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"RIEMANN ZETA FUNCTIONS Louis de Branges de Bourcia∗ Abstract. A Riemann zeta function is a function which is analytic in the complex plane, with the possible exception of a simple pole at one, and which has characteristic Euler product and functional identity. Riemann zeta functions originate in an adelic generalization of the Laplace transformation which is deﬁned using a theta function. Hilbert spaces, whose elements are entire functions, are obtained on application of the Mellin transformation. Maximal dissipative transformations are constructed in these spaces which have implications for zeros of zeta functions. The zeros of a Riemann zeta function in the critical strip are simple and lie on the critical line. The Euler zeta function and Dirichlet zeta functions are examples of Riemann zeta functions. A Riemann zeta function is represented by a Dirichlet series ζ(s) = τ (n)n−s in the half–plane Rs > 1 with summation over the positive integers n which are relatively prime to a given positive integer ρ. A Riemann zeta function has an analytic extension to the complex plane with the possible exception of a simple pole at s = 1. Riemann zeta functions are divided into two classes according to Euler product and functional identity. Riemann zeta functions originate in Fourier analysis either on a plane or on a skew-plane. The Euler product for the zeta function of a plane is a product ζ(s)−1 = (1 − χ(p)p−s ) taken over the primes p which are not divisors of ρ. The identity |τ (p)| = 1 holds for every such prime p. The functional identity for the zeta function of a skew-plane states that the functions 1 1 (π/ρ)− 2 ν− 2 s Γ( 1 ν + 1 s)ζ(s) 2 2 and ∗ (π/ρ)− 2 ν− 2 + 2 s Γ( 1 ν + 2 1 1 1 1 2 − 1 s)ζ(1 − s− )− 2 Research supported by the National Science Foundation 1 2 L. DE BRANGES DE BOURCIA April 21, 2003 of s are linearly dependent for ν equal to zero or one. The Euler product for the zeta function of a skew–plane is a product ζ(s)−1 = (1 − τ (p)p−s + [τ (p)2 − τ (p2 )]p−2s ) taken over the primes p which are not divisors of ρ. The inequality |τ (p)| ≤ 2 holds for every such prime p with τ (p)2 − τ (p2 ) of absolute value one. The functional identity for the zeta function of a skew–plane states that the functions (2π/ρ)− 2 ν−s Γ( 1 ν + s)ζ(s) 2 1 and (2π/ρ)− 2 ν−1+s Γ( 1 ν + 1 − s)ζ(1 − s) 2 1 are linearly dependent for some odd positive integer ν. The Euler zeta function is a Riemann zeta function for a plane. The other Riemann zeta functions for a plane are Dirichlet zeta functions. The Riemann zeta functions for a skew-plane are examples of zeta functions for which the Ramanujan hypothesis [3], [4] is satisﬁed. Zeta functions originate in Fourier analysis on locally compact rings. The locally compact ﬁeld of real numbers is obtained by completion of the ﬁeld of rational numbers in a topology which is compatible with additive and multiplicative structure. Other locally compact ﬁelds are constructed by completion of subrings of the rational numbers admitting topologies compatible with additive and multiplicative structure. If r is a positive integer, a corresponding subring consists of the rational numbers which have integral product with some positive integer whose prime divisors are divisors of r. The ring admits a topology for which addition and multiplication are continuous as transformations of the Cartesian product of the ring with itself into the ring. The r–adic topology is determined by its neighborhoods of the origin. Basic neighborhoods are the ideals of the integers generated by positive integers whose prime divisors are divisors of r. The r–adic line is the completion of the ring in the resulting uniform structure. Addition and multiplication have continuous extensions as transformations of the Cartesian product of the r–adic line with itself into the r–adic line. The r–adic line is a commutative ring which is canonically isomorphic to the Cartesian product of the p–adic lines taken over the prime divisors p of r. Each p–adic line is a locally compact ﬁeld. An element of the r–adic line is said to be integral if it belongs to the closure of the integers in the r–adic topology. The integral elements of the r–adic line form a compact neighborhood of the origin for the r–adic topology. An invertible integral element of the r–adic line is said to be a unit if its inverse is integral. The r–adic modulus of an invertible element ξ of the r–adic line is the unique positive rational number |ξ|− , which represents an element of the r–adic line, such that |ξ|−ξ is a unit. The r–adic modulus of a noninvertible element of the r–adic line is zero. Haar measure for the r–adic line is normalized so that the set of integral elements has measure one. Multiplication by an element