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The main objective of this paper is to present a new extension of the familiar Mathieu series and the alternating Mathieu series S(r) and $${{\widetilde{S}}}(r)$$ S ~ ( r ) which are denoted by $${\mathbb {S}}_{\mu ,\nu }(r)$$ S μ , ν ( r ) and $$\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)$$ S ~ μ , ν ( r ) , respectively. The computable series expansions of their related integral representations are obtained in terms of the exponential integral $$E_1$$ E 1 , and convergence rate discussion is provided for the associated series expansions. Further, for the series $${\mathbb {S}}_{\mu ,\nu }(r)$$ S μ , ν ( r ) and $$\widetilde{{\mathbb {S}}}_{\mu ,\nu }(r)$$ S ~ μ , ν ( r ) , related expansions are presented in terms of the Riemann Zeta function and the Dirichlet Eta function, also their series built in Gauss’ $${}_2F_1$$ 2 F 1 functions and the associated Legendre function of the second kind $$Q_\mu ^\nu $$ Q μ ν are given. Our discussion also includes the extended versions of the complete Butzer–Flocke–Hauss Omega functions. Finally, functional bounding inequalities are derived for the investigated extensions of Mathieu-type series.

Let λ ( s ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) s , β ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) s , and η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s be the Dirichlet lambda function, its alternating form, and the Dirichlet eta function, respectively. According to a recent historical book by Varadarajan ( [25, p. 70] ), these three functions were investigated by Euler under the notations N ( s ) , L ( s ) , and M ( s ) , respectively. In this paper, we shall present some additional properties for them. That is, we obtain a number of infinite families of linear recurrence relations for λ ( s ) at positive even integer arguments λ ( 2 m ) , convolution identities for special values of λ ( s ) at even arguments and special values of β ( s ) at odd arguments, and a power series expansion for the alternating Hurwitz zeta function J ( s , a ) , which involves a known one for η ( s ) .

We investigate the representations of integrals involving the product of the Legendre-chi function and the tanh−1x or arctanx functions. The investigation will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.

We discuss some local analytic properties of the ring of Dirichlet series. We obtain mainly the equivalence between the irreducibility in the analytic ring and in the formal one. In the same way we prove that the ring of analytic Dirichlet series is integrally closed in the ring of formal Dirichlet series. Finally we introduce the notion of standard basis in these rings and we give a finitely generated ideal which does not admit standard bases.

In this paper we investigate the representation of integrals involving the Legendre Chi function. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods and results of this paper are based on well-known theorems on the number of zeros for complex value functions (Jensen, Titchmarsh, Rouche theorems), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta Function, except leveraging its implied symmetry for non-trivial zeros on the critical strip., Subj-Class: Number Theory; submitted to the Annals of Mathematics for publication

This self-contained book will benefit beginners as well as researchers. It is devoted to Diophantine approximation, the analytic theory of Dirichlet series, and some connections between these two domains, which often occur through the Kronecker approximation theorem. Accordingly, the book is divided into seven chapters, the first three of which present tools from commutative harmonic analysis, including a sharp form of the uncertainty principle, ergodic theory and Diophantine approximation to be used in the sequel. A presentation of continued fraction expansions, including the mixing property of the Gauss map, is given. Chapters four and five present the general theory of Dirichlet series, with classes of examples connected to continued fractions, the famous Bohr point of view, and then the use of random Dirichlet series to produce non-trivial extremal examples, including sharp forms of the Bohnenblust-Hille theorem. Chapter six deals with Hardy-Dirichlet spaces, which are new and useful Banach spaces of analytic functions in a half-plane. Finally, chapter seven presents the Bagchi-Voronin universality theorems, for the zeta function, and r-tuples of L functions. The proofs, which mix hilbertian geometry, complex and harmonic analysis, and ergodic theory, are a very good illustration of the material studied earlier.

Using the Hurwitz zeta and the alternating Hurwitz zeta function, ζ ( s , a ) and ζ ⁎ ( s , a ) , it was shown through classical analysis and in a straightforward and unified manner that a s ζ ( s , a ) with a > 0 and s > 1 is strictly log-convex in s on ( 1 , ∞ ) , whereas a s ζ ⁎ ( s , a ) for a , s > 0 is strictly concave in s on ( 0 , ∞ ) . As an immediate consequence, convexity properties of the Riemann zeta function as well as the Dirichlet beta, eta and lambda function were deduced.

