Transcendental values of the digamma function. (English) Zbl 1222.11097
Summary: Let \(\psi(x)\) denote the digamma function, that is, the logarithmic derivative of Euler’s \(\Gamma\)-function. Let \(q\) be a positive integer greater than 1 and \(\gamma\) denote Euler’s constant. We show that all the numbers
\[
\psi(a/q)+\gamma,\quad (a,q)=1,\quad 1\leq a\leq q,
\]
are transcendental. We also prove that at most one of the numbers
\[
\gamma,\quad\psi(a/q),\quad (a,q)=1,\quad 1\leq a\leq q,
\]
is algebraic.
MSC:
| 11J91 | Transcendence theory of other special functions |
| 11J86 | Linear forms in logarithms; Baker’s method |
| 11J81 | Transcendence (general theory) |
| 33B15 | Gamma, beta and polygamma functions |
| 11M35 | Hurwitz and Lerch zeta functions |
References:
| [1] | Adhikari, S.; Saradha, N.; Shorey, T. N.; Tijdeman, R., Transcendental infinite sums, Indag. Math. (N.S.), 12, 1, 1-14 (2001) · Zbl 0991.11043 |
| [2] | Apostol, T. M., Introduction to Analytic Number Theory, Undergrad. Texts Math. (1976), Springer: Springer New York · Zbl 0335.10001 |
| [3] | Artin, E., Galois Theory, Notre Dame Mathematical Lectures Number, vol. 2 (1970) |
| [4] | Baker, A., Transcendental Number Theory (1975), Cambridge Univ. Press · Zbl 0297.10013 |
| [5] | Baker, A.; Birch, B.; Wirsing, E., On a problem of Chowla, J. Number Theory, 5, 224-236 (1973) · Zbl 0267.10065 |
| [6] | Chowla, S., The non-existence of non-trivial relations between the roots of a certain irreducible equation, J. Number Theory, 2, 120-123 (1970) · Zbl 0211.07005 |
| [7] | Chudnovsky, G. V., Algebraic independence of constants connected with the exponential and elliptic functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 8, 698-701 (1976) |
| [8] | Livingston, A. E., The series \(\sum_1^\infty f(n) / n\) for periodic \(f\), Canad. Math. Bull., 8, 413-432 (1965) · Zbl 0129.02801 |
| [9] | Gauss, C. F., Disquisitiones generales circa seriem infinitam etc., Comment. Gottingen. Comment. Gottingen, Werke, 3, 122-162 (1876), See also |
| [10] | Grinspan, P., Measures of simultaneous approximation for quasi-periods of abelian varieties, J. Number Theory, 94, 1, 136-176 (2002) · Zbl 1030.11033 |
| [11] | Hurwitz, A., Einige Eigenschaften der Dirichlet’schen Funktionen \(F(s) = \sum(D / n) n^{- s}\), die bei der Bestimmung der Klassenzahlen Binärer quadratischer Formen auftreten, Z. Math. Phys., 27, 86-101 (1882) |
| [12] | Kontsevich, M.; Zagier, D., Periods, (Mathematics Unlimited - 2001 and Beyond (2001), Springer: Springer Berlin), 771-808 · Zbl 1039.11002 |
| [13] | Jensen, J. L.W. V., An elementary exposition of the Gamma function, Ann. of Math. (2), 17, 124-166 (1916) · JFM 46.0563.02 |
| [14] | Lehmer, D. H., Euler constants for arithmetical progressions, Acta Arith., 27, 125-142 (1975), See also Selected Papers of D.H. Lehmer, vol. 2, pp. 591-608 · Zbl 0302.12003 |
| [15] | Mahler, K., Applications of a theorem of A.B. Shidlovski, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 305, 149-173 (1968) · Zbl 0164.05702 |
| [16] | Nesterenko, Yu. V., Modular functions and transcendence, Mat. Sb., 187, 9, 65-96 (1996) · Zbl 0898.11031 |
| [17] | Okada, T., On certain infinite sums for periodic arithmetical functions, Acta Arith., 40, 143-153 (1982) · Zbl 0402.10035 |
| [18] | N. Saradha, On non-vanishing infinite sums, in: Proceedings of the Conference in Honour of K. Ramachandra’s 70th Birthday, in press; N. Saradha, On non-vanishing infinite sums, in: Proceedings of the Conference in Honour of K. Ramachandra’s 70th Birthday, in press · Zbl 1234.11131 |
| [19] | Schneider, T., Zur Theorie der Abelschen Funktionen und Integrale, J. Reine Angew. Math., 183, 110-128 (1941) · JFM 67.0147.02 |
| [20] | Vasilév, K. G., On the algebraic independence of the periods of abelian integrals, Mat. Zametki, 60, 5, 681-691 (1996), 799 · Zbl 0936.11043 |
| [21] | Waldschmidt, M., Transcendence of periods: The state of the art, Q. J. Pure Appl. Math., 2, 2, 199-227 (2006) |
| [22] | Washington, L., Introduction to Cyclotomic Fields (1975), Springer |
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