ξ of the r–adic line multiplies Haar measure by the RIEMANN ZETA FUNCTIONS 3 r-adic modulus. The function exp(2πiξ) of ξ in the r–adic line is deﬁned by continuity from values when ξ is a rational number which represents an element of the r–adic line. The Euclidean line is the locally compact ring of real numbers. The Euclidean modulus of an element ξ of the Euclidean line is its absolute value |ξ|. A unit of the Euclidean line is an element of absolute value one. Haar measure for the Euclidean line is Lebesgue measure. Multiplication by an element ξ of the Euclidean line multiplies Haar measure by a factor of the Euclidean modulus |ξ|. The function exp(2πiξ) of ξ in the Euclidean line is continuous. The r–adelic line is a locally compact ring which is the Cartesian product of the Euclidean line and the r–adic line. An element ξ of the r–adelic line has a Euclidean component ξ+ and an r–adic component ξ− . The Euclidean modulus of an element ξ of the r–adelic line is the Euclidean modulus |ξ|+ of its Euclidean component ξ+ . The r–adic modulus of an element ξ of the r–adic line is the r–adic modulus |ξ|− of its r–adic component ξ− . The r–adelic modulus of an element ξ of the r–adelic line is the product |ξ| of its Euclidean modulus |ξ|+ and its r–adic modulus |ξ|− . An element of the r–adelic line is said to be a unit if its Euclidean modulus and its r–adic modulus are one. An element of the r–adelic line is said to be unimodular if its r–adelic modulus is one. Haar measure for the r–adelic line is the Cartesian product of Haar measure for the Euclidean line and Haar measure for the r–adic line. Multiplication by an element of the r–adelic line multiplies Haar measure by the r–adelic modulus. The function exp(2πiξ) = exp(2πiξ+ )/ exp(2πiξ− ) of ξ in the r–adelic line is the quotient of the function exp(2πiξ+ ) of the Euclidean component and of the function exp(2πiξ− ) of the r–adic component. A principal element of the r–adelic line is an element whose Euclidean and r–adic components are represented by equal rational numbers. The principal elements of the r–adelic line form a discrete subring of the r–adelic line. The identity exp(2πiξ) = 1 holds for every principal element ξ of the r–adelic line. A principal element of the r–adelic line is unimodular if it is nonzero. The adic line is a locally compact ring which is a restricted inverse limit of the r–adic lines. The ring is a completion of the ﬁeld of rational numbers in a topology for which addition and multiplication are continuous as transformations of the Cartesian product of the ﬁeld with itself into the ﬁeld. The adic topology is determined by its neighborhoods of the origin. Basic neighborhoods are the ideals of the integers which are generated by positive integers. The adic line is the completion of the ﬁeld in the resulting uniform structure. Addition and multiplication have continuous extensions as transformations of the Cartesian product of the adic line with itself into the adic line. The adic line is canonically isomorphic to a subring of the Cartesian product of the p–adic lines taken over all primes p. An element of the Cartesian product represents an element of the adic line if, and only if, its p–adic component is integral for all but a ﬁnite number of primes p. An element of the adic line is said to be integral if its p–adic component is integral for every 4 L. DE BRANGES DE BOURCIA April 21, 2003 prime p. The integral elements of the adic line form a compact neighborhood of the origin for the adic topology. An invertible integral element of the adic line is said to be a unit if its inverse is integral. The adic modulus of an invertible element ξ of the adic line is the unique positive rational number |ξ|− such that |ξ|− ξ is a unit. The adic modulus of a noninvertible element of the adic line is zero. Haar measure for the adic line is normalized so that the set of integral elements has measure one. Multiplication by an element of the adic line multiplies Haar measure by the adic modulus. The function exp(2πiξ) of ξ in the adic line is deﬁned by continuity from its values when ξ is a rational number. The adelic line is a locally compact ring which is the Cartesian product of the Euclidean line and the adic line. An element ξ of the adelic line has a Euclidean component ξ+ and an adic component ξ− . The Euclidean modulus of an element ξ of the adelic line is the Euclidean modulus |ξ|+ of its Euclidean component ξ+ . The adic modulus of an element ξ of the adelic line is the adic modulus |ξ|− of its adic component ξ− . The adelic modulus of an element ξ of the adelic line is the product |ξ| of its Euclidean modulus |ξ|+ and its adic modulus |ξ|−. An element of the adelic line is said to be a unit if its Euclidean modulus and its adic modulus are one. An