Sur des formules de Atle Selberg

Delange, Hubert

doi:10.4064/aa-19-2-105-146

Cited by 62 publications

(59 citation statements)

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“…Our results are unlfo in k not exceeding a fed power of log x, and in j H. Delange [2] obtained comparable results for fed k.…”

supporting

confidence: 73%

Extensions of some formulae of A. Selberg

Spiro

1985

International Journal of Mathematics and Mathematical Sciences

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ABSTRACT. This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.

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“…Our results are unlfo in k not exceeding a fed power of log x, and in j H. Delange [2] obtained comparable results for fed k.…”

supporting

confidence: 73%

Extensions of some formulae of A. Selberg

Spiro

1985

International Journal of Mathematics and Mathematical Sciences

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“…A great number of papers have been published on the Selberg type divisor problem including [17,18,21,24,40,41,42,47,50,51,52], etc. and references therein.…”

Section: The Rieger-nowak-nakaya Resultsmentioning

confidence: 99%

“…Another example is the generating function considered by Delange [18]. Let f (n) be an additive function which takes only positive integer values and χ(n) is a bounded multiplicative function satisfying…”

Section: Other Complex Powers Of L-functionsmentioning

confidence: 99%

Abel-Tauber Process and Asymptotic Formulas

et al. 2017

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Abstract. The Abel-Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration-differentiation process. In this article, we use the Abel-Tauber process to establish an interesting asymptotic expansion for the Riesz sums of arithmetic functions with best possible error estimate. The novelty of our paper is that we incorporate the Selberg type divisor problem in this process by viewing the contour integral as part of the residual function. The novelty also lies in the uniformity of the error term in the additional parameter which varies according to the cases. Generalization of the famous Selberg Divisor problem to arithmetic progression has been made by Rieger [163][164][165][166][167][168][169][170][171][172][173] studied the related subject of reciprocals of an arithmetic function and obtained an asymptotic formula with the Vinogradov-Korobov error estimate with the main term as a finite sum of logarithmic terms. We shall also elucidate the situation surrounding these researches and illustrate our results by rich examples.

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“…H. Delange [2] gives an asymptotic expansion of F k , ε {x) in the form Proof. The proof goes on the similar lines as those of H. Delange [2], using the zero free region due to Vinogradov-Korobov.…”

Section: Conversely If R K>ε (X) < Xmentioning

confidence: 99%

“…3. R. Balasubramanian and K. Ramachandra [1] have observed an asymptotic formula for Σnd(n)< x 1. by a method similar to that of Selberg [8] or H. Delange [2]. THEOREM …”

Section: Introductionmentioning

confidence: 99%