Abstract: This chapter aims at providing a self-contained theory of the Zeta and related functions, which will be required in each of the next chapters. We first introduce (and investigate the various properties and relationships satisfied by) the multiple Hurwitz Zeta function ζ n (s, a) (n∈ℕ) ζ n ( s , a ) ( n ∈ ℕ ) and consider its relatively more familiar special case when n = 1, that is, the Hurwitz (or generalized) Zeta function ζ(s, a) ζ ( s , a ) We then deal rather systematically with the Riemann Zeta function which (for the main purpose of this book) happens to be the most important member of the significantly large family of Zeta functions considered in this chapter. Other functions (introduced in this chapter) include the Polylogarithm functions, Legendre's Chi function, Clausen's integral (or Clausen's function), the Hurwitz-Lerch Zeta function, and so on.

Abstract: The coupled derivative Schrodinger equation is studied in the framework of the Riemann-Hilbert problem and a compact N-soliton solution formula is found. Taking advantage of this result, some properties for single soliton solution and asymptotic analysis of N-soliton solution are explored. As a by-product, a coupled Fokas-Lenells equation together its N-soliton solution is presented.

Abstract: We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p_N(\theta) of large N\times N random unitary (CUE) matrices; i.e. the extreme value statistics of p_N(\theta) when N \rightarrow\infty. In addition, we argue that it leads to multifractal-like behaviour in the total length \mu_N(x) of the intervals in which |p_N(\theta)|>N^x, x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta-function \zeta(s) over stretches of the critical line s=1/2+it of given constant length, and present the results of numerical computations of the large values of \zeta(1/2+it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

Abstract: The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we also give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function zeta(s) does not have any zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.

Abstract: We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twist of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann Hypothesis over finite fields, we prove general estimates with power-saving for such sums. We then derive various concrete applications.

Abstract: The spectral operator was introduced by Lapidus and van Frankenhuijsen (2006 Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings) in their reinterpretation of the earlier work of Lapidus and Maier (1995 J. Lond. Math. Soc. 52 15?34) on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this review, we present the rigorous functional analytic framework given by Herichi and Lapidus (2012) and within which to study the spectral operator. Furthermore, we give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is quasi-invertible (or equivalently, that its truncations are invertible) if and only if the Riemann zeta function ?(s) does not have any zeros on the vertical line Re(s) = c. Hence, it is not invertible in the mid-fractal case when , and it is quasi-invertible everywhere else (i.e. for all c ? (0, 1) with ) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension and c = 1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical in honour of Stuart Dowker?s 75th birthday devoted to ?Applications of zeta functions and other spectral functions in mathematics and physics?.

Abstract: We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an `index-free' proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.

Abstract: Boundary value problems for integrable nonlinear evolution PDEs formulated on the half-line can be analyzed by the unified method introduced by one of the authors and used extensively in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann?Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving four scalar functions of k, called spectral functions. Two of these functions depend on the initial data, whereas the other two depend on all boundary values. The most difficult step of the new method is the characterization of the latter two spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. For certain boundary conditions, called linearizable, this can be achieved by simply using algebraic manipulations. Here, we first present an effective characterization of the spectral functions in terms of the given initial and boundary data for the general case of non-linearizable boundary conditions. This characterization is based on the analysis of the so-called global relation and on the introduction of the so-called Gelfand?Levitan?Marchenko representations of the eigenfunctions defining the spectral functions. We then concentrate on the physically significant case of t-periodic Dirichlet boundary data. After presenting certain heuristic arguments which suggest that the Neumann boundary values become periodic as t ? ?, we show that for the case of the NLS with a sine-wave as Dirichlet data, the asymptotics of the Neumann boundary values can be computed explicitly at least up to third order in a perturbative expansion and indeed at least up to this order are asymptotically periodic.

Abstract: Derdzinski and Shen’s theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new “Codazzi deviation tensor”, with a geometric significance. The general properties are studied, with their implications on Pontryagin forms. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mappings. Compatibility is extended to generalized curvature tensors, with an application to Weyl’s tensor and general relativity.

Abstract: We construct the conserved charge of generic gravity theories built on arbitrary contractions of the Riemann tensor (but not on its derivatives) for asymptotically (anti)--de Sitter spacetimes. Our construction is a generalization of the Abbott-Deser-Tekin charges of linear and quadratic gravity theories in cosmological backgrounds. As an explicit example we find the energy and angular momentum of the Banados-Teitelboim-Zanelli black hole in the ($2+1$)-dimensional Born-Infeld gravity.

