book [6]. Two other excellent reference books are [7] and [2].

Abstract: We generalize earlier studies on the Laplacian for a bounded open domain $\Omega\subset\mathbb R^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this same domain. Using single layer and double layer potentials we can prove a number of new relations which hold when one chooses independently Dirichlet or Neumann boundary conditions for the interior and exterior problem. This relation is provided by a very simple set of $\zeta$-functions, which involve the single and double layer potentials. We also provide Krein spectral formulas for all the cases considered and give a numerical algorithm to compute the $\zeta$-function.

Abstract: Let M denote a hyperbolic Riemann surface of finite volume, and let KM(t,x,y) be the heat kernel associated to the hyperbolic Laplacian which acts on the space of smooth functions on M. If M is compact, then we have the equality¶¶M∫KM(t, x, x)dμ(x) = ∞n=0Σe-λnt,¶where {λn} is the set of eigenvalues of the Laplacian. If M is not compact, then it is well known that the heat kernel exists yet is not of trace class. In this paper we will define a regularized heat trace associated to any hyperbolic Riemann surface of finite volume, compact or non-compact. After we have defined the regularized heat trace, we study the asymptotic behavior of the regularized heat trace on a family of degenerating hyperbolic Riemann surfaces. Our results involve pointwise convergence and uniformity of asymptotic expansions in the pinching parameters. In particular, we study uniformity of long time asymptotics of the regularized heat trace minus the contribution from the small eigenvalues by analyzing the Poisson kernel and Dirichlet heat kernel in a finite cylindrical neighborhood of the pinching geodesics. As applications of our results, we are able to study asymptotic expansions of the Selberg zeta function and spectral zeta function on degenerating families, both improving known results in the compact setting and proving new results in the non-compact situation. Results from this article have been extended to the setting of degenerating hyperbolic three manifolds of finite volume in [DJ1] and [DJ2].

Abstract: A consistent procedure for regularization of divergences and for the subsequent renormalization of the string tension is proposed in the framework of the one-loop calculation of the interquark potential generated by the Polyakov-Kleinert string. In this way, a justification of the formal treatment of divergences by analytic continuation of the Riemann and Epstein-Hurwitz zeta functions is given. A spectral representation for the renormalized string energy at zero temperature is derived, which enables one to find the Casimir energy in this string model at nonzero temperature very easy.

Abstract: The global additive and multiplicative properties of Laplace type operators acting on irreducible rank 1 symmetric spaces are considered. The explicit form of the zeta function on product spaces and of the multiplicative anomaly is derived.

Abstract: This is an expanded version. We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of the above mentioned relations and explicit computations of the various zeta values and Dedekind sums involved.

Abstract: We find examples of duality among quantum theories that are related to arithmetic functions by identifying distinct Hamiltonians that have identical partition functions at suitably related coupling constants or temperatures. We are led to this after first developing the notion of partial supersymmetry-in which some, but not all, of the operators of a theory have superpartners-and using it to construct fermionic and parafermionic thermal partition functions, and to derive some number theoretic identities. In the process, we also find a bosonic analogue of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function.

Abstract: We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to infinite-dimensional representation of the Lorentz group have many interesting properties. Especially, a relationship exists between the zero zeta function condition and the absence of trivial representations in the wave function.

Abstract: Two types (binomial and exponential types) of power series, together with a related sum, associated with the Riemann zeta-function (s) will be investigated by using Mellin-Barnes type integrals. As for generalizations of these sums we shall introduce hypergeometric type generating functions of (s) and derive their basic properties.

Abstract: Geometric zeta functions of Ihara and Hashimoto are generalized to higher rank. The $p$-adic version of the Patterson conjecture is proven.