Dirichlet's series associated with some power series

There are presented sufficient conditions for the entire Dirichlet series with monotonically increasing sequence of exponents providing validity of Borel-type relation outside some set $E$ of finite $h$-measure, i.e. $m_h E=\int_E dh(x)<+\infty$ with a positive continuously differentiable on $[0,+\infty)$ function $h,$ whose derivative $h'$ increases to infinity.The corresponding Borel-type relation states that the logarithm of supremum of the Dirichlet series along an imaginary line behaves like as logarithm of maximal term of the Dirichlet series.The conditions are given as the convergence of some auxiliary series constructed from the values of the function $h'$ and the sequence of exponents.

Abstract This study examines the time-dependent behavior of reinforced and prestressed concrete structures using rate-type formulations. The integral-type approach requires storing the full load history and reprocessing all past stress increments at each time step, resulting in high computational cost. This cost can be substantially reduced by adopting a rate-type formulation, which approximates the creep function with a Dirichlet series derived from a Kelvin chain approximation. These series are capable of accurately reproducing creep functions obtained from laboratory tests or prescribed by design codes. The stress–strain rate-type law for aging concrete is expressed by a second-order differential equation, but a single rate-type incremental equation is derived from a Kelvin chain approximation, considering the superposition principle. The paper details the numerical procedure used to integrate the rate-type equations over time, using a fixed time discretization. A relaxation model for prestressing steel is also adopted, accounting for the influence of varying strain histories on stress relaxation—an aspect often neglected in traditional approaches. The proposed formulation improves numerical efficiency while maintaining accuracy in long-term structural analysis, as verified through a representative example.

Abstract Serre famously showed that almost all plane conics over have no rational point. We investigate versions of this over global function fields, focusing on a specific family of conics over which illustrates new behavior. We obtain an asymptotic formula using harmonic analysis, which requires a Tauberian theorem over function fields for Dirichlet series with branch point singularities.

Abstract In this paper, we prove a lower bound result for extremely large values of $L(\frac{1}{2},\chi _p)$ with prime numbers $p\equiv 1\pmod 8$, based on generalizing the twisted first moment result of Baluyot–Pratt to the case of short Dirichlet series with large coefficients.

An asymptotic formula is derived for the summatory function <img src=image/13443653_02.gif>, where <img src=image/13443653_03.gif> denotes the total number of prime factors of <img src=image/13443653_16.gif> counted with multiplicity, and <img src=image/13443653_04.gif> is a multiplicative arithmetical function satisfying <img src=image/13443653_05.gif> for primes <img src=image/13443653_06.gif> and non-negative integers <img src=image/13443653_07.gif>, where <img src=image/13443653_08.gif> and <img src=image/13443653_01.gif> for <img src=image/13443653_09.gif>. The study builds on a rich history in analytic number theory, including classical results by Dirichlet on the divisor function <img src=image/13443653_10.gif>, and refinements using zeta-function estimates, as well as probabilistic approaches like the Erdős–Kac theorem extended to <img src=image/13443653_03.gif> (distinct primes) over <img src=image/13443653_11.gif>-free and <img src=image/13443653_11.gif>-full numbers. However, prior research has largely overlooked the multiplicity in <img src=image/13443653_03.gif> and its twisting by broad classes of multiplicative functions beyond divisors, particularly for square-full integers. The analysis covers three distinct cases: when <img src=image/13443653_04.gif> belongs to the subclass <img src=image/13443653_12.gif> (where <img src=image/13443653_13.gif> for all primes <img src=image/13443653_06.gif>); when <img src=image/13443653_04.gif> is in the broader class <img src=image/13443653_15.gif> but not in <img src=image/13443653_12.gif>; and when <img src=image/13443653_16.gif> is square-full with <img src=image/13443653_14.gif>. Examples of such functions include the number of non-isomorphic Abelian groups of order <img src=image/13443653_16.gif>, the number of square-full divisors of <img src=image/13443653_16.gif>, the divisor function <img src=image/13443653_17.gif>, and the <img src=image/13443653_07.gif>-fold divisor function <img src=image/13443653_18.gif>. The results are obtained using Dirichlet series <img src=image/13443653_19.gif>, which admit an Euler product decomposition due to multiplicativity, enabling analytic continuation via differentiation with respect to an auxiliary parameter <img src=image/13443653_20.gif>, contour integration, and estimates for the Riemann zeta function, as well as analytic continuation techniques. The following results were obtained: for case (i), <img src=image/13443653_21.gif>; for case (ii), <img src=image/13443653_22.gif>; and for square-full <img src=image/13443653_16.gif>, <img src=image/13443653_23.gif>. The work is theoretical in nature. The results of this study can be applied in further research in number theory, group theory, and discrete mathematics, with potential applications in algorithmic number theory (e.g., efficient computation of group orders) and cryptographic protocols relying on prime factorizations.

