Abstract: Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Abstract: Preface.- Hyperbolic surfaces.- Geometry of H.- Fuchsian groups.- Geometric finiteness.- Classification of hyperbolic ends.- Length spectrum and Selberg's zeta function.- Review of the Compact Case.- Spectral theory for compact manifolds.- Selberg's trace formula for compact surfaces.- Consequences of the trace formula.- Review of the finite-volume case.- Finite-volume hyperbolic surfaces.- Spectral theory.- Selberg's trace formula.- Scattering Theory in Model Cases.- Spectral theory of H.- Scattering theory on H.- Hyperbolic cylinders.- Funnels.- Parabolic cylinder.- Scattering Theory for infinite-volume hyperbolic surfaces.- Compactification.- Continuation of the resolvent.- Resolvent asymptotics and the stretched product.- Structure of the resolvent kernel.- Discrete and continuous spectrum.- Generalized eigenfunctions.- Scattering matrix.- Structure of kernels in the conformally compact case.- Resonances and scattering poles.- Multiplicities of resonances.- Scattering poles.- Half-integer points.- Coincidence of resonances and scattering poles.- Upper bound on the density of resonances.- Infinite-volume spectral geometry.- Relative scattering determinant.- Regularized traces.- The resolvent 0-trace calculation.- Structure of Selberg's zeta function.- The Poisson formula for resonances.- Application.- Lower bounds on the density.- Weyl formula for the scattering phase.- The length spectrum.- Finiteness of isospectral classes.- Appendix A Functional analysis.- Basic spectral theory.- Analytic Fredholm theorem.- Operator residues and multiplicities.- Appendix B Asymptotic expansions.- References.- Index.

Abstract: We use a smoothed version of the explicit formula to find an accurate pointwise approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function which involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. For the second and fourth moments, we establish all of the steps in our approach rigorously. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory

Abstract: In this paper we provide a geometric description of the possi- ble poles of the Igusa local zeta function Z�(s, f) associated to an analytic mapping f = (f1, . . . , fl) : U(� K n ) ! K l , and a locally constant function Φ, with support in U, in terms of a log-principalizaton of the K (x) ideal If = (f1, . . . , fl). Typically our new method provides a much shorter list of possible poles compared with the previous methods. We determine the largest real part of the poles of the Igusa zeta function, and then as a corollary, we ob- tain an asymptotic estimation for the number of solutions of an arbitrary sys- tem of polynomial congruences in terms of the log-canonical threshold of the subscheme given by If . We associate to an analytic mapping f = (f1, . . . , fl) a Newton polyhedron Γ (f) and a new notion of non-degeneracy with respect to Γ (f). The novelty of this notion resides in the fact that it depends on one Newton polyhedron, and Khovanskii's non-degeneracy notion depends on the Newton polyhedra of f1, . . . , fl . By constructing a log-principalization, we give an explicit list for the possible poles of Z�(s, f), l � 1, in the case in which f is non-degenerate with respect to Γ (f).

Abstract: Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein–Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat–Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed.

Abstract: In this paper, we investigate a new class of analytic functions defined by the convolution operator J s, b (f) involving the Hurwitz–Lerch Zeta function, which was introduced recently by H M Srivastava and A A Attiya [Integral Transforms and Special Functions, 18, 2007, 207–216] Coefficient inequalities, distortion theorems and other properties of functions belonging to this class are derived We also indicate briefly the relevant connections of some of the results presented here with those that were proven in earlier investigations

Abstract: Let G be a finite simple group. We show that the commutator map $a : G \times G \to G$ is almost equidistributed as the order of G goes to infinity. This somewhat surprising result has many applications. It shows that for a subset X of G we have $a^{-1}(X)/|G|^2 = |X|/|G| + o(1)$, namely $a$ is almost measure preserving. From this we deduce that almost all elements $g \in G$ can be expressed as commutators $g = [x,y]$ where x,y generate G. This enables us to solve some open problems regarding T-systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T-systems in G with two generators tends to infinity as the order of G goes to infinity. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators. Some of our results apply for more general finite groups, and more general word maps. Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function plays a key role in the proofs.

Abstract: We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study of the associated height zeta function and its spectral expansion.