Abstract: Let a be an integer different from 0, 1, or a perfect square We consider a conjecture of Erd˝ os which states that #f pV'a p/D rg " r " for any " > 0, where 'a p/ is the order of a modulo p In particular, we see what this conjecture says about Artin's primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture We also extend work of Goldfeld related to divisors of pC a and the order of a modulo p §1 Introduction Let p be a prime number We know that Z= pZ/ Dhai for' p 1/ such a2Z= pZ/ , where'n/ is the Euler totient function When Z= pZ/ Dhai we say that a is a primitive root modulo p In 1927, Artin asked a similar question: let a be a non-zero integer which is not 1 or a square, and define

Abstract: The new (2+1)-dimensional generalized KdV equation which exists the bilinear form is mainly discussed. We prove that the equation does not admit the Painleve property even by taking the arbitrary constant a=0. However, this result is different from Radha and Lakshmanan’s work. In addition, based on Hirota bilinear method, periodic wave solutions in terms of Riemann theta function and rational solutions are derived, respectively. The asymptotic properties of the periodic wave solutions are analyzed in detail.

Abstract: We extend a remarkable theorem of Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. We show that the Codazzi equation can be replaced by a more general algebraic condition. The ensuing extension applies both to the Riemann and to generalized curvature tensors.

Abstract: A long-standing conjecture in quantum field theory due to Broadhurst and Kreimer states that the amplitudes of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms. The zig-zag graphs therefore provide the only infinite family of primitive graphs in $\phi^4_4$ theory (in fact, in any renormalisable quantum field theory in four dimensions) whose amplitudes are now known.

Abstract: In this paper we prove, assuming the Generalized Riemann Hypothesis, the Andr?e-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH.

Abstract: Fractional Operations Confluences Series Expansion and Contiguity Relation Fuchsian Differential Equation and Generalized Riemann Scheme Reduction of Fuchsian Differential Equations Deligne - Simpson Problem A Kac - Moody Root System Expression of Local Solutions Monodromy Reducibility Shift Operators Connection Problem Examples Further Problems

Abstract: Hardy's Z-function, related to the Riemann zeta-function ζ(s), was originally utilised by G. H. Hardy to show that ζ(s) has infinitely many zeros of the form ½+it. It is now amongst the most important functions of analytic number theory, and the Riemann hypothesis, that all complex zeros lie on the line ½+it, is perhaps one of the best known and most important open problems in mathematics. Today Hardy's function has many applications; among others it is used for extensive calculations regarding the zeros of ζ(s). This comprehensive account covers many aspects of Z(t), including the distribution of its zeros, Gram points, moments and Mellin transforms. It features an extensive bibliography and end-of-chapter notes containing comments, remarks and references. The book also provides many open problems to stimulate readers interested in further research.

Abstract: This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.

Abstract: The Riemann Hypothesis has been of central interest to mathematicians for a long time and many unsuccessful attempts have been made to either prove or disprove it. Since the Riemann zeta function is defined as a sum of the infinite number of items, in this paper, we look at the Riemann Hypothesis using a new applied approach to infinity allowing one to easily execute numerical computations with various infinite and infinitesimal numbers in accordance with the principle `The part is less than the whole' observed in the physical world around us. The new approach allows one to work with functions and derivatives that can assume not only finite but also infinite and infinitesimal values and this possibility is used to study properties of the Riemann zeta function and the Dirichlet eta function. A new computational approach allowing one to evaluate these functions at certain points is proposed. Numerical examples are given. It is emphasized that different mathematical languages can be used to describe mathematical objects with different accuracies. The traditional and the new approaches are compared with respect to their application to the Riemann zeta function and the Dirichlet eta function. The accuracy of the obtained results is discussed in detail.

Abstract: Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n) 5040 (\gamma is Euler's constant). It is a natural question in this direction to find a first integer, if exists, which violates this inequality. Following this process, we introduce a new sequence of numbers and call it as extremely abundant numbers. In this paper we show that the Riemann hypothesis is true, if and only if, there are infinitely many of these numbers. Moreover, we investigate some of their properties together with superabundant and colossally abundant numbers.

