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Sep 10, 2024 · In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x + O(x^\beta)$ for some $\rho>0$ and $\beta<1$.
Sep 10, 2024 · In this paper we obtain a mean value theorem for a general Dirichlet series f (s)= \sum_ {j=1}^\infty a_j n_j^ {-s} with positive coefficients for which the counting function A (x) = \sum_ {n_ {j}\le x}a_ {j} satisfies A (x)=\rho x + O (x^\beta) for some \rho>0 and \beta<1.
- arXiv:2409.06301 [math.NT]
- 30B50, 11M41
- To appear in Quart. J. Math
Sep 3, 2024 · The algebra of Dirichlet series \(\mathcal {A}({{\mathbb {C}}}_+)\) consists on those Dirichlet series convergent in the right half-plane \({{\mathbb {C}}}_+\) and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis.
3 days ago · In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
Sep 4, 2024 · As we progress in the book we will develop a more general theory of Green’s functions for ordinary and partial differential equations. Much of this theory relies on understanding that the Green’s function really is the system response function to a point source.
2 days ago · Glossary. Let Ω be a subset of n -dimensional Euclidean space ℝ n with a smooth boundary ∂Ω; let also L[D] be a elliptic partial differential operator of the second order. There are known two Dirichlet problems: either interior or inner Dirichlet problem or exterior or outer Dirichlet problem.
Sep 10, 2024 · A Mean Value Theorem for general Dirichlet Series. In this paper we obtain a mean value theorem for a general Dirichlet series f (s) = ∑ j = 1 ∞ a j n j − s with positive coefficients for which the counting function A (x) = ∑ n j ≤ x a j satisfies A (x) = ρ x + O (x β) for some ρ> 0 and β <1.
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