element of the adelic line is said to be unimodular if its adelic modulus is one. Haar measure for the adelic line is the Cartesian product of Haar measure for the Euclidean line and Haar measure for the adic line. Multiplication by an element of the adelic line multiplies Haar measure by the adelic modulus. The function exp(2πiξ) = exp(2πiξ+ )/ exp(2πiξ− ) of ξ in the adelic line is deﬁned as the ratio of the function exp(2πiξ+) of ξ+ in the Euclidean line and the function exp(2πiξ− ) of ξ− in the adic line. A principal element of the adelic line is an element whose Euclidean and adic components are represented by equal rational numbers. The principal elements of the adelic line form a discrete subring of the adelic line. An element ξ of the adelic line is a principal element if, and only if, the identity exp(2πiξη) = 1 holds for every principal element η of the adelic line. A principal element of the adelic line is unimodular if it is nonzero. The Fourier transformation for the adelic line is an isometric transformation whose domain and range are the space of square integrable functions with respect to Haar measure for the adelic line. The transformation takes a function f (ξ) of ξ in the adelic line into a function g(η) of η in the adelic line when the identity g(η) = f (ξ) exp(2πiηξ)dξ is formally satisﬁed. The integral is accepted as the deﬁnition of the transformation when the integral with respect to Haar measure for the adelic line is absolutely convergent. The identity |f (ξ)|2dξ = |g(ξ)|2dξ RIEMANN ZETA FUNCTIONS 5 then holds with integration with respect to Haar measure for the adelic line. The identity f (η) = g(ξ) exp(−2πiηξ)dξ holds with integration with respect to Haar measure for the adelic line when the integral is absolutely convergent. The Poisson summation formula f (ξ) = g(ξ) holds with summation over the principal elements of the adelic line when both integrals are absolutely convergent. The Euclidean plane is the locally compact ﬁeld of complex numbers. The complex conjugation of the Euclidean plane is the automorphism ξ into ξ − of order two whose ﬁxed ﬁeld is the Euclidean line. The Euclidean modulus of an element ξ of the Euclidean plane is its absolute value |ξ|. An element of the Euclidean plane is said to be a unit if its Euclidean modulus is one. Haar measure for the Euclidean plane is Lebesgue measure. Multiplication by an element of the Euclidean plane multiplies Haar measure by the square of the Euclidean modulus. The Euclidean skew–plane is a locally compact ring in which every nonzero element is invertible. The Euclidean skew–plane is an algebra over the Euclidean plane which is generated by an element j which satisﬁes the identity j 2 = −1 and the identity jγ = γ − j for every element γ of the Euclidean plane. The elements of the Euclidean skew–plane are of the form α + jβ with α and β elements of the Euclidean plane. The conjugation of the Euclidean skew–plane is the anti–automorphism ξ into ξ − of order two which takes α + jβ into α− − jβ for all elements α and β of the Euclidean plane. The Euclidean plane is a subﬁeld of the Euclidean skew–plane on which the conjugation of the Euclidean skew–plane agrees with the conjugation of the Euclidean plane. The Euclidean line is the ﬁxed ﬁeld of the conjugation of the Euclidean skew–plane. If ξ is an element of the Euclidean skew–plane, ξ ∗ ξ is a nonnegative element of the Euclidean line which is nonzero if, and only if, ξ is nonzero. The Euclidean modulus of an element ξ of the Euclidean skew–plane is the nonnegative square root |ξ| of ξ − ξ. A unit of the Euclidean skew–plane is an element of Euclidean modulus one. Haar measure for the Euclidean skew–plane is the Cartesian product of the Haar measures for component Euclidean planes. Multiplication by an 6 L. DE BRANGES DE BOURCIA April 21, 2003 element of the Euclidean skew–plane multiplies Haar measure by the fourth power of the Euclidean modulus. A theorem of Lagrange states that every positive integer is a sum of four squares of integers. If n is a positive integer, the equation n = ξ −ξ has solutions ξ = (α + iβ) + j(γ + iδ) in the elements of the Euclidean skew–plane whose components α, β, γ, and δ are all integers or all halves of odd integers. The solutions form a group of order twenty-four when n is equal to one. The r–adic skew–plane is an algebra of dimension four over the r–adic line which is generated by the same units as i and j as the Euclidean skew–plane. The elements of the r–adic skew–plane are of the form (α + iβ) + j(γ + iδ) for elements α, β, γ, and δ of the r–adic line. The conjugation of the r–adic skew–plane is the anti–automorphism ξ into ξ − of order two which takes (α + iβ) + j(γ + iδ) into (α − iβ) − j(γ + iδ) for all elements α, β, γ, and δ of the r–adic line. The