Some identities involving convolutions of Dirichlet characters and the Möbius function

Roy, Arindam; Zaharescu, Alexandru; Zaki, Mohammad

doi:10.1007/s12044-015-0256-1

Cited by 8 publications

(4 citation statements)

References 11 publications

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“…They arise from the Z * -function and for which the distribution law of zeros is known. Such a case has been treated in [7], [23] and [46]. The authors in the last two papers move the line of integration to the left of the critical strip and apply the functional equation to derive a modular relation with residual function as the infinite sum over non-trivial zeros in the critical strip.…”

Section: Infinitely Many Poles Casementioning

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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Section: Infinitely Many Poles Casementioning

confidence: 99%

Abel-Tauber Process and Asymptotic Formulas

et al. 2017

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“…In [12], Dixit, Roy, and Zaharescu obtained such an analog for Hecke forms. In [24], Roy, Zaharescu, and Zaki obtained a result of the type (1.3) where the Möbius function is replaced by a convolution of Dirichlet characters with the Möbius function. Further results of this kind have also been obtained by Dixit [7,8], Kühn, Robles, and Roy [16], Dixit, Roy, and Zaharescu [13], Agarwal, Garg, and Maji [1], Dixit, Gupta, and Vatwani [9], etc.…”

Section: Introductionmentioning

confidence: 99%

Riesz-type criteria for L-functions in the Selberg class

Gupta¹,

Vatwani²

2023

Can. J. Math.-J. Can. Math.

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We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type ${G^{n , 0}_{0 , n}}$ .

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“…In [12], Dixit, Roy and Zaharescu obtained such an analogue for Hecke forms. In [24], Roy, Zaharescu, and Zaki obtained a result of the type (1.3) where the M öbius function is replaced by a convolution of Dirichlet characters with the M öbius function. Further results this kind have also been obtained by Dixit [8,9], K ühn, Robles and Roy [16], Dixit, Roy and Zaharescu [13], Agarwal, Garg and Maji [1], Dixit, Gupta and Vatwani [10], etc.…”

Section: Introductionmentioning

confidence: 99%

Riesz type criteria for $L$-functions in the Selberg class

Gupta¹,

Vatwani²

2022

Preprint

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We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type G n 0 0 n .

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Some identities involving convolutions of Dirichlet characters and the Möbius function

Abstract: In this paper, we present some identities involving convolutions of Dirichlet characters and the Möbius function, which are related to a well known identity of Ramanujan, Hardy and Littlewood.

Cited by 8 publications

References 11 publications

Abel-Tauber Process and Asymptotic Formulas

Abel-Tauber Process and Asymptotic Formulas

Riesz-type criteria for L-functions in the Selberg class

Riesz type criteria for $L$-functions in the Selberg class

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