Abstract: Quantization of the Riemann zeta-function is proposed. We treat the Riemann zeta-function as a symbol of a pseudodifferential operator and study the corresponding classical and quantum field theories. This approach is motivated by the theory of p-adic strings and by recent works on stringy cosmological models. We show that the Lagrangian for the zeta-function field is equivalent to the sum of the Klein-Gordon Lagrangians with masses defined by the zeros of the Riemann zeta-function. Quantization of the mathematics of Fermat-Wiles and the Langlands program is indicated. The Beilinson conjectures on the values of L-functions of motives are interpreted as dealing with the cosmological constant problem. Possible cosmological applications of the zeta-function field theory are discussed.

Abstract: Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficientsλn = P ρ 1 − � 1 − 1 � n , (n = 1,2, ...), in which ρ runs over the nontrivial zeros of the Riemann zeta function. We define similar coefficientsλn(π) associated to principal automorphic L-functions L(s, π) over GL(N). We relate these cofficients to values of Weil’s quadratic functional associated to the representation π on a suitable set of test functions. The positivity of the real parts of these coefficients is a necessary and sufficient condition for the Riemann hypothesis for L(s, π) to hold. We derive an unconditional asymptotic formula for the coefficients λn(π), in terms of the zeros of L(s, π). Assuming the Riemann hypothesis for L(s, π), we deduce that λn(π) = N 2 nlog n + C1(π)n + O( √ nlog n), where C1(π) is a real-valued constant and the implied constant in the remainder term depends on π. We also show that there exists a entire function Fπ(z) of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis there is an (essentially) unique interpolation function having exponential type at most π, and this function restricted to the real axis has a (tempered) distributional Fourier transform whose support is a countable set in [−π, π] having 0 as its only limit point.

Abstract: Gauss in 1812, in his celebrated memoir on the hypergeometric series, presented a remarkable formula for the psi (or digamma) function, ?(z), at rational arguments z, which can be expressed in terms of elementary functions. Davis in 1935 extended Gauss's result to the polygamma functions by using a known series representation of ?(n)(z) in an elementary yet technical way. K?lbig in 1996, in his CERN technical report, also gave two extensions to ?(n)(z) by using the series definition of polylogarithm function and the above-known series representation. Here we aim at deriving general formulae expressing ?(n)(z) as rational arguments in terms of other functions, which will be obtained in two ways. In addition, several special cases are also considered and, as a by-product of our main results, we derive, in a simple and unified manner, all formulae given by Gauss, Davis and K?lbig. Finally, it should be noted that all our results, in view of the relationship between ?(n)(z) and the Hurwitz zeta function, ?(s, a), could be rewritten in the representation of ?(s, a).

Abstract: We uncover a new type of unitary operation for quantum mechanics on the half-line which yields a transformation to ``Hyperbolic phase space''. We show that this new unitary change of basis from the position x on the half line to the Hyperbolic momentum $p_\eta$, transforms the wavefunction via a Mellin transform on to the critial line $s=1/2-ip_\eta$. We utilise this new transform to find quantum wavefunctions whose Hyperbolic momentum representation approximate a class of higher transcendental functions, and in particular, approximate the Riemann Zeta function. We finally give possible physical realisations to perform an indirect measurement of the Hyperbolic momentum of a quantum system on the half-line.

Abstract: In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model.

Abstract: We compute the asymptotics of the fourth moment of the Riemann zeta function times an arbitrary Dirichlet polynomial of length $T^{{1/11} - \epsilon}$

Abstract: We present three remarks on Goldbach's problem. First we suggest a refinement of Hardy and Littlewood's conjecture for the number of representations of $2n$ as the sum of two primes positing an estimate with a very small error term. Next we show that if a strong form of Goldbach's conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Finally we show that an averaged strong form of Goldbach's conjecture is equivalent to the Generalized Riemann Hypothesis; as well as a similar equivalence to estimates for the number of ways of writing integers as the sum of $k$ primes.

Abstract: We show that the real parts of the poles of the Igusa zeta function of a monomial ideal can be computed from the torus-invariant divisors on the normalized blow-up of the affine space along the ideal. Moreover, we show that every such number is a root of the Bernstein-Sato polynomial associated to the monomial ideal.

Abstract: The Apery-like numbers J2(n) associated to the special value ζQ(2) of the spectral zeta function ζQ(s) for the non-commutative harmonic oscillator Q have remarkable modular form interpretation. In fact, we show that the differential equation satisfied by the generating function w2(t) of J2(n) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t of this family can be regarded as a modular function for the congruent subgroup Γ0(4). Further, we see that the function w2(t) is regarded as a Γ0(4)-modular form of weight 1 in the variable τ by taking t as the classical Legendre modular function λ(τ).

