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An asymptotic expansion related to the Dickman function

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Abstract

We prove a conjecture of Broadhurst on asymptotic expansions of certain polylogarithm type functions related to the Dickman function.

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References

  1. Balazard, M.: Unimodalité de la distribution du nombre de diviseurs premiers d’un entier. Ann. Inst. Fourier 40(2), 255–270 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Broadhurst, D.: Dickman polylogarithms and their constants. Preprint, arXiv:1004.0519v1

  3. Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable. Oxford University Press, Oxford (1970)

    MATH  Google Scholar 

  4. Dickman, K.: On the frequency of numbers containing prime factors of a certain relative magnitude. Ark. Mat. Astron. Fys. 22, 1–14 (1930)

    Google Scholar 

  5. Hildebrand, A., Tenenbaum, G.: On the number of prime factors of an integer. Duke Math. J. 56, 471–501 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hildebrand, A., Tenenbaum, G.: On a class of difference-differential equations arising in number theory. J. Anal. Math. 61, 145–179 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa Publishing House, New Delhi (1988)

    MATH  Google Scholar 

  8. Sathe, L.G.: On a problem of Hardy on the distribution of integers having a given number of prime factors. III and IV. J. Indian Math. Soc. 18, 27–42 and 43–81 (1954)

    MathSciNet  MATH  Google Scholar 

  9. Selberg, A.: Note on a paper by L.G. Sathe. J. Indian Math. Soc. 18, 83–87 (1954)

    MathSciNet  MATH  Google Scholar 

  10. Smida, H.: Sur les puissances de convolution de la fonction de Dickman. Acta Arith. 59(2), 123–143 (1991)

    MathSciNet  MATH  Google Scholar 

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Correspondence to K. Soundararajan.

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The author was supported in part by NSF grant DMS-0500711.

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Soundararajan, K. An asymptotic expansion related to the Dickman function. Ramanujan J 29, 25–30 (2012). https://doi.org/10.1007/s11139-011-9304-3

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  • DOI: https://doi.org/10.1007/s11139-011-9304-3

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