Integrals of certain Dirichlet series
نویسندگان
چکیده
منابع مشابه
A Class of Dirichlet Series Integrals
In this note we extend the solution to a recent Monthly problem to analyze a broad class of Dirichlet series and illustrate the result in action in various ways. More precisely, in [6] the following integral evaluation is obtained: ∞ 0 3 − 2 √ 2 cos (t log 2) |ζ (1/2 + it)| 2 t 2 + 1/4 dt = π log 2. (1) This somewhat recondite-looking result transpires to be a case of a rather pretty class of e...
متن کاملThe Critical Values of Certain Dirichlet Series
We investigate the values of several types of Dirichlet series D(s) for certain integer values of s, and give explicit formulas for the value D(s) in many cases. The easiest types of D are Dirichlet L-functions and their variations; a somewhat more complex case involves elliptic functions. There is one new type that includes ∑∞ n=1(n +1) for which such values have not been studied previously. 2...
متن کاملSpecial values of trigonometric Dirichlet series and Eichler integrals
We provide a general theorem for evaluating trigonometric Dirichlet series of the form ∑ n>1 f(πnτ) ns , where f is an arbitrary product of the elementary trigonometric functions, τ a real quadratic irrationality and s an integer of the appropriate parity. This unifies a number of evaluations considered by many authors, including Lerch, Ramanujan and Berndt. Our approach is based on relating th...
متن کاملExact Estimates for Integrals Involving Dirichlet Series with Nonnegative Coefficients
We consider the Dirichlet series ∞ ∑ k=2 akk −1−x =: f(x), x > 0, with coefficients ak ≥ 0 for all k. Among others, we prove exact estimates of certain weighted Lp-norms of f on the unit interval (0, 1) for any 0 < p <∞, in terms of the coefficients ak . Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for int...
متن کاملAnalytic properties of a certain multiple Dirichlet series
We consider a certain multiple Dirichlet series which is a generalization of that introduced in Masri, and we prove the meromorphic continuation to the whole space. Also, using certain functional relations and the technique of chaging variables introduced in Akiyama, Egami and Tanigawa, we prove that " the possible singularities " is indeed " the true singularities " .
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Revista Colombiana de Matemáticas
سال: 2020
ISSN: 2357-4100,0034-7426
DOI: 10.15446/recolma.v54n1.89775