The periodic zeta-function $\zeta(s; a)$, $s = \sigma + it$, $a = \{a_m \in \mathbb{C} : m \in \mathbb{N}\}$, in the half-plane $\sigma &gt; 1$ is defined by Dirichlet series with periodic coefficients $a_m$, and has the meromorphic continuation to the whole complex plane. The function $\zeta(s; a)$ is a generalization of the Riemann zeta-function and Dirichlet $L$-functions. In the paper, using only the periodicity of the sequence $a$, we obtain that the shifts $\zeta(s + i\tau; a)$, $\tau \in \mathbb{R}$, approximate a certain class of analytic functions, defined in the strip $\{s \in \mathbb{C} : 1/2 &lt; \sigma &lt; 1\}$. For $T^{23/70} \leqslant H \leqslant T^{1/2}$, the set of such shifts has a positive lower density in the interval $[T, T + H]$, $T \to \infty$. The case of positive density is also discussed. For the proof, the mean square estimate in short intervals for the Hurwitz zeta-function, and probabilistic limit theorems are applied.

Abstract Let $$\mathscr {H}^\infty $$ H ∞ be the set of all Dirichlet series $$\textstyle f\!=\!{{\sum \limits _{n=1}^\infty }} a_nn^{-s}$$ f = ∑ n = 1 ∞ a n n - s (where $$a_n\!\in \mathbb {C}$$ a n ∈ C for all $$n\!\in \! \mathbb {N}\!=\!\{1,2,3,\cdots \}$$ n ∈ N = { 1 , 2 , 3 , ⋯ } ) that converge at each s in the half-plane $$\mathbb {C}_0\!:=\!\{s\!\in \! \mathbb {C}\!:\! \text {Re}(s)\!&gt;\!0\}$$ C 0 : = { s ∈ C : Re ( s ) &gt; 0 } , such that $$\Vert f\Vert _{\infty }\!=\!\sup _{s\in \mathbb {C}_0}\!|f(s)|\!&lt;\!\infty $$ ‖ f ‖ ∞ = sup s ∈ C 0 | f ( s ) | &lt; ∞ . Then $$\mathscr {H}^\infty $$ H ∞ is a Banach algebra with pointwise operations and the supremum norm $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ , and has been studied in earlier works. The article introduces a new family of Banach subalgebras $$\mathscr {H}^\infty _{S}$$ H S ∞ of $$\mathscr {H}^\infty $$ H ∞ . For $$S\!\subset \! \mathbb {N}$$ S ⊂ N , let $$\mathscr {H}^\infty _{S}$$ H S ∞ be the set of all elements $$\textstyle {{\sum \limits _{n=1}^\infty }} a_nn^{-s}\in \mathscr {H}^\infty $$ ∑ n = 1 ∞ a n n - s ∈ H ∞ such that for all $$n\in \mathbb {N}\setminus S$$ n ∈ N \ S , $$a_n\!=\!0$$ a n = 0 . Then $$\mathscr {H}^\infty _{S}$$ H S ∞ is a unital Banach subalgebra of $$\mathscr {H}^\infty $$ H ∞ with the $$\Vert \cdot \Vert _\infty $$ ‖ · ‖ ∞ norm if and only if S is a multiplicative subsemigroup of $$\mathbb {N}$$ N containing 1. It is shown that for such S , $$\mathscr {H}^\infty _{S}$$ H S ∞ is the multiplier algebra of $$\mathscr {H}^2_S$$ H S 2 , where $$\mathscr {H}^2_S$$ H S 2 is the Hilbert space of all $$ \textstyle f\!=\!{{\sum \limits _{n\in S}}} a_nn^{-s}$$ f = ∑ n ∈ S a n n - s such that $$\Vert f\Vert _2\!:=\!({{\sum \limits _{n\in S}}} |a_n|^2)^{\frac{1}{2}}\!&lt;\!\infty $$ ‖ f ‖ 2 : = ( ∑ n ∈ S | a n | 2 ) 1 2 &lt; ∞ . A characterisation of the group of units in $$\mathscr {H}^\infty _{S}$$ H S ∞ is given, by showing an analogue of the Wiener 1/ f theorem for $$\mathscr {H}^\infty _{S}$$ H S ∞ . If S has an infinite set of generators allowing a unique representation of each element of S , then it is shown that the Bass stable rank of $$\mathscr {H}^\infty _S$$ H S ∞ is infinite.

