Browsing by Author "Sahoo, Jagannath"

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series [Formula presented]. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.

The authors would like to inform the reader that the statement of Theorem 1.1 in [1] should be modified as follows. In addition to the assumption of absolute convergence for [Formula presented]. Moreover, the conclusion of Theorem 1.1 should be modified to the following: [Formula presented] such as the one imposed by Tenenbaum [2] in his discussion of the Selberg–Delange method. The most expedient way of arriving at the asymptotic formula (without the error term) in Theorem 1.1 is to first observe that [Formula presented] respectively. Using the elementary result [Formula presented] of the above theorem, which of course is the content of the third author's paper [3]. We thank Frederik Broucke for pointing out these lacunae in our paper.

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for can be obtained with 'nice' test functions f(n), provided a(n) is an 'arithmetic function'. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

Alladi's duality identities (1977) provide a fundamental relation between the smallest and the -th largest prime factors of integers. In this paper, we establish these dualities in the setting of global function fields, extending a result of Duan, Wang, and Yi (2021) to higher orders. We apply this to study a function field analogue of the sum , when restricted to integers whose smallest prime factor lies in an arbitrary subset of primes possessing a natural density. These results demonstrate how the second-order duality identity governs the asymptotic behaviour of these weighted Möbius sums in the function field setting.