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

In this paper, we study the distribution of zeros of the ordinary Dirichlet polynomials which are generated by an equivalence relation introduced by Harald Bohr. Through the use of completely multiplicative functions, we construct equivalent Dirichlet polynomials which have the same critical strip, where all their zeros are situated, and satisfy the same topological property consisting of possessing zeros arbitrarily near every vertical line contained in some substrips inside their critical strip. We also show that the real projections of the zeros of the partial sums of the alternating zeta function, for some particular cases, are dense in their critical intervals. The second author's research was partially supported by Generalitat Valenciana under project GV/2015/035.

Let be a continuous function which increases to on an infinite interval of the form . A necessary and sufficient condition is found on a sequence increasing to which ensures that for each Dirichlet series of the form , , which is absolutely convergent in the following relation holds: where and are the maximum modulus and maximum term of the series, respectively. Bibliography: 10 titles.

The main purpose of this paper is to present a multi–parameter study of the familiar Mathieu series and the alternating Mathieu series S ( r ) and S ˜ ( r ) . The computable series expansions of the their related integral representations are obtained in terms of higher transcendental hypergeometric functions like Lauricella’s hypergeometric function F C ( m ) [ x ] , Fox–Wright Ψ function, Srivastava–Daoust S generalized Lauricella function, Riemann Zeta and Dirichlet Eta functions, while the extensions concern products of Bessel and modified Bessel functions of the first kind, hyper–Bessel and Bessel–Clifford functions. Auxiliary Laplace–Mellin transforms, bounding inequalities for the hyper–Bessel and Bessel–Clifford functions are established- which are also of independent but considerable interest. A set of bounding inequalities are presented either for the hyper–Bessel and Bessel–Clifford functions which are to our best knowledge new, or also for all considered extended Mathieu–type series. Next, functional bounding inequalities, log–convexity properties and Turan inequality results are presented for the investigated extensions of multi–parameter Mathieu–type series. We end the exposition by a thorough discussion closes the exposition including important details, bridges to occuring new questions like the similar kind multi–parameter treatment of the complete Butzer–Flocke–Hauss Ω function which is intimately connected with the Mathieu series family.

We study the Dirichlet problem of the n-dimensional complex Monge—Ampere equation det(u ij ) = F/|z|2α, where 0 < α < n. This equation comes from La Nave—Tian’s continuity approach to the Analytic Minimal Model Program.

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a new flavour: we give the Dirichlet series representation to solution $f=f(s,w)$ of the functional equation $L(s-wf)=\exp(f)$, where $L(s)$ is the L-function corresponding to a completely multiplicative function. Our result seems to be a Dirichlet series analogue of the well known Lagrange-B\"urmann formula for power series. The proof is probabilistic in nature and is based on Kendall's identity, which arises in the fluctuation theory of L\'evy processes., Comment: 12 pages, 1 figure

We consider a broad family of zeta functions which includes the classical zeta functions of Riemann and Hurwitz, the beta and eta functions of Dirichlet, and the Lerch transcendent, as well as the Arakawa–Kaneko zeta functions and the recently introduced alternating Arakawa–Kaneko zeta functions. We construct their [Formula: see text]-adic analogues and indicate the many strong connections between the complex and [Formula: see text]-adic versions. As applications, we focus on the alternating case and show how certain families of alternating odd harmonic number series can be expressed in terms of Riemann zeta and Dirichlet beta values.

With the help of transformation formulas of Dirichlet L-series, we generalize some classical formulas for the values $$\zeta (2N+1)$$ given by Ramanujan. This will be done by constructing generalized Dirichlet series of the form $$\sum \nolimits _{n=1}^\infty a_n n^{-s/b}$$ where $$b > 0$$ is an integer, which have similar transformation properties as Dirichlet L-functions and by considering their Mellin transforms using contour integration methods.

On the Hilbert space of Dirichlet series with square summable coefficients, there are no non-normal complex symmetric composition operators induced by non-constant symbols. This is in sharp contrast with the phenomenon on the classical Hardy space over the unit disk.