topology of the r–adic skew–plane is the Cartesian product of topologies of coordinate r–adic lines. If ξ is an element of the r–adic skew–plane, ξ − ξ is an element of the r–adic line which is invertible if, and only if, ξ is invertible. The r–adic modulus of an element ξ of the r–adic skew–plane is the nonnegative square root |ξ|− of the r–adic modulus of ξ − ξ. An integral element of the r–adic skew–plane is an element ξ such that ξ − ξ is an integral element of the r–adic line. The integral elements of the r–adic skew–plane form a compact subring which is a neighborhood of the origin for the r–adic topology. A unit of the r–adic skew–plane is an invertible integral element whose inverse is integral. An element ξ of the r–adic skew–plane is a unit if, and only if, ξ − ξ is a unit of the r–adic line. Haar measure for the r–adic skew– plane is normalized so that the set of integral elements has measure one. Multiplication by an element of the r–adic skew–plane multiplies Haar measure by the fourth power of the r–adic modulus. The r–adelic skew–plane is a locally compact ring which is the Cartesian product of the Euclidean skew–plane and the r–adic skew–plane. An element ξ of the r–adelic skew–plane has a Euclidean component ξ+ and an r–adic component ξ− . The conjugation of the r– adelic skew–plane is the anti–automorphism ξ into ξ − of order two such that the Euclidean component of ξ − is obtained from the Euclidean component of ξ under the conjugation RIEMANN ZETA FUNCTIONS 7 of the Euclidean skew–plane and the r–adic component of ξ − is obtained from the r–adic component of ξ under the conjugation of the r–adic skew–plane. The Euclidean modulus of an element ξ of the r–adelic skew–plane is the Euclidean modulus |ξ|+ of its Euclidean component ξ+ . The r–adic modulus of an element ξ of the r–adelic skew–plane is the r–adic modulus |ξ|− of its r–adic component ξ− . The r–adelic modulus of an element ξ of the r–adelic skew–plane is the product |ξ| of its Euclidean modulus and its r–adic modulus. An element of the r–adelic skew–plane is said to be a unit if its Euclidean modulus and its r–adic modulus are one. An element of the r–adelic skew–plane is said to be unimodular if its r–adelic modulus is one. Haar measure for the r–adelic skew–plane is the Cartesian product of Haar measure for the Euclidean skew–plane and Haar measure for the r–adic skew–plane. Multiplication by an element of the r–adelic skew–plane multiplies Haar measure by the fourth power of the r–adelic modulus. A principal element of the r– adelic skew–plane is an element whose coordinates with respect to the canonical basis are principal elements of the r–adelic line. The principal elements of the r–adelic skew–plane form a closed subring whose nonzero elements are unimodular and invertible. The adic skew–plane is an algebra of dimension four over the adic line which is generated by the same units i and j as the Euclidean skew–plane. The elements of the adic skew– plane are of the form (α + iβ) + j(γ + iδ) for elements α, β, γ, and δ of the adic line. The conjugation of the adic skew–plane is the anti–automorphism ξ into ξ − of order two which takes (α + iβ) + j(γ + iδ) into (α − iβ) − j(γ + iδ) for all elements α, β, γ, and δ of the adic line. The topology of the adic skew–plane is the Cartesian product of topologies of coordinate adic lines. If ξ is an element of the adic skew–plane, ξ − ξ is an element of the adic line which is invertible if, and only if, ξ is invertible. The adic modulus of an element ξ of the adic skew–plane is the nonnegative square root |ξ|− of the adic modulus of ξ − ξ. An integral element of the adic skew–plane is an element ξ such that ξ − ξ is an integral element of the adic line. The integral elements of the adic skew–plane form a compact subring which is a neighborhood of the origin for the adic topology. A unit of the adic skew–plane is an invertible integral element whose inverse is integral. An element ξ of the adic skew–plane is a unit if, and only if, ξ − ξ is a unit of the adic line. Haar measure for the adic skew–plane is normalized so that the set of integral elements has measure one. Multiplication by an element of the adic skew–plane multiplies Haar measure by the fourth power of the adic modulus. The adelic skew–plane is a locally compact ring which is the Cartesian product of the Euclidean skew–plane and the adic skew–plane. An element ξ of the adelic skew–plane has a Euclidean component ξ+ and an adic component ξ− . The conjugation of the adelic skew–plane is the anti–automorphism ξ into ξ − of order two such that the Euclidean component of ξ − is obtained from the Euclidean component of ξ under the conjugation 8 L. DE BRANGES DE BOURCIA April 21, 2003 of the..."

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