Abstract
1. In 1947 R. Brauer [428] found a proof of Artin’s conjecture on the divisibility of Dedekind zeta-functions for Galois extensions, showing first that in Artin’s theorem about linear combinations of characters induced by cyclic subgroups the rational coefficients may be taken to be nonnegative. As a corollary he obtained for normal extensions L / K a representation of \((\zeta _L(s)/\zeta _K(s))^n\) as a product of Abelian L-functions. He pointed out that from the truth of Artin’s conjecture on the integrality of Artin L-functions this corollary would hold also for non-normal extensions.
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Notes
- 1.
In a footnote on p. 243 Brauer acknowledged the priority of Aramata, adding “I publish my proof ...since it seems to be somewhat simpler”.
- 2.
Richard Howard Hudson (1945–2016), professor at the University of South Carolina.
- 3.
Atle Selberg (1917–2007), professor in Princeton. See [163].
- 4.
Harold Nathaniel Shapiro (1922–2013), professor at the New York University.
- 5.
Raymond Ayoub (1923–2013), professor at the Penn State University.
- 6.
Tate’s thesis has been published only after 17 years.
- 7.
Hel Braun (1914–1986), professor in Göttingen and Hamburg. See [3938].
- 8.
Srinivasacharya Raghavan (1934–2014), professor at Tata Institute of Fundamental Research.
- 9.
In 1936 Rademacher [3366] announced some results on the Waring problem in totally real fields, but no details were published.
- 10.
Chidambaran Padmanabhan Ramanujam (1938–1973), professor at Tata Institute of Fundamental Research. See [3379].
- 11.
Rosemarie S. Stemmler (1930–2011), professor at the Purdue University.
- 12.
Aleksandr Osipovič Gelfond (1906–1968), professor in Moscow. See [2581].
- 13.
Kurt Heegner (1893–1965). See [3228].
- 14.
Naum Ilič Feldman (1918–1994), professor in Moscow. See [2224].
- 15.
Kiyoshi Iseki (1919–2011), professor in Osaka.
- 16.
This problem is closely related to the problem of moments of the values at \(s=1\) of Dirichlet L-functions associated with real characters \(\chi _k(x)=\left( \frac{-k}{x}\right) \).
- 17.
Mark Borisovič Barban (1935–1968). See [4221].
- 18.
Takuro Shintani (1943–1980), professor in Tokyo. See [1956].
- 19.
Edward Charles Titchmarsh (1899–1963), professor in Liverpool and Oxford. See [592].
- 20.
The later paper [3792] of Siegel contains a weaker result.
- 21.
George William Whaples (1914–1981), professor at the University of Massachusetts in Amherst and Indiana University. See [1025, 1026].
- 22.
Gerhard Hochschild (1915–2010), professor at the University of Illinois at Urbana and the University of California at Berkeley. See [3573].
- 23.
Samuel Eilenberg (1913–1998), professor at the University of Michigan, University of Indiana and the Columbia University. See [210].
- 24.
Shokichi Iyanaga (1906–2006), professor in Tokyo and the Gakushuin University.
- 25.
Heinrich-Wolfgang Leopoldt (1927–2011), professor in Karlsruhe. See [3510].
- 26.
These fields occur already in a paper of Scholz [3679] published in 1940.
- 27.
Stanley Gurak (1949–2010), professor at the University of San Diego.
- 28.
Eric Stephen Barnes (1924–2000), professor in Adelaide.
- 29.
George Erskine Cooke (1942–1976), professor at the Cornell University and the University of Maryland.
- 30.
Jerzy Urbanowicz (1951–2012), professor in Warsaw. See [3623].
- 31.
Carl Störmer (1874–1957), professor in Kristiania (Oslo). See [482].
- 32.
Wilhelm Ljunggren (1905–1973), professor in Oslo.
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Narkiewicz, W. (2018). The Forties. In: The Story of Algebraic Numbers in the First Half of the 20th Century. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-03754-3_6
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