Riv. Mat. Univ. Parma, Vol. 13, No. 1, 2022
Carlo Viola [a]
Linear independence of periods
Pages: 243-266
Received: 20 August 2021
Accepted in revised form: 2 December 2021
Mathematics Subject Classification: 11J72, 11J82, 20B05, 32A10.
Keywords: Periods, irrationality, linear independence, permutation group method, saddle-point method.
Authors address:
[a]: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy.
Dedicated to Roberto Dvornicich on the occasion of his seventieth birthday
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Abstract:
We present a survey on some classical and more recent results concerning
irrationality or \(\mathbb Q\)-linear independence of periods. We also discuss
the Rhin-Viola permutation group method, and the Pinna-Viola saddle-point
method in \(\mathbb C^N\).
References
- [AR]
-
K. Alladi and M. L. Robinson,
Legendre polynomials and irrationality,
J. Reine Angew. Math. 318 (1980), 137-155.
MR0579389
- [A]
-
R. Apéry,
Irrationalité de \(\zeta(2)\) et \(\zeta(3)\),
Astérisque 61 (1979), 11-13.
MR3363457
- [B]
-
F. Beukers,
A note on the irrationality of \(\zeta(2)\) and \(\zeta(3)\),
Bull. London Math. Soc. 11 (1979), no. 3, 268-272.
MR0554391
- [Br]
-
F. Brown,
Irrationality proofs for zeta values, moduli spaces and dinner parties,
Mosc. J. Comb. Number Theory 6 (2016), no. 2-3, 102-165.
MR3607782
- [DV]
-
R. Dvornicich and C. Viola,
Some remarks on Beukers' integrals,
Number Theory, Vol. II (Budapest, 1987),
Colloq. Math. Soc. János Bolyai, 51, Amsterdam, 1990, 637-657.
MR1058238
- [F]
-
S. Fischler,
Groupes de Rhin-Viola et intégrales multiples,
J. Théor. Nombres Bordeaux 15 (2003), no. 2, 479-534.
MR2140865
- [H]
-
M. Hata,
\(\mathbb C^2\)-saddle method and Beukers' integral,
Trans. Amer. Math. Soc. 352 (2000), no. 10, 4557-4583.
MR1641099
- [KZ]
-
M. Kontsevich and D. Zagier,
Periods,
in ''Mathematics unlimited - 2001 and beyond'', Springer, Berlin, 2001, 771-808.
MR1852188
- [Ma]
-
Yu. I. Manin,
Cubic Forms: algebra, geometry, arithmetic,
North-Holland Math. Library, 4, North-Holland, Amsterdam, 1974.
MR0460349
- [M1]
-
R. Marcovecchio,
The Rhin-Viola method for \(\log2\),
Acta Arith. 139 (2009), no. 2, 147-184.
MR2539543
- [M2]
-
R. Marcovecchio,
Multiple Legendre polynomials in diophantine approximation,
Int. J. Number Theory 10 (2014), no. 7, 1829-1855.
MR3256854
- [Pe]
-
O. Perron,
Über Summengleichungen und Poincarésche Differenzengleichungen,
Math. Ann. 84 (1921), 1-15.
MR1512016
- [PV]
-
F. Pinna and C. Viola,
The saddle-point method in \(\mathbb C^N\) and the generalized Airy functions,
Bull. Soc. Math. France 147 (2019), no. 2, 221-257.
MR3982276
- [Po]
-
H. Poincaré,
Sur les équations linéaires aux différentielles ordinaires et aux différences finies,
Amer. J. Math. 7 (1885), 203-258.
MR1505385
- [P]
-
A. van der Poorten,
A proof that Euler missed \(\dots\) Apéry's proof of the irrationality of \(\zeta(3)\),
Math. Intelligencer 1 (1978/79), no. 4, 195-203.
MR0547748
- [RV1]
-
G. Rhi and C. Viol,
On a permutation group related to \(\zeta(2)\),
Acta Arith. 77 (1996), no. 1, 23-56.
MR1404975
- [RV2]
-
G. Rh and C. Vio,
The group structure for \(\zeta(3)\),
Acta Arith. 97 (2001), no. 3, 269-293.
MR1826005
- [RV3]
-
G. Rhin and C. Viola,
The permutation group method for the dilogarithm,
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), no. 3, 389-437.
MR2185863
- [RV4]
-
G. Rhin and C. Viola,
Linear independence of \(1\), \(\rm Li_1\) and \(\rm Li_2\),
Mosc. J. Comb. Number Theory 8 (2019), no. 1, 81-96.
MR3864310
- [V]
-
C. Viola,
Birational transformations and values of the Riemann zeta-function,
J. Théor. Nombres Bordeaux 15 (2003), no. 2, 561-592.
MR2140868
- [VZ]
-
C. Viola and W. Zudilin,
Linear independence of dilogarithmic values,
J. Reine Angew. Math. 736 (2018), 193-223.
MR3769989
- [W]
-
M. Waldschmidt,
Transcendence of periods: the state of the art,
Pure Appl. Math. Q. 2 (2006), no. 2, 435-463.
MR2251476
- [WZ]
-
H. S. Wilf and D. Zeilberger,
An algorithmic proof theory for hypergeometric (ordinary and ''\(q\)'') multisum/integral identities,
Invent. Math. 108 (1992), no. 3, 575-633.
MR1163239
- [ZZ]
-
D. Zeilberger and W. Zudilin,
Automatic discovery of irrationality proofs and irrationality measures,
Int. J. Number Theory 17 (2021), no. 3, 815-825.
MR4254778
- [Z1]
-
W. Zudilin,
Well-poised hypergeometric service for Diophantine problems of zeta values,
J. Théor. Nombres Bordeaux 15 (2003), no. 2, 593-626.
MR2140869
- [Z2]
-
W. Zudilin,
Arithmetic of linear forms involving odd zeta values,
J. Théor. Nombres Bordeaux 16 (2004), no. 1, 251-291.
MR2145585
- [Z3]
-
W. Zudilin,
Two hypergeometric tales and a new irrationality measure of \(\zeta(2)\),
Ann. Math. Qué. 38 (2014), no. 1, 101-117.
MR3249415
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