대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제60권6호
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  • Pages.1651-1672
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  • 2023
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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RESULTS ON THE ALGEBRAIC DIFFERENTIAL INDEPENDENCE OF THE RIEMANN ZETA FUNCTION AND THE EULER GAMMA FUNCTION

  • Xiao-Min Li (Department of Mathematics Ocean University of China) ;
  • Yi-Xuan Li (Department of Mathematics Ocean University of China)
  • 투고 : 2022.11.29
  • 심사 : 2023.02.17
  • 발행 : 2023.11.30
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초록

In 2010, Li-Ye [13, Theorem 0.1] proved that P(ζ(z), ζ'(z), . . . , ζ(m)(z), Γ(z), Γ'(z), Γ"(z)) ≢ 0 in ℂ, where m is a non-negative integer, and P(u0, u1, . . . , um, v0, v1, v2) is any non-trivial polynomial in its arguments with coefficients in the field ℂ. Later on, Li-Ye [15, Theorem 1] proved that P(z, Γ(z), Γ'(z), . . . , Γ(n)(z), ζ(z)) ≢ 0 in z ∈ ℂ for any non-trivial distinguished polynomial P(z, u0, u1, . . ., un, v) with coefficients in a set Lδ of the zero function and a class of nonzero functions f from ℂ to ℂ ∪ {∞} (cf. [15, Definition 1]). In this paper, we prove that P(z, ζ(z), ζ'(z), . . . , ζ(m)(z), Γ(z), Γ'(z), . . . , Γ(n)(z)) ≢ 0 in z ∈ ℂ, where m and n are two non-negative integers, and P(z, u0, u1, . . . , um, v0, v1, . . . , vn) is any non-trivial polynomial in the m + n + 2 variables u0, u1, . . . , um, v0, v1, . . . , vn with coefficients being meromorphic functions of order less than one, and the polynomial P(z, u0, u1, . . . , um, v0, v1, . . . , vn) is a distinguished polynomial in the n + 1 variables v0, v1, . . . , vn. The question studied in this paper is concerning the conjecture of Markus from [16]. The main results obtained in this paper also extend the corresponding results from Li-Ye [12] and improve the corresponding results from Chen-Wang [5] and Wang-Li-Liu-Li [23], respectively.

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과제정보

This work is supported in part by the NSFC (No. 11171184), the NSF of Shandong Province, China (No. ZR2019MA029) and the FRFCU (No. 3016000841964007).

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