Abstract: We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the nonuniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

Abstract: We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N), \] where $\rho=1/2+i\gamma$ runs over the non-trivial zeros of the Riemann zeta function $\zeta(s)$.

Abstract: Boundary value problems for integrable nonlinear evolution PDEs formulated on the finite interval can be analyzed by the unified method introduced by one of the authors and extensively used in the literature. The implementation of this general method to this particular class of problems yields the solution in terms of the unique solution of a matrix Riemann?Hilbert problem formulated in the complex k-plane (the Fourier plane), which has a jump matrix with explicit (x, t)-dependence involving six scalar functions of k, called the spectral functions. Two of these functions depend on the initial data, whereas the other four depend on all boundary values. The most difficult step of the new method is the characterization of the latter four spectral functions in terms of the given initial and boundary data, i.e. the elimination of the unknown boundary values. Here, we present an effective characterization of the spectral functions in terms of the given initial and boundary data. We present two different characterizations of this problem. One is based on the analysis of the so-called global relation, on the analysis of the equations obtained from the global relation via certain transformations leaving the dispersion relation of the associated linearized PDE invariant and on the computation of the large k asymptotics of the eigenfunctions defining the relevant spectral functions. The other is based on the analysis of the global relation and on the introduction of the so-called Gelfand?Levitan?Marchenko representations of the eigenfunctions defining the relevant spectral functions. We also show that these two different characterizations are equivalent and that in the limit when the length of the interval tends to infinity, the relevant formulas reduce to the analogous formulas obtained recently for the case of boundary value problems formulated on the half-line.

Abstract: On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in random matrix theory. In an appendix we put forward some general theorems regarding our knowledge of the zeta zeroes in the mesoscopic regime.

Abstract: A scale invariant model of statistical mechanics is applied to derive invariant forms of conservation equations. A modified form of Cauchy stress tensor for fluid is presented that leads to modified Stokes assumption thus a finite coefficient of bulk viscosity. The phenomenon of Brownian motion is described as the state of equilibrium between suspended particles and molecular clusters that themselves possess Brownian motion. Physical space or Casimir vacuum is identified as a tachyonic fluid that is “stochastic ether” of Dirac or “hidden thermostat” of de Broglie, and is compressible in accordance with Planck’s compressible ether. The stochastic definitions of Planck h and Boltzmann k constants are shown to respectively relate to the spatial and the temporal aspects of vacuum fluctuations. Hence, a modified definition of thermodynamic temperature is introduced that leads to predicted velocity of sound in agreement with observations. Also, a modified value of JouleMayer mechanical equivalent of heat is identified as the universal gas constant and is called De Pretto number 8338 which occurred in his mass-energy equivalence equation. Applying Boltzmann’s combinatoric methods, invariant forms of Boltzmann, Planck, and Maxwell-Boltzmann distribution functions for equilibrium statistical fields including that of isotropic stationary turbulence are derived. The latter is shown to lead to the definitions of (electron, photon, neutrino) as the mostprobable equilibrium sizes of (photon, neutrino, tachyon) clusters, respectively. The physical basis for the coincidence of normalized spacings between zeros of Riemann zeta function and the normalized Maxwell-Boltzmann distribution and its connections to Riemann Hypothesis are examined. The zeros of Riemann zeta function are related to the zeros of particle velocities or “stationary states” through Euler’s golden key thus providing a physical explanation for the location of the critical line. It is argued that because the energy spectrum of Casimir vacuum will be governed by Schrodinger equation of quantum mechanics, in view of Heisenberg matrix mechanics physical space should be described by noncommutative spectral geometry of Connes. Invariant forms of transport coefficients suggesting finite values of gravitational viscosity as well as hierarchies of vacua and absolute zero temperatures are described. Some of the implications of the results to the problem of thermodynamic irreversibility and Poincare recurrence theorem are addressed. Invariant modified form of the first law of thermodynamics is derived and a modified definition of entropy is introduced that closes the gap between radiation and gas theory. Finally, new paradigms for hydrodynamic foundations of both Schrodinger as well as Dirac wave equations and transitions between Bohr stationary states in quantum mechanics are discussed. Key-Words: Kinetic theory of ideal gas; Thermodynamics; Statistical mechanics; Riemann Hypothesis; TOE.