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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 1, Pages 39–50 (Mi izv1139)  

This article is cited in 4 scientific papers (total in 4 papers)

Waring's problem with the Ramanujan $\tau$-function

M. Z. Garaeva, V. C. Garciaa, S. V. Konyaginb

a National Autonomous University of Mexico, Institute of Mathematics
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We prove that for every integer $N$ the Diophantine equation $\sum_{i=1}^{74000}\tau(n_i)=N$, where $\tau(n)$ is the Ramanujan $\tau$-function, has a solution in positive integers $n_1, n_2,…, n_{74000}$ satisfying the condition $\max_{1\le i\le 74000}n_i {\ll}|N|^{2/11}+1$. We also consider similar questions in residue fields modulo a large prime $p$.

DOI: https://doi.org/10.4213/im1139

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English version:
Izvestiya: Mathematics, 2008, 72:1, 35–46

Bibliographic databases:

UDC: 511.34
MSC: 11B83, 11B50, 11P32
Received: 12.07.2006

Citation: M. Z. Garaev, V. C. Garcia, S. V. Konyagin, “Waring's problem with the Ramanujan $\tau$-function”, Izv. RAN. Ser. Mat., 72:1 (2008), 39–50; Izv. Math., 72:1 (2008), 35–46

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Garaev M.Z., Garcia V.C., Konyagin S.V., “The Waring problem with the Ramanujan $\tau$-function. II”, Canad. Math. Bull., 52:2 (2009), 195–199  crossref  mathscinet  zmath  isi  elib  scopus
    2. P. V. Snurnitsyn, “On the Basic Properties of the Ramanujan $\tau$-Function”, Math. Notes, 90:5 (2011), 723–729  mathnet  crossref  crossref  mathscinet  isi
    3. Snurnitsyn P.V., “O predstavimosti tselykh chisel znacheniyami funktsii Ramanudzhana”, Vestn. Mosk. un-ta. Ser. 1. Matem. Mekh., 2011, no. 6, 49–52  mathscinet  zmath  elib
    4. P. V. Snurnitsyn, “Additivnaya zadacha s funktsiei Ramanudzhana”, Chebyshevskii sb., 12:4 (2011), 112–128  mathnet  mathscinet
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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