Let L ( s , f j ) with j ∈ { 1 , 2 } be a non-constant meromorphic function of finite order having a convergent Dirichlet series representation of the form L ( s , f j ) = ∑ n ≥ 1 f j ( n ) n s with j ∈ { 1 , 2 } in some right half-plane, where f j : Z → C is a q-periodic arithmetical function, and let a and b be two distinct finite complex values. If L ( s , f 1 ) and L ( s , f 2 ) share a CM and b IM, then L ( s , f 1 ) = L ( s , f 2 ) identically in C . Moreover, when L ( s , f j ) , ( j = 1 , 2 ) , admits an analytic continuation as a meromorphic function of finite order, we show some relations between L ( s , f 1 ) , L ( s , f 2 ) if they share the set { a , b } .

Abstract In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full‐rank sublattices of a given quadratic lattice in a hyperbolic plane—that is, a nondegenerate isotropic quadratic space of dimension 2. We derive explicit formulas for the associated zeta functions and obtain a combinatorial way to compute them. Their analytic properties lead to the intriguing consequence that a large proportion of proper classes are one‐lattice classes.

On the values of the Dirichlet series associated with certain hyperharmonic numbers

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest H1,11,1↔H0,22,0, with H denoting the Fox H-function.

Topological structure of the space of composition operators on the Hardy space of Dirichlet series

Let [Formula: see text] be the error term of the sum of [Formula: see text] for [Formula: see text], where [Formula: see text]. In this paper we study the behavior of [Formula: see text] in the case [Formula: see text] by applying analytic properties of the Dirichlet series [Formula: see text]. Especially, we shall give an explicit upper bound for the error term of [Formula: see text] by means of exponent pairs.

Functional Large Deviation Principle and Functional Laws of Iterated Logarithm for Random Dirichlet Series

Let P,Q:Rn→R be two polynomials. This paper studies the existence of the following Łojasiewicz inequality at infinity: |Q(x)|θ≥c|P(x)| for ∥x∥≫1, where c and θ are positive constants. We provide a condition under which the Łojasiewicz inequality holds, and the exponent is computed explicitly in terms of the Newton polyhedra of the two polynomials. On the way, we give some criteria for the convergence of some integrals of rational functions, and describe the domain of convergence of multidimensional Dirichlet series associated with polynomials in terms of Newton polyhedra of polynomials defining the series.

We introduce area operators Aμ,l in the Dirichlet series setting for l&gt;0 and positive Borel measures μ on the right half-plane C0. It is proved that if μ is a Carleson measure on C0, then for 0&lt;p&lt;∞, the area operator Aμ,l is bounded from the Hardy space Hp0 of Dirichlet series vanishing at +∞ to some Lp-space. We also give an application of our methods to Volterra operators.

Identities for the product of two Dirichlet series satisfying Hecke's functional equation

abstract: In this paper and its sequel (in preparation), we investigate the precise relationship between the quadratic affine Weyl group multiple Dirichlet series in the sense of the works by Chinta and Gunnells (2007) and Bucur and Diaconu (2010), and those defined axiomatically by Whitehead. In particular, we show that the axiomatic quadratic Weyl group multiple Dirichlet series of type $\smash{D_4^{(1)}}$ over rational function fields of odd characteristic admits meromorphic continuation to the interior of the corresponding complexified Tits cone. We shall also determine the polar divisor of this function, and compute the residue at each of its poles. As a consequence, we obtain an \textit{exact} formula for a weighted 4-th moment of quadratic Dirichlet $L$-functions over rational function fields; we shall also derive an asymptotic formula for this weighted moment that is expected to generalize to any global field.

Abstract This paper is concerned with random holomorphic functions over the infinite‐dimensional polydisc. The celebrated Billard's theorem states, among other things, that for a random holomorphic function in a single variable—obtained by placing a random sign before each Taylor coefficient—the boundedness of the function can be automatically improved to continuity. We extend Billard's theorem in three aspects: first, we extend it to the case of infinitely many variables. Second, we establish an inverse result, that is, not merely a sufficient condition. Such results are rare in the literature. In particular, we define Billard's property (BP) for random variables and prove: (1) all symmetric random variables have BP; (2) a square‐integrable random variable has BP if and only if it is centered. Third, another novelty of our results is that they encompass applications to random Dirichlet series.