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We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula., Spanish MICINN, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

We define the class of Left Located Divisor (LLD) meromorphic functions, their vertical order m(0)(f) and their convergence exponent d(f). When m0(f) <= d(f) we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example, Dirichlet series, the Gamma-function, and trigonometric functions., Spanish MICINN, Depto. de Álgebra, Geometría y Topología, Fac. de Ciencias Matemáticas, TRUE, pub

Given two arbitrary almost periodic functions with Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip V, where both functions assume the same set of values on every open vertical substrip included in V, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be ⁎-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.

This paper deals with the sets of real projections of zeros of analytic almost periodic functions defined in a vertical strip. By using our equivalence relation introduced in the context of the complex functions which can be represented by a Dirichlet-like series, this work provides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents of an analytic almost periodic function are linearly independent over the rational numbers, such a set has no isolated points.

Given two arbitrary almost periodic functions with Fourier exponents which are linearly independent over the rational numbers, we prove that the existence of a common open vertical strip V, where both functions assume the same set of values on every open vertical substrip included in V, is a necessary and sufficient condition for both functions to have the same region of almost periodicity and to be ⁎-equivalent or Bohr-equivalent. This result represents the converse of Bohr's equivalence theorem for this particular case.

This paper deals with the sets of real projections of zeros of analytic almost periodic functions defined in a vertical strip. By using our equivalence relation introduced in the context of the complex functions which can be represented by a Dirichlet-like series, this work provides practical results in order to determine whether a real number belongs to the closure of such a set. Its main result shows that, in the case that the Fourier exponents of an analytic almost periodic function are linearly independent over the rational numbers, such a set has no isolated points.

This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely formulated in the case of existence of an integral basis for the set of exponents of the associated Dirichlet series.

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain.

In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isomorphism between Jacobi forms of integral weight and half-integral weight is also obtained.

This paper presents a full generalization of Bohr’s equivalence theorem for the case of almost periodic functions, which improves a recent result that was uniquely formulated in the case of existence of an integral basis for the set of exponents of the associated Dirichlet series.

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain.

[EN] We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

[EN] We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

Based on a generalization of Bohr’s equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr’s equivalence theorem extended to the case of these classes of functions that are associated with countable sets of distinct real numbers having an integral basis.

Correction to: Ramanujan J (2018) 46:245–267 https://doi.org/10.1007/s11139-017-9950-1

Based on a generalization of Bohr’s equivalence relation for general Dirichlet series, in this paper we study the sets of values taken by certain classes of equivalent almost periodic functions in their strips of almost periodicity. In fact, the main result of this paper consists of a result like Bohr’s equivalence theorem extended to the case of these classes of functions that are associated with countable sets of distinct real numbers having an integral basis.

Correction to: Ramanujan J (2018) 46:245–267 https://doi.org/10.1007/s11139-017-9950-1

In this paper, I present a formalisation of a large portion of Apostol’s Introduction to Analytic Number Theory in Isabelle/HOL. Of the 14 chapters in the book, the content of 9 has been mostly formalised, while the content of 3 others was already mostly available in Isabelle before. The most interesting results that were formalised are: - The Riemann and Hurwitz zeta functions and the Dirichlet L functions - Dirichlet’s theorem on primes in arithmetic progressions - An analytic proof of the Prime Number Theorem - The asymptotics of arithmetical functions such as the prime omega function, the divisor count sigma_0(n), and Euler’s totient function phi(n)

We present an abstract approach to the abscissas of convergence of vector-valued Dirichlet series. As a consequence we deduce that the abscissas for Hardy spaces of Dirichlet series are all equal. We also introduce and study weak versions of the abscissas for scalar-valued Dirichlet series.

We study random unconditionality of Dirichlet series in vector-valued Hardy spaces Hp(X). It is shown that a Banach space X has type 2 (respectively, cotype 2) if and only if for every choice (xn)n⊂X it follows that (xnn−s)n is random unconditionally convergent (respectively, divergent) in H2(X). The analogous question on Hp(X) spaces for p≠2 is also explored. We also provide explicit examples exhibiting the differences between the unconditionality of (xnn−s)n in Hp(X) and that of (xnzn)n in Hp(X). © 2019 Elsevier Inc.

[EN] A classical result of Harald Bohr linked the study of convergent and bounded Dirichlet series on the right half plane with bounded holomorphic functions on the open unit ball of the space c(0) f complex null sequences. Our aim here is to show that many questions in Dirichlet series have very natural solutions when, following Bohr's idea, we translate these to the infinite dimensional setting. Some are new proofs and other new results obtained by using that point of view.

[EN] The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Frechet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

[EN] The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Frechet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not a Q-algebra. Composition operators on this space are also studied.

