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Complex Powers of L-functions and Integers Without Small Prime Factors

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Abstract

The Selberg type divisor problem (Selberg in J Indian Math Soc 18:83–87, 1954, Serre in A course in arithmetic, Springer, Berlin–Heidelberg, 1973), pertains to the study on the coefficients of the complex power of the zeta-function as has been exhibited in Banerjee et al. (Kyushu J Math 71:363–385). The original objective of Selberg was to apply the results to the problem related to the prime number theorem. However, the complex powers turn out to be of independent interest and have applications in studying the mean values of the zeta and other L-functions. We consider the zeta-function (á la Hecke) associated to the normalised cusp forms for the full modular group in the quest of continuing the research started in Laurinčikas and Steuding (Cent Eur Sci J Math 2:1–18, 2004) on the summatory function of the complex power in the form of the Dirichlet convolution of the coefficients of such zeta-functions. The convolution includes the one with the identity and we may cover the results in Laurinčikas and Steuding (Cent Eur Sci J Math 2:1–18, 2004). We treat the general case of the Selberg class zeta-functions and the modular form case is an example. It is interesting to note that one of the main roles is played by the integers without large and small prime factors which have been studied extensively. Our approach has a merit of attaining both, the result on the summatory function as well as a result entailing the classical result on distrobution of integers without small prime factors together.

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Acknowledgements

The authors would like to thank the referee for his/her careful checking of the MS and many enlightening suggestions, which improved the presentation and visibility of the paper. The earlier part of the paper was completed while the second author was staying with Professor \(\mathrm{H}^2\) Chans’. He is deeply indebted to the hospitality of the whole family and the consoling scenery through the window. Without occasional cessation of the work by throwing a glance at the jungle, the work could not have been done with this much comfort.

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Authors and Affiliations

  1. Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad, 211019, India

    K. Chakraborty

  2. Kerala School of Mathematics, Kunnamangalam, Kozhikode, Kerala, 673571, India

    K. Chakraborty

  3. Faculty of Engineering, Kyushu Inst. Techn, Sensuicho 1-1, Tobata, Kitakyushu, 804-8555, Japan

    S. Kanemitsu

  4. Department of Mathematics, Vilnius University Naugarduko, 24 2006, Vilnius, Lithuania

    A. Laurinčikas

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  1. K. Chakraborty
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  2. S. Kanemitsu
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  3. A. Laurinčikas
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Correspondence to K. Chakraborty.

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To Professor Ramachandran Balasubramanian on the occasion of his 70th birthday, with friendship and respect.

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Chakraborty, K., Kanemitsu, S. & Laurinčikas, A. Complex Powers of L-functions and Integers Without Small Prime Factors. Mediterr. J. Math. 19, 167 (2022). https://doi.org/10.1007/s00009-022-02037-y

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  • DOI: https://doi.org/10.1007/s00009-022-02037-y

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