Let π and π′ be irreducible cuspidal automorphic representations of GL m (𝔸ℚ) and GL m′ (𝔸ℚ), respectively. Denote by a π×π′ (n) the nth coefficient of the Dirichlet series of the Rankin–Selberg L-function associated with π and π′. In this paper, we obtain the upper bound for the sum Σ n ≤ x a π × π'(n).

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.

In this paper we consider a class of Dirichlet series with multiplicative coefficients which define functions holomorphic in the right half of the complex plane, and for which there are sequences of Dirichlet polynomials that converge uniformly to these functions in any rectangle within the critical strip. We call such polynomials approximating Dirichlet polynomials. We study the properties of the approximating polynomials, in particular, for those Dirichlet series, whose coefficients are determined by nonprincipal generalized characters, i.e. finite-valued numerical characters which do not vanish on almost all prime numbers and whose summatory function is bounded. These developments are interesting in connection with the problem of the analytical continuation of such Dirichlet series to the entire complex plane, which in turn is tied with the solution of a well-known Chudakov hypothesis about every generalized character being a Dirichlet character.

The fractional derivative of the Dirichlet eta function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function on the critical strip. Its converg ...

Here we obtain the meromorphic continuation of some classical Dirichlet series by means of elementary and simple translation formulae for these series. We are also able to determine the poles and the residues by this method. The motivation to our work originates from an idea of Ramanujan which he used to derive the meromorphic continuation of the Riemann zeta function.

The spaces of entire functions represented by Dirichlet series have been studied by Hussein and Kamthan and others. In the present paper we consider the space $X$ of all entire functions defined by vector-valued Dirichlet series and study the properties of a sequence space which is defined using the type of an entire function represented by vector-valued Dirichlet series. The main result concerns with obtaining the nature of the dual space of this sequence space and coefficient multipliers for some classes of vector-valued Dirichlet series.

We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things, to prove a theorem about general Dirichlet series having as corollary the Speiser's theorem. We have also proved that those functions do not possess multiple zeros of order higher than $2$ and the double zeros have very particular locations. Moreover, their derivatives have only simple zeros. With these results at hand, we revisited GRH for a simplified proof., 22 pages, 9 figures

The article investigates the growth of multiple Dirichlet series. The lower order and the linear order of n-tuple Dirichlet series in ℂn are defined and some relations between them and the coefficients and exponents of n-tuple Dirichlet series are obtained.

Les processus de Dirichlet matriciels, en reference a leur mesure reversible, apparaissent de maniere naturelle dans de nombreux modeles differents en probabilite. En utilisant la langage des operateurs de diffusion et la theorie des equations de bord, nous decrivont les processus de Dirichlet sur le simplexe matriciel et proposont deux modeles pour les processus de Dirichlet matriciels, qui peuvent etre realises par les projections diverses, par le movement brownien sur le groupe unitaire special et par les processus de Wishart.

In this paper we consider the behavior of funcions defined by Dirichlet series with multiplicative coefficients and with bounded summatory function when approaching the imaginary axis. We show that the points of the imaginary axis are also the points of continuity in a broad sense of functions defined by Dirichlet series with multiplicative coefficients which are determined by nonprincipal generalized characters. This result is particularly interesting in its connection with a solution of Chudakov hyphotesis, which states that any finite-valued numerical character, which does not vanish on all prime numbers and has bounded summatory function, is a Dirichlet character. The proof of the main result in this paper is based on the method of reduction to power series, basic principles of which were developed by prof. Kuznetsov in the early 1980s. Ths method establishes a connection between analytical properties of Dirichlet series and boundary properties of the corresponding power series (i.e. a power series with the same coefficients as the Dirichlet series). This allows to obtain new results both for the Dirichlet series and for the power series. In our case this method allowed us to prove the main result using the properties of the power series with multiplicative coefficients determined by the nonprincipal generalized characters, which also were obtained in this work.

We give character analogues of a generalization of a result due to Ramanujan, Hardy and Littlewood, and provide Riesz-type criteria for Riemann Hypotheses for the Riemann zeta function and Dirichlet L -functions. We also provide analogues of the general theta transformation formula and of recent generalizations of the transformation formulas of W.L. Ferrar and G.H. Hardy for real primitive Dirichlet characters.

In this paper, we present some identities involving convolutions of Dirichlet characters and the Mobius function, which are related to a well known identity of Ramanujan, Hardy and Littlewood.