Abstract: In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the Riemann-Liouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schr\"odinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.5-3.7) and conjecture about additional bounds.

Abstract: This paper intends to give a mathematical explanation for results on the zeta function of some families of varieties recently obtained in the context of mirror symmetry. In the process we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in ℙn. In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the p-adic Picard–Fuchs equation associated with monomial deformations of Fermat hypersurfaces. As a byproduct we obtain Poincare duality for the rigid cohomology of certain singular affine varieties.

Abstract: This paper studies the Schrodinger operator with Morse potential on a right half line [u, \infty) and determines the Weyl asymptotics of eigenvalues for constant boundary conditions. It obtains information on zeros of the Whittaker function $W_{\kappa, \mu}(x)$ for fixed real parameters $\kappa, x$, with x positive, viewed as an entire function of the complex variable $\mu$. In this case all zeros lie on the imaginary axis, with the exception, if $k<0$, of a finite number of real zeros. We obtain an asymptotic formula for the number of zeros of modulus at most T of form $N(T) = (2/\pi) T \log T + f(u) T + O(1)$. Some parallels are noted with zeros of the Riemann zeta function.

Abstract: In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function ζ ˆ ( s ) (resp. the zeta function ζ Y ( s ) of a smooth projective curve Y over a finite field F q , q = p f )) one could possibly associate a foliated Riemannian laminated space ( S Q , F , g , ϕ t ) (resp. ( S Y , F , g , ϕ t ) ) endowed with an action of a flow ϕ t whose primitive compact orbits should correspond to the primes of Q (resp. Y ). Precise conjectures were stated in our report [E. Leichtnam, An invitation to Deninger's work on arithmetic zeta functions, in: Geometry, Spectral Theory, Groups, and Dynamics, in: Contemp. Math. vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 201–236] on Deninger's work. The existence of such a foliated space and flow ϕ t is still unknown except when Y is an elliptic curve (see Deninger [C. Deninger, On the nature of explicit formulas in analytic number theory, a simple example, in: Number Theoretic Methods, Iizuka, 2001, in: Dev. Math., vol. 8, Kluwer Acad. Publ., Dordrecht, 2002, pp. 97–118]). Being motivated by this latter case, we introduce a class of foliated laminated spaces ( S = L × R + ∗ q Z , F , g , ϕ t ) where L is locally D × Z p m , D being an open disk of C . Assuming that the leafwise harmonic forms on L are locally constant transversally, we prove a Lefschetz trace formula for the flow ϕ t acting on the leafwise Hodge cohomology H τ j ( 0 ⩽ j ⩽ 2 ) of ( S , F ) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over F q . We also prove that the eigenvalues of the infinitesimal generator of the action of ϕ t on H τ 1 have real part equal to 1 2 . Moreover, we suggest in a precise way that the flow ϕ t should be induced by a renormalization group flow “a la K. Wilson”. We show that when Y is an elliptic curve over F q this is indeed the case. It would be very interesting to establish a precise connection between our results and those of Connes (p. 553 in [A. Connes, Noncommutative Geometry Year 2000, in: Special Volume GAFA 2000 Part II, pp. 481–559], p. 90 in [A. Connes, Symetries Galoisiennes et Renormalisation, in: Seminaire Bourbaphy, Octobre 2002, pp. 75–91]) and Connes–Marcolli [A. Connes, M. Marcolli, Q -lattices: quantum statistical mechanics and Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. I, Springer-Verlag, 2006, pp. 269–350; A. Connes, M. Marcolli, From physics to number theory via noncommutative geometry. Part II: renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. II, Springer-Verlag, 2006, pp. 617–713] on the Galois interpretation of the renormalization group.

Abstract: Green function for periodic boundary value problem of 2M-th order or- dinary differential equation is found by symmetric orthogonalization method under a suitable solvability condition. As an application, the best constants and the best functions of the Sobolev inequalities in a certain series of Hilbert spaces are found and expressed by means of the well-known Bernoulli polynomials. This result has clarified the variational meaning of the special values ζ(2M )( M =1 , 2, 3,···) of Riemann zeta function ζ(z).