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Elements of Analytic Number Theory

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Elements of Analytic Number Theory ELEMENTS OF ANALYTIC NUMBER THEORY P. S. Kolesnikov, E. P. Vdovin Lecture course Novosibirsk, Russia 2013 Contents Chapter 1. Algebraic and transcendental numbers4 x 1.1. Field of algebraic numbers. Ring of algebraic integers4 1. Preliminary information4 2. Minimal polynomial6 3. Algebraic complex numbers9 4. Algebraic integers 11 x 1.2. Diophantine approximations of algebraic numbers 13 1. Diophantine approximation of degree ν 13 2. Dirichlet approximation theorem 15 3. Liouville theorem on Diophantine approximation of algebraic numbers 18 x 1.3. Transcendentality of e and π 20 1. Hermite identity 20 2. Transcendentality of e 23 3. Symmetrized n-tuples 25 4. Transcendentality of π 26 x 1.4. Problems 29 Chapter 2. Asymptotic law of distribution of prime numbers 30 x 2.1. Chebyshev functions 31 1. Definition and estimates 31 2. Equivalence of the asymptotic behavior of Chebyshev functions and of the prime-counting function 32 3. Von Mangoldt function 34 x 2.2. Riemann function: Elementary properties 35 1. Riemann function in Re z > 1 35 2. Distribution of the Dirichlet series of a multiplicative function 36 3. Convolution product and the M¨obiusinversion formula 37 4. Euler identity 38 5. Logarithmic derivative of the Riemann function 39 2 Contents 3 6. Expression of the integral Chebyshev function via the Riemann function 40 x 2.3. Riemann function: Analytic properties 43 1. Analytic extension of the Riemann function 43 2. Zeros of the Riemann function 47 3. Estimates of the logarithmic derivative 48 4. Proof of the Prime Number Theorem 51 x 2.4. Problems 55 Chapter 3. Dirichlet Theorem 56 x 3.1. Finite abelian groups and groups of characters 56 1. Finite abelian groups 56 2. Characters 58 3. Characters modulo m 60 x 3.2. Dirichlet series 60 1. Convergence of L-series 60 2. Landau Theorem 66 3. