Giovanni Enrico Eugenio Vacca

Giovanni Vacca

Giovanni Enrico Eugenio Vacca (born November 18, 1872 in Genoa , † January 6, 1953 in Rome ) was an Italian mathematician .

He studied mathematics in Genoa and received his doctorate in 1897 under GB Negri . In November 1897 he became an assistant to Giuseppe Peano , whom he supported in writing his form . He also dealt with the history of mathematics and science. In 1899 he studied the unpublished manuscripts of Gottfried Wilhelm Leibniz in Hanover , some of which he published in 1903.

A major contribution by Vacca was that in 1910 and 1926, after Euler, he gave a second type of series expansion (the Vacca series named after him) with rational terms for Euler's constant :

${\displaystyle {\begin{aligned}\gamma &=\sum _{k=2}^{\infty }(-1)^{k}{\frac {\left\lfloor \log _{2}k\right\rfloor }{k}}={\tfrac {1}{2}}-{\tfrac {1}{3}}+2\left({\tfrac {1}{4}}-{\tfrac {1}{5}}+{\tfrac {1}{6}}-{\tfrac {1}{7}}\right)+3\left({\tfrac {1}{8}}-\dots -{\tfrac {1}{15}}\right)+\dots \\\zeta (2)+\gamma &=\sum _{k=2}^{\infty }\left({\frac {1}{\lfloor {\sqrt {k}}\rfloor ^{2}}}-{\frac {1}{k}}\right)=\sum _{k=2}^{\infty }{\frac {k-\lfloor {\sqrt {k}}\rfloor ^{2}}{k\lfloor {\sqrt {k}}\rfloor ^{2}}}={\tfrac {1}{2}}+{\tfrac {2}{3}}+{\tfrac {1}{2^{2}}}\left({\tfrac {1}{5}}+{\tfrac {2}{6}}+{\tfrac {3}{7}}+{\tfrac {4}{8}}\right)+{\tfrac {1}{3^{2}}}\left({\tfrac {1}{10}}+\dots +{\tfrac {6}{15}}\right)+\dots \end{aligned}}}$

and notes:

There is some hope that this series can be of some use in the proof of the irrationality of , a very difficult problem, proposed, but not resolved, in the Correspondence, recently published, between Hermite and Stieltjes . ${\displaystyle \gamma }$

He also published a complex iteration for the circle number in 1910 : ${\displaystyle \pi \ }$

${\displaystyle x_{0}=i,\quad x_{n+1}={\frac {x_{n}+|x_{n}|}{2}}}$with . ${\displaystyle \lim _{n\to \infty }x_{n}={\frac {2}{\pi }}}$

However, because of the necessary long number arithmetic, it is not well suited for the numerical calculation of , because the efficiency is significantly worse compared to the known Borwein iterations . With each iteration step, only about half a decimal place is gained. It is . ${\displaystyle \pi \ }$ ${\displaystyle x_{23}=0{,}636619772367573902289405968390...+0{,}00000011920928955078125...i}$

Web links

• John J. O'Connor, Edmund F. Robertson :  Giovanni Enrico Eugenio Vacca. In: MacTutor History of Mathematics archive .
• Kramer, Stefan. " Euler's Constant γ = 0.577 ... Its Mathematics and History. "

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1. ^ G. Vacca. A new analytical expression for the number π and some historical considerations , Bulletin of the American Mathematical Society (Ser. 2), 1910, vol.16, pp.368-369, JFM 41.0496.03