Roger Apéry

Roger Apéry (born November 14, 1916 in Rouen , † December 18, 1994 in Caen ) was a French mathematician .

His mother was French and his father was Greek . After studying at the École normal supérieure from 1936, and a year as a prisoner of war in World War II, he became a lecturer in Rennes . In 1949 he became a professor at Caen University . He stayed here until his retirement. He died in 1994 after a long illness.

In 1979 he surprised the mathematical world with his unexpected proof of the irrationality of the Apéry constant named after him :

${\displaystyle \zeta (3)=1+{\tfrac {1}{2^{3}}}+{\tfrac {1}{3^{3}}}+{\tfrac {1}{4^{3}}}+{\tfrac {1}{5^{3}}}+\cdots }$

He was essentially using the rapidly converging series

${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{3}{\binom {2k}{k}}}}}$.

An appreciation of this achievement becomes clear from two perspectives:

• The corresponding problem for larger odd exponents (5, 7, 9, 11, ...) is still unsolved. Since 1979, many mathematicians have been working with these so-called aperitif sequences and looking for alternative proofs that may be transferable to other odd powers ( Frits Beukers , Alf van der Poorten , Marc Prévost , Keith Ball , Tanguy Rivoal , Wadim Zudilin and others).
• The series development used by Apery was already proven by Andrei Andrejewitsch Markow around 1890 and in 1903 independently of Ernst Reichenbächer (1881-1944). In 1953 it was rediscovered by the student Margrethe Munthe Hjortnaes (* 1927) and presented at the 12th Mathematicians' Congress in Lund , nevertheless the series remained generally unknown until 1978, although the great mathematicians Viggo Brun , Niels Erik Nørlund , Sigmund Selberg (1910-1994 ) and Carl Størmer were present. Knowledge of the series was reserved for individual mathematicians.
• A corresponding series expansion for ζ (2) was proven in 1918 by Konrad Knopp and Issai Schur :

${\displaystyle \zeta (2)=3\sum _{k=1}^{\infty }{\frac {1}{k^{2}{\binom {2k}{k}}}}}$.

Carl Ludwig Siegel once gave the following assessment of Apery's evidence:

"You can only hold the proof like a crystal in front of you"

- orally by Wilhelm Maak handed

Fonts

• La géométrie algébrique , Bulletin de la Société Mathématique de France 71, 1943, pp. 46–66 (French; in Numdam: [1] )
• Irrationalité de et ${\displaystyle \zeta (2)}$ ${\displaystyle \zeta (3)}$ . Astérisque 61, 1979, pp. 11-13 (French)
• Sur certaines séries entières arithmétiques , Groupe d'étude d'analysis ultramétrique 9 No. 16, 1982 (French; in Numdam: [2] )

literature

• Alfred van der Poorten : A proof that Euler missed… Apéry's proof of the irrationality of ζ (3). An informal report , The Mathematical Intelligencer 1, December 1979, pp. 195–203 (English; PDF file, 205 kB, on Alf van der Poortens homepage ; PDF; 210 kB)
• François Apéry: Roger Apéry, 1916–1994: a radical mathematician , The Mathematical Intelligencer 18 No. 2, June 1996, pp. 54–61 (English; online )
• Tom Apostol: A Proof that Euler missed (PDF; 1.3 MB)