Examples of beurling prime systems
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Faez A Al-Maamori
Abstract
A generalized prime system 𝓟 is a sequence of positive reals p1, p2, p3, … satisfying 1 < p1 ≤ p2 ≤ ⋯ ≤ pn ≤ ⋯ and for which pn → ∞ as n → ∞. The {pn} are called generalized primes (or Beurling primes) with the products
In this article we generalise Balanzario’s result [BALANZARIO, E.: An example in Beurling’s theory of primes, Acta Arith. 87 (1998), 121–139] by adapting his method to show that for any 0 < α < 1 there is a continuous g-prime system for which
and
We use the method developed by Diamond, Montgomery and Vorhauer [DIAMOND, H.—MONTGOMERY, H.—VORHAUER, U.: Beurling primes with large oscillation, Math. Ann. 334 (2006), 1–36] and Zhang [ZHANG, W.: Beurling primes with RH and Beurling primes with large oscillation, Math. Ann. 337 (2007), 671–704] to prove (by using some measure theoretical results) that there is a discrete system of Beurling primes satisfying (0.1) and (0.2) which is similar to the continuous system.
Finding discrete example is typically more challenging since one cannot control the various growth rates (of π𝓟(x), 𝓝𝓟(x) and ζ𝓟(s)) so easily.
References
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© 2017 Mathematical Institute Slovak Academy of Sciences
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Artikel in diesem Heft
- 10.1515/ms-2015-0200
- Zero-divisor graphs of lower dismantlable lattices I
- Some results on the intersection graph of submodules of a module
- Class number parities of compositum of quadratic function fields
- Examples of beurling prime systems
-
Connection between multiplication theorem for Bernoulli polynomials and first factor
- On permutational invariance of the metric discrepancy results
- Evaluation of sums containing triple aerated generalized Fibonomial coefficients
- Linear algebraic proof of Wigner theorem and its consequences
- A note on groups with finite conjugacy classes of subnormal subgroups
- Groups with the same complex group algebras as some extensions of psl(2, pn)
- Klee-Phelps convex groupoids
- On analytic functions with generalized bounded Mocanu variation in conic domain with complex order
- Weak interpolation for the lipschitz class
- Generalized Padé approximants for plane condenser and distribution of points
- Three-variable symmetric and antisymmetric exponential functions and orthogonal polynomials
- Positive solutions of nonlocal integral BVPS for the nonlinear coupled system involving high-order fractional differential
- Existence of positive solutions for a nonlinear nth-order m-point p-Laplacian impulsive boundary value problem
- Dirichlet boundary value problem for differential equation with ϕ-Laplacian and state-dependent impulses
- On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term
- Homoclinic solutions for ordinary (q, p)-Laplacian systems with a coercive potential
- Semi-equivelar maps on the torus and the Klein bottle with few vertices
- A problem considered by Friedlander & Iwaniec and the discrete Hardy-Littlewood method
