Introduction to analytic number theory. (Einführung in die analytische Zahlentheorie.) (German) Zbl 0154.04402

Lecture Notes in Mathematics. 29. Berlin-Heidelberg-New York: Springer-Verlag. vi, 199 S. (1966).
The present Lecture Note is a nice and well-planned introduction to diverse methods and topics in analytic theory of numbers. There are eleven chapters in the Note.
Chap. I: Der Fundamentalsatz der elementaren Zahlentheorie.
Three proofs for the fundamental theorem of number theory, i. e. the uniqueness of the canonical decomposition of natural numbers (the first proof is by complete induction, the second depends on the study of the solubility of some linear Diophantine equations, and the third is an application of the theory of Farey series). Two proofs for Euclid’s second theorem to the effect that the number of primes is infinite (one is well known and the other uses the fact that no two Fermat numbers have a common divisor greater than 1).
Chap. II: Kongruenzen.
Euler’s function \(\varphi(m)\). The Fermat-Euler theorem. Multiplicative (number-theoretic) functions. The number of solutions of a polynomial congruence (in the simplest case of prime modulus).
Chap. III: Die rationale Approximation einer irrationalen Zahl. Der Satz von Hurwitz.
Rational approximations to a real number: results obtained by Dirichlet’s „Schubfachprinzip“ and by the use of Farey series. The theorem of Hurwitz (proof is by the use of Ford circles).
Chap. IV : Quadratische Reste, und die Darstellbarkeit einer positiven ganzen Zahl als Summe von vier Quadraten.
Legendre’s symbol. Wilson’s theorem. Euler’s criterion. Every positive integer \(n\) can be represented as a sum of two squares, if and only if in the canonical decomposition of \(n\) all prime factors of the form \(4k + 3\) have even exponents. Lagrange’s theorem: every positive integer is a sum of four squares.
Chap. V: Das quadratische Reziprozitätsgesetz.
Gaussian sums and the reciprocity formula for them. The derivation of the quadratic reciprocity law from the reciprocity formula for Gaussian sums. A proof of the second complementary law (by the use of the Lagrange resolvent).
Chap. VI: Zahlentheoretische Funktionen und Gitterpunkte.
The fundamental properties, the order and the average order of some important number-theoretic functions, \(r(n)\), \(d(n)\), \(\sigma(n)\), \(\varphi(n)\). Möbius’ inversion formulae. Definition and some simple properties of the Riemann zeta-function.
Chap. VII: Der Satz von Chebyshev über die Verteilung der Primzahlen.
Usual proof for the theorem of Chebyshev (Tchebysheff): the order of \(\pi(x)\), is \(x/\log x\). Proof of the Bertrand postulate (P. Erdős). Proof that the superior, respectively inferior, limits for \(x\to\infty\) of the functions \(\pi(x)/(x/\log x)\), \(\vartheta(x)/x\), \(\psi(x)/x\) are equal. The Mertens’ formulae.
Chap. VIII: Die Weylsche „Gleichverteilung von Zahlen mod 1“, und der Satz von Kronecker.
Two forms of Kronecker’s theorem on simultaneous inhomogeneous Diophantine approximation (proof by H. Bohr – B. Jessen). Weyl’s criteria for the uniform distribution of sequences of real numbers (mod 1) and of sequences of real finite-dimensional vectors (mod 1).
Chap. IX: Der Satz von Minkowski über Gitterpunkte in konvexen Bereichen.
Two proofs for the theorem of Minkowski (let \(S\) be a convex region in \(R^n\), symmetric about \(0\) and of volume \(V > 2^n\); then \(S\) contains a lattice point other than \(0)\): one is due to Siegel and the other to Blichfeldt-Birkhoff. Some applications of Minkowski’s theorem.
Chap. X : Der Dirichletsche Satz von den Primzahlen in einer arithmetischen Progression.
An (ordinary) analytic proof with residue characters and simple properties of Dirichlet series, of the theorem of Dirichlet: if \((k,l) = 1\) then there exist infinitely many primes \(p \equiv l\pmod k\).
Chap. XI: Der Primzahlsatz.
An analytic proof with the Riemann zeta-function and Wiener-Ikehara’s Tauberian theorem of the prime-number theorem: \[ \lim_{x\to\infty} \pi(x)/(x/\log x) = 1. \] (The additive theory of numbers, in particular the problem of partitions, has been excluded from the topics in this Lecture Note.)


11-02 Research exposition (monographs, survey articles) pertaining to number theory
11Hxx Geometry of numbers
11Jxx Diophantine approximation, transcendental number theory
11Mxx Zeta and \(L\)-functions: analytic theory
11Nxx Multiplicative number theory
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