Density of primes in a substitution class

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equation_99: Tschebotareff's density theorem of 1926: for an irreducible normal equation of degree n and a substitution S_lambda of its Galois group whose class has n_lambda members, the primes belonging to that class have Dirichlet density n_lambda/n.

satz_p225: Tschebotareff's sharpening of Hilbert's Zahlbericht Satz 152: in a normal field containing the l-th roots of unity, integers alpha_1, ..., alpha_t multiplicatively independent modulo l-th powers have arbitrarily prescribed l-th power residue symbols at infinitely many prime ideals.

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N. Tschebotareff, “Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören,” Mathematische Annalen 95 (1926), 191–228, DOI 10.1007/BF01206606. The article is in German; the author line on the scan reads “N. Tschebotareff in Odessa.”

The rendered PDF pages 2–5 (printed pp. 191–194) set up an irreducible normal equation of degree $n$, its Galois substitution $S$, and the congruence conditions used to sort primes by substitution class. The density quantity in equation (3) is the source's limit

$$\lim_{s\to 1^+} \frac{\sum_p p^{-s}}{\lg(1/(s-1))}.$$

The source writes the limit as $\lim_{s=1}$. The $1/(s-1)$ logarithm makes the modern analytic reading one-sided, as displayed above. The early statements include Satz 1 (printed p. 193) on the congruences (2a) and definitions of the relevant substitution-class and density terminology.

The main result is equation (99) of § 5 (printed p. 225). Let $S_\lambda$ lie in the Galois group $G$ of the normal equation (1), a group of order $n$, and let $n_\lambda$ be the number of distinct substitutions in the class (conjugacy class) of $S_\lambda$. Then the primes belonging to the class of $S_\lambda$ have density

$$\lim_{s\to 1^+}\frac{\sum_p p^{-s}}{\lg(1/(s-1))}=\frac{n_\lambda}{n},$$

the liminf and limsup in (98) being equal. Frobenius had proved only the corresponding density $k_\lambda n_\lambda/n$ for the union of the $k_\lambda$ classes making up the Abteilung (division) of $S_\lambda$ (§ 1, Hauptsatz and equation (20)). § 6 proves a sharpened form of a theorem of Hilbert on prime ideals with prescribed $l$-th power residue symbols.

The Göttingen scan read for this card holds the digitizing library's terms sheet followed by the article's printed pages 191 through 228, ending with "(Eingegangen am 5. 9. 1924.)"; the article pages are images with no text layer. The terms sheet prints, in part, "are protected by copyright. Publication and/or broadcast in any form (including electronic) requires prior written permission" and "Reproductions of material on the web site may not be made for or donated to other repositories, nor may be further reproduced without written permission from the Goettingen State- and University Library."

Sources: EuDML bibliographic record and the Göttingen digital library record http://resolver.sub.uni-goettingen.de/purl?PPN235181684_0095.

Read status: claims checked for the definitions on pp. 191--194, the Hauptsatz of § 1 (p. 195) with (20) (p. 198), the result (99) of § 5 with (93) to (98) (pp. 223--225) and the Satz of § 6 with its proof (pp. 225--228), read clause by clause on the page images; the proofs of §§ 2--5 were read for structure only. Nothing here is independently reviewed.

Results.

• Equation (99) (§ 5, p. 225): the primes belonging to the class of $S_\lambda$ have density $n_\lambda/n$.
• Satz of § 6 (p. 225): in a normal field containing the $l$-th roots of unity, integers independent modulo $l$-th powers take arbitrarily prescribed $l$-th power residue symbols at infinitely many prime ideals.

Bears on. None: the paper concerns the distribution of primes over the substitution classes of a Galois group, and it mentions no Erdős problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.