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HYPERTRANSCENDENCE OF $L$ -FUNCTIONS FOR $\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$

  • HIROFUMI NAGOSHI (a1)
Abstract

We generalise a result of Hilbert which asserts that the Riemann zeta-function ${\it\zeta}(s)$ is hypertranscendental over $\mathbb{C}(s)$ . Let ${\it\pi}$ be any irreducible cuspidal automorphic representation of $\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We establish a certain type of functional difference–differential independence for the associated $L$ -function $L(s,{\it\pi})$ . This result implies algebraic difference–differential independence of $L(s,{\it\pi})$ over $\mathbb{C}(s)$ (and more strongly, over a certain field ${\mathcal{F}}_{s}$ which contains $\mathbb{C}(s)$ ). In particular, $L(s,{\it\pi})$ is hypertranscendental over $\mathbb{C}(s)$ . We also extend a result of Ostrowski on the hypertranscendence of ordinary Dirichlet series.

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[1]Bank, S. and Kaufman, R., ‘An extension of Hölder’s theorem concerning the Gamma function’, Funkcial. Ekvac. Ser. Int. 19 (1976), 5363.
[2]Bank, S. and Kaufman, R., ‘A note on Hölder’s theorem concerning the Gamma function’, Math. Ann. 232 (1978), 115120.
[3]Chiang, Y. and Feng, S., ‘Difference independence of the Riemann zeta function’, Acta Arith. 125 (2006), 317329.
[4]Godement, R. and Jacquet, H., Zeta Functions of Simple Algebras, Lecture Notes in Mathematics, 260 (Springer, Berlin, 1972).
[5]Goldberg, A. A. and Ostrovskii, I. V., Value Distribution of Meromorphic Functions, Translations of Mathematical Monographs, 236 (American Mathematical Society, Providence, RI, 2008).
[6]Gunning, R. and Rossi, H., Analytic Functions of Several Complex Variables (AMS Chelsea, Providence, RI, 2009), reprint of the 1965 original.
[7]Hilbert, D., ‘Mathematical problems’, Bull. Amer. Math. Soc. 8 (1902), 437479 (reprinted); Bull. Amer. Math. Soc. (N.S.) 37(4) (2000), 407–436.
[8]Hölder, O., ‘Ueber die Eigenschaft der Gammafunction keiner algebraischen Differentialgleichung zu genügen’, Math. Ann. 28 (1886), 113.
[9]Jacquet, H. and Shalika, J. A., ‘On Euler products and the classification of automorphic representations I’, Amer. J. Math. 103 (1981), 499558.
[10]Karatsuba, A. A. and Voronin, S. M., The Riemann Zeta-Function (Walter de Gruyter, Berlin, 1992).
[11]Laine, I., Nevanlinna Theory and Complex Differential Equations, de Gruyter Studies in Mathematics, 15 (Walter de Gruyter, Berlin, 1993).
[12]Liao, L. and Yang, C., ‘On some new properties of the gamma function and the Riemann zeta function’, Math. Nachr. 257 (2003), 5966.
[13]Liu, J., Wang, Y. and Ye, Y., ‘A proof of Selberg’s orthogonality for automorphic L-functions’, Manuscripta Math. 118 (2005), 135149.
[14]Mijajlović, Ž. and Malešević, B., ‘Differentially transcendental functions’, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), 193201.
[15]Nagoshi, H., ‘On the universality for L-functions attached to Maass forms’, Analysis 25 (2005), 122.
[16]Nagoshi, H., ‘The universality of L-functions attached to Maass forms’, Adv. Stud. Pure Math. 49 (2007), 289306.
[17]Ostrowski, A., ‘Über Dirichletsche Reihen und algebraische Differentialgleichungen’, Math. Z. 8 (1920), 241298.
[18]Reich, A., ‘Zetafunktionen und Differenzen-Differentialgleichungen’, Arch. Math. 38 (1982), 226235.
[19]Reich, A., ‘On hypertranscendental functions’, in: Topics in Classical Number Theory (Colloquium Budapest, 1981), Vol. II, Colloquia Mathematica Societatis János Bolyai, 34 (1984), 13491369.
[20]Reich, A., ‘Über Dirichletsche Reihen und holomorphe Differentialgleichungen’, Analysis 4 (1984), 2744.
[21]Rudnick, Z. and Sarnak, P., ‘Zeros of principal L-functions and random matrix theory’, Duke Math. J. 81 (1996), 269322.
[22]Stadigh, V. E. E., ‘Ein Satz Ueber Funktionen, Die Algebraische Differentialgleichungen Befriedigen, Und Ueber Die Eigenschaft Der Function ${\it\zeta}(s)$ Keiner Solchen Gleichung Zu Genügen’; Dissertation, Helsinki, 1902. http://daten.digitale-sammlungen.de/0006/bsb00067906/images.
[23]Steuding, J., Value-Distribution of L-Functions, Lecture Notes in Mathematics, 1877 (Springer, Berlin, 2007).
[24]Voronin, S. M., ‘On the distribution of nonzero values of the Riemann zeta-function’, Proc. Steklov Inst. Math. 128 (1972), 153175.
[25]Voronin, S. M., ‘On differential independence of 𝜁-functions’, Soviet Math. Dokl. 14 (1973), 607609.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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