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Statement

Setting (pp. 191--194). Let $f(x)=0$ be an irreducible normal equation of degree $n$ with roots $x_1,\ldots,x_n$, and let $G$ be its Galois group, which has order $n$. For a prime ideal $\mathfrak P$ of the field generated by the roots that divides the rational prime $p$ but not the discriminant of the equation, Satz 1 (p. 193) gives congruences $x_i^p\equiv x_{\alpha_i}\pmod{\mathfrak P}$ for $i=1,\ldots,n$, and the substitution $S$ sending $i$ to $\alpha_i$ lies in $G$ (Satz 2). Then $\mathfrak P$ belongs to $S$ (Definition 1); the class of $S$ is the set of all $T^{-1}ST$ with $T\in G$ (Definition 2); and, since the conjugates of $\mathfrak P$ belong to the conjugates of $S$ (Satz 3), $p$ belongs to the class of $S$ (Definition 3). The density of a set of primes is the value of $\lim_{s=1}\sum_p p^{-s}/\lg\frac1{s-1}$, the sum running over the primes of the set (Definition 7, p. 194; also (3), p. 191).

Theorem (§ 5, pp. 216--225; result (99), p. 225). Let the irreducible normal equation (77) of degree $n$ have in its group a substitution $S_\lambda$ of order $f_\lambda$ (78), and let $n_\lambda$ be the number of distinct substitutions in the class of $S_\lambda$ (notation of p. 194). Write $p_1$ for the primes belonging to the class of $S_\lambda$. Then (p. 225, display (99), quoted)

$$\liminf_{s=1}\frac{\sum_p p_1^{-s}}{\lg\frac1{s-1}} =\limsup_{s=1}\frac{\sum_p p_1^{-s}}{\lg\frac1{s-1}} =\lim_{s=1}\frac{\sum_p p_1^{-s}}{\lg\frac1{s-1}} =\frac{n_\lambda}{n},$$

"und somit ist die gesuchte Dichtigkeit gefunden": the set of primes belonging to the class of $S_\lambda$ has density $n_\lambda/n$. The print writes the limits as $s=1$; since $\lg\frac1{s-1}$ is real only for $s>1$, they are read here as $s\to1^+$.

Context. The Hauptsatz of § 1 (p. 195), with its formula (20) (p. 198), is Frobenius's result as the paper re-proves it: the primes belonging to the Abteilung (division) of $S_\lambda$, the set of all $TS_\lambda^iT^{-1}$ with $T\in G$ and $i$ prime to the order of $S_\lambda$ (Definition 5, p. 194), have density $k_\lambda n_\lambda/n$, where $k_\lambda$ is the number of classes in that Abteilung. The paper records (p. 198) that Frobenius did not succeed in showing that a single class has density $n_\lambda/n$; (99) supplies this. On p. 213 the paper notes, in § 4, that once the main result of § 5 is obtained the word "Abteilung" may be replaced everywhere by "Klasse".

Proof pointer

§ 5, pp. 216--225. Choose $k$ primes $l_1,\ldots,l_k$ of the form $f_\lambda x+1$, prime to the discriminant $D$, and adjoin to the field of the roots $k$ cyclic fields of degree $f_\lambda$ built from the $l_i$-th roots of unity (79); these are disjoint from one another and from the field of the roots, and the compositum is normal of degree $n\cdot f_\lambda^k$ (p. 216). The primes of the Abteilung of $S_\lambda$ are distributed over the $f_\lambda^k$ residue "complexes" of § 2, using the uniform-distribution Hauptsatz of § 3 (p. 211, its proof completed in § 4, p. 215) and Sätze 10--13. Counting the primes of the single class of $S_\lambda$ that lie in the primitive parts of the rays gives the lower bound (92) with a factor $1-a/Q^k$, and letting $k$ grow gives (93): the liminf is at least $n_\lambda/n$ for every class of the Abteilung (pp. 223--224). Frobenius's formula (96) for the whole Abteilung, with (93) applied to the other $k_\lambda-1$ classes, gives the limsup bound (97), and (98) closes the gap (pp. 224--225). Not reconstructed here.

Read depth

Claims checked: the setting on pp. 191--195, the Hauptsatz of § 1 and (20), the opening of § 5 and the displays (93) to (99) were read clause by clause on the page images of the print. The intermediate steps of § 5 and the proofs of §§ 2--4 were read for structure only. Nothing here is independently reviewed.

Dependencies

None in the corpus. The paper uses Frobenius's 1896 results (Sätze 1--4, quoted from his Berlin Academy paper), Kronecker's formula (Satz 5, proof cited from Landau) and Dirichlet's theorem on primes in progressions.

Source. N. Tschebotareff, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören, Math. Ann. 95 (1926), 191--228, doi:10.1007/BF01206606; the edition read is named on the source card.

Bears on

None: the paper mentions no Erdős problem, and no problem page cites it.