On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis

There has been a surge of interest of late in an old result of Nyman and Beurling giving a Hilbert space formulation of the Riemann hypothesis. Many authors have contributed to this circle of ideas, culminating in a beautiful refinement due to Baez-Duarte. The purpose of this little survey is to dis-entangle the resulting web of complications, and reveal the essential simplicity of the main results.     

Linear relations of theta series of genera of quadratic forms

In this paper we study linear relations among theta series of genera of positive definite n-ary quadratic forms with given level D,2D,4D and 8D for square free D. We obtain a basis for the space generated by genus theta series. This forms a basis of Eisenstein space.     

A note on the non-holomorphic Eisenstein series

We give a sufficient condition of bounded growth for the non-holomorphic Eisenstein series on SL ₂(ℤ). The C ^∞-automorphic forms of bounded growth are introduced by Sturm (Duke Math. J. 48(2), 327–350, 1981) in the study of automorphic L-functions. We also give a Laplace-Mellin transform of the Fourier coefficients of the Eisenstein series. The transformation constructs a projection of the Eisenstein series to the space of holomorphic cusp forms.      

Analysis of orthogonality and of orbits in affine iterated function systems

We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractor X supporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert space has an orthonormal basis (ONB) of exponentials . We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra for X. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum for X. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3. Research supported in part by the National Science Foundation DMS 0457491.     

Fourier frequencies in affine iterated function systems

We examine two questions regarding Fourier frequencies for a class of iterated function systems (IFS). These are iteration limits arising from a fixed finite families of affine and contractive mappings in Rd, and the “IFS” refers to such a finite system of transformations, or functions. The iteration limits are pairs (X,μ) where X is a compact subset of Rd (the support of μ), and the measure μ is a probability measure determined uniquely by the initial IFS mappings, and a certain strong invariance axiom. The two questions we study are: (1) existence of an orthogonal Fourier basis in the Hilbert space L²(X,μ); and (2) explicit constructions of Fourier bases from the given data defining the IFS.     

Spectral synthesis on torsion groups

Spectral synthesis on Abelian torsion groups is proved.     

Eisenstein series associated with Γ<Subscript>0</Subscript>(2)

In this paper, we define the normalized Eisenstein series ℘, e, and associated with Γ₀(2), and derive three differential equations satisfied by them from some trigonometric identities. By using these three formulas, we define a differential equation depending on the weights of modular forms on Γ₀(2) and then construct its modular solutions by using orthogonal polynomials and Gaussian hypergeometric series. We also construct a certain class of infinite series connected with the triangular numbers. Finally, we derive a combinatorial identity from a formula involving the triangular numbers.      

Moment inequalities and the Riemann hypothesis

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann ξ-function are real. On writingξ(x/2)=8 ∫ ₀ ^∞ Φ(t) cos(xt)dt, it is known that a necessary condition that the Riemann hypothesis be valid is that the moments \(\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt}\) satisfy the Turán inequalities (*) $$(\hat b_m (\lambda ))^2 > \left( {\frac{{2m - 1}}{{2m + 1}}} \right)\hat b_{m - 1} (\lambda )\hat b_{m + 1} (\lambda )(m \geqslant 1,\lambda \geqslant 0).$$ We give here a constructive proof that log \(\Phi (\sqrt t )\) is strictly concave for 0 <t < ∞, and with this we deduce in Theorem 2.4 a general class of moment inequalities which, as a special case, establishes that the inequalities (*) are in fact valid for all real λ. As the case λ=0 of (*) corresponds to the Pólya conjecture of 1927, this gives a new proof of the Pólya conjecture.     

亚正常加权移位算子的谱刻画

在这篇文章里，我们给出了亚正常单侧与双侧加权移们算子的谱及其各部分的完全刻画，推广了已有文献中的相关结果。     

Equivalents of the Riemann hypothesis

We obtain necessary and sufficient conditions for the Riemann hypothesis for the Riemann zeta-function, in terms of the functional distribution of quadratic Dirichlet L-functions. Received: 29 November 2004     

Spectral synthesis of Jordan-like operators

Necessary conditions and sufficient conditions are given for an operator acting on a separable Hilbert space whose root spaces are pairwise orthogonal and have dense linear span to admit spectral synthesis; that is, for each of its closed invariant subspaces to be the closed linear span of the root vectors it contains.     

Spectral synthesis of diagonal operators and representing systems for the space of entire functions

In this paper, we study continuous linear operators on spaces of functions analytic on disks in the complex plane having as eigenvectors the monomials zn whose associated eigenvalues λn are distinct. In particular, we show that under mild conditions, such a diagonal operator has non-spectral invariant subspaces (that is, closed invariant subspaces which are not the closed linear span of collections of monomials) if and only if every entire function of a particular growth rate is representable as a generalized Dirichlet series .     

The GHS inequality and the Riemann hypothesis

LetV(t) be the even function on (–, ) which is related to the Riemann xi-function by (x/2)=4 _– exp(ixt–V(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.Communicated by Richard Varga.     

Diophantine equations and the generalized Riemann hypothesis

We show that, for a listable set P of polynomials with integer coefficients, the statement “for all roots θ of all polynomials in P, the generalized Riemann hypothesis for Q(θ) holds” is Diophantine. That is, the statement is equivalent to the unsolvability of a particular Diophantine equation. This is achieved by finding a decidable property P such that the aforementioned statement may be written in the form “P holds for all natural numbers”.     

Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ?²(X), and the energy space HE. In particular, we prove that these operators are always essentially self-adjoint on ?²(X), but may fail to be essentially self-adjoint on HE. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme.     

On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L²(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L²(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A₁, first observed by Everitt, and the self-adjoint operator A^1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A).     

Representations of Integers as Sums of 32 Squares

In this paper, we derive a new explicit formula for r ₃₂(n), where r _k(n) is the number of representations of n as a sum of k squares. For a fixed integer k, our method can be used to derive explicit formulas for r _8k (n). We conclude the paper with various conjectures that lead to explicit formulas for r _2k (n), for any fixed positive integer k > 4.     

Ramanujan's Eisenstein series and new hypergeometric-like series for

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π².