We establish the central convergence properties of ordinary Dirichlet series, including the classical result by Bohr, providing uniform convergence of the series where it has a bounded analytic continuation. We also derive a lower bound for the supremum of Dirichlet polynomials using Kronecker's theorem, of which we see one proof. With this knowledge and some probability theory we can follow the work of Queffélec and Boas proving the existence of random series, with terms ±n^-s, with certain convergence properties. In particular Boas work is a probabilistic version of what Bohnenblust and Hille did, namely showing that the estimate of the distance of the abscissae of absolute and uniform convergence - estimated from above by 1/2 - is sharp. An abscissa denotes the vertical lines Re(s) to the right of which the Dirichlet series converges and to the left of which it diverges (in some sense of convergence), and is throroughly introduced in Chapter 1., En funktion som har fått mycket uppmärksamhet bland matematiker är den så kallade Riemanns zeta-funktion, som definieras med hjälp av en summa av oändligt antal termer, en så kallad serie: zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ... Här är s=a+ib en komplex variabel med en reell del a och en imaginär del, där i har egenskapen i^2=-1. Värdet på den här serien kommer givetvis bero på vilket värde på s man använder, och det kan ge serien både ett ändligt och ett oändligt värde. T.ex. så har man lyckats visa att zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ... = pi^2/6 men också att zeta(1) = 1+ 1/2 + 1/3 + 1/4 + ... = ∞, d.v.s. blir oändligt stort. Man kan visa att om s = a + ib har realdel a > 1 så har serien ett ändligt värde. I övriga fall säger man att serien inte är definierad, och då behöver man ett annat sätt att ge mening till zeta-funktionen. Anledningen till att den här funktionen är så uppmärksammad är att den har en nära koppling till primtalen, och det har länge varit ett olöst problem att hitta alla dess nollställen, dvs de s sådana att zeta(s) = 0. Riemann själv formulerade en hypotes om att alla (icke-triviala) nollställen finns längs en linje s = 1/2 + ib, och Hardy har bevisat att längs den här linjen finns det oändligt många nollställen. Däremot är det ingen hittills som har lyckats visa att hypotesen faktiskt är sann eller hittat ett motbevis genom ett nollställe utanför linjen. Många matematiker har funderat på det här problemet och The Clay Mathematics Institute har till och med utlyst en belöning på 1 miljon dollar till den som löser det, som en del av deras sju prisbelönta Milleniumproblem. Ett försök att bättre förstå den här funktionen har varit att undersöka en mer allmän serie a_1/1 + a_2/2^s + a_3/3^s + a_4/4^s + ... I fallet av zeta-funktionen så noterar vi att a_n = 1, för alla n. Den här mer allmänna serien kallas en Dirichlet-serie. Det visar sig att alla serier som har den formen delar en hel del gemensamma egenskaper, som man därför hoppas kan

We establish the central convergence properties of ordinary Dirichlet series, including the classical result by Bohr, providing uniform convergence of the series where it has a bounded analytic continuation. We also derive a lower bound for the supremum of Dirichlet polynomials using Kronecker's theorem, of which we see one proof. With this knowledge and some probability theory we can follow the work of Queffélec and Boas proving the existence of random series, with terms ±n^-s, with certain convergence properties. In particular Boas work is a probabilistic version of what Bohnenblust and Hille did, namely showing that the estimate of the distance of the abscissae of absolute and uniform convergence - estimated from above by 1/2 - is sharp. An abscissa denotes the vertical lines Re(s) to the right of which the Dirichlet series converges and to the left of which it diverges (in some sense of convergence), and is throroughly introduced in Chapter 1., En funktion som har fått mycket uppmärksamhet bland matematiker är den så kallade Riemanns zeta-funktion, som definieras med hjälp av en summa av oändligt antal termer, en så kallad serie: zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + ... Här är s=a+ib en komplex variabel med en reell del a och en imaginär del, där i har egenskapen i^2=-1. Värdet på den här serien kommer givetvis bero på vilket värde på s man använder, och det kan ge serien både ett ändligt och ett oändligt värde. T.ex. så har man lyckats visa att zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ... = pi^2/6 men också att zeta(1) = 1+ 1/2 + 1/3 + 1/4 + ... = ∞, d.v.s. blir oändligt stort. Man kan visa att om s = a + ib har realdel a > 1 så har serien ett ändligt värde. I övriga fall säger man att serien inte är definierad, och då behöver man ett annat sätt att ge mening till zeta-funktionen. Anledningen till att den här funktionen är så uppmärksammad är att den har en nära koppling till primtalen, och det har länge varit ett olöst problem att hitta alla dess nollställen, dvs de s sådana att zeta(s) = 0. Riemann själv formulerade en hypotes om att alla (icke-triviala) nollställen finns längs en linje s = 1/2 + ib, och Hardy har bevisat att längs den här linjen finns det oändligt många nollställen. Däremot är det ingen hittills som har lyckats visa att hypotesen faktiskt är sann eller hittat ett motbevis genom ett nollställe utanför linjen. Många matematiker har funderat på det här problemet och The Clay Mathematics Institute har till och med utlyst en belöning på 1 miljon dollar till den som löser det, som en del av deras sju prisbelönta Milleniumproblem. Ett försök att bättre förstå den här funktionen har varit att undersöka en mer allmän serie a_1/1 + a_2/2^s + a_3/3^s + a_4/4^s + ... I fallet av zeta-funktionen så noterar vi att a_n = 1, för alla n. Den här mer allmänna serien kallas en Dirichlet-serie. Det visar sig att alla serier som har den formen delar en hel del gemensamma egenskaper, som man därför hoppas kan

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c0 contains an isomorphic copy of ℓ1. Moreover, we can have this copy of ℓ1 in such a way that every non-zero element of it fails to be analytic at precisely the same point., Ministerio de Educación, Cultura y Deporte, Depto. de Análisis Matemático y Matemática Aplicada, Fac. de Ciencias Matemáticas, TRUE, pub

This paper demonstrates the application of a generalised layered linear viscoelastic (LVE) analysis for estimating the structural response of flexible pavements. A comparison of the direct layered viscoelastic responses with approximate solutions based on the linear elastic (LE) and LVE collocation methods was also carried out. The different approaches were implemented by extending a layered elastic program with an improved computational performance. The LE and LVE collocation methods were further extended for analysis of pavements under moving loads. The methods were illustrated by analysing a pavement structure subjected to moving wheel loads of 30, 50, 60 and 80kN using a Heavy Vehicle Simulator (HVS). The various responses (stresses and strains) in the pavement, at pavement temperatures of 0, 10 and 20 degrees C, were measured using various types of sensors installed in the structure. It was shown that the approximated LVE solution based on the LE collocation method agreed very well with the measurements and is computationally the least expensive., QC 20151203

[EN] We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c(o) contains an isomorphic copy of l(1). Moreover, we can have this copy of l(1) in such a way that every non-zero element of it fails to be analytic at precisely the same point.

[EN] We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m-homogeneous non-analytic polynomials on c(o) contains an isomorphic copy of l(1). Moreover, we can have this copy of l(1) in such a way that every non-zero element of it fails to be analytic at precisely the same point.

[EN] Denote by Ω(n) the number of prime divisors of n ∈ N (counted with multiplicities). For x ∈ N define the Dirichlet-Bohr radius P L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n≤x ann −s we have X n≤x |an|r Ω(n) ≤ sup t∈R X n≤x ann −it . We prove that the asymptotically correct order of L(x) is (log x) 1/4x −1/8 . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa

[EN] Denote by Ω(n) the number of prime divisors of n ∈ N (counted with multiplicities). For x ∈ N define the Dirichlet-Bohr radius P L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n≤x ann −s we have X n≤x |an|r Ω(n) ≤ sup t∈R X n≤x ann −it . We prove that the asymptotically correct order of L(x) is (log x) 1/4x −1/8 . Following Bohr’s vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa

[EN] In 1931 H.F.Bohnenblust and E.Hille published a very important paper in which not only did they solve a long standing problem on convergence of Dirichlet series, but also gave a general version of a celebrated inequality of Littlewood. Although it is full of extremely valuable mathematical ideas, the paper has been overlooked for a long time and even today we feel that it does not get the credit it deserves. This may be caused by the not always accessible style that makes that the ideas are sometimes hidden. It is our intention to try to study the paper from a modern point of view and to bring to light the valuable aspects we believe it has.

Write an integer as finite products of ordered factors belonging to a given subset P of integers larger than one. A very general central limit theorem is derived for the number of ordered factors in random factorizations for any subset P containing at least two elements. The method of proof is very simple and relies in part on Delange's Tauberian theorems and an interesting Tauberian technique for handling Dirichlet series associated with odd centered moments., Correction in: Electronic Journal of Probability, vol. 18, pages 1-3, Article Number: 16DOI: 10.1214/EJP.v18-2297

[EN] Recent results on Dirichlet series Sigma(n) a(n) 1/n(s), s is an element of C, with coefficients a(n) in an infinite dimensional Banach space X show that the maximal width of uniform but not absolute convergence coincides for Dirichlet series and for m-homogeneous Dirichlet polynomials. But a classical non-trivial fact fue to Bohnenblust and Hille shows that if X is one dimensional, this maximal width heavily depends on the degree m of the Dirichlet polynomials. We carefully analyze this phenomenon, in particular in the setting of l(p)-spaces.

[EN] Each Dirichlet series D=∑∞n=1an1nsD=∑n=1∞an1ns, with variable s∈Cs∈C and coefficients an∈Can∈C, has a so called Bohr strip, the largest strip in CC on which DD converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals 1/21/2. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments.

[EN] Each Dirichlet series D=∑∞n=1an1nsD=∑n=1∞an1ns, with variable s∈Cs∈C and coefficients an∈Can∈C, has a so called Bohr strip, the largest strip in CC on which DD converges absolutely but not uniformly. The classical Bohr-Bohnenblust-Hille theorem states that the width of the largest possible Bohr strip equals 1/21/2. Recently, this deep work of Bohr, Bohnenblust and Hille from the beginning of the last century was revisited by various authors. New methods from different fields of modern analysis (e.g. probability theory, number theory, functional and Fourier analysis) allow to improve the Bohr-Bohnenblust-Hille cycle of ideas, and to extend it to new settings, in particular to Dirichlet series which coefficients in Banach spaces. We survey on various aspects of these new developments.

[EN] Recent results on Dirichlet series Sigma(n) a(n) 1/n(s), s is an element of C, with coefficients a(n) in an infinite dimensional Banach space X show that the maximal width of uniform but not absolute convergence coincides for Dirichlet series and for m-homogeneous Dirichlet polynomials. But a classical non-trivial fact fue to Bohnenblust and Hille shows that if X is one dimensional, this maximal width heavily depends on the degree m of the Dirichlet polynomials. We carefully analyze this phenomenon, in particular in the setting of l(p)-spaces.

This thesis makes a survey of converse theorems for Dirichlet series. A converse theo-rem gives sufficient conditions for a Dirichlet series to be the Dirichlet series attachedto a modular form. Such Dirichlet series have special properties, such as a functionalequation and an Euler product. Sometimes these properties characterize the modularform completely, i.e. they are sufficient to prove the proper transformation behaviourunder some discrete group. The problem dates back to Hecke and Weil, and has morerecently been treated by Conrey et.al. The articles surveyed are: "An extension of Hecke's converse theorem", by B. Conrey and D. Farmer "Converse theorems assuming a partial Euler product", by D. Farmer and K.Wilson "A converse theorem for ¡0(13)", by B. Conrey, D. Farmer, B. Odgers and N.Snaith The results and the proofs are described. The second article is found to contain anerror. Finally an alternative proof strategy is proposed.

This thesis makes a survey of converse theorems for Dirichlet series. A converse theo-rem gives sufficient conditions for a Dirichlet series to be the Dirichlet series attachedto a modular form. Such Dirichlet series have special properties, such as a functionalequation and an Euler product. Sometimes these properties characterize the modularform completely, i.e. they are sufficient to prove the proper transformation behaviourunder some discrete group. The problem dates back to Hecke and Weil, and has morerecently been treated by Conrey et.al. The articles surveyed are: "An extension of Hecke's converse theorem", by B. Conrey and D. Farmer "Converse theorems assuming a partial Euler product", by D. Farmer and K.Wilson "A converse theorem for ¡0(13)", by B. Conrey, D. Farmer, B. Odgers and N.Snaith The results and the proofs are described. The second article is found to contain anerror. Finally an alternative proof strategy is proposed.

In this paper we show how a wide class of overconvergence phenomena can be described in terms of infinite order differential operators, and that we can provide a multi-dimensional analog for such phenomena.

Mellin transforms and Dirichlet series are useful in quantifying periodicity phenomena present in recursive divide-and-conquer. This note illustrates the techniques by providing a precise analysis of the standard top-down recursive mergesort algorithm, in the average case, as well as in the worst and best cases. It also derives the variance and shows that the cost of mergesort has a Gaussian limiting distribution. The approach is applicable to a number of divide-and-conquer recurrences.

unavailable.

In this paper we wish to propose a totally new approach to the theory of Dirichlet series, with the hope that our method will be able to shed new light on some old problems in the area.

For p ≥ 2 we obtain bounds for L p-norms of the Fourier transform of real parts of simple partial fractions. For even p our estimate is sharp. We also prove a new inequality for L p-norms of simple partial fractions which in some cases is stronger than the corresponding inequality obtained by V. Yu. Protasov. © Allerton Press, Inc., 2012.

A necessary and sufficient condition for the series {equation presented}, to converge in Lp(R), p > 1, is obtained. © 2011 RAS(DoM) and LMS.

A necessary and sufficient condition for the series {equation presented}, to converge in Lp(R), p > 1, is obtained. © 2011 RAS(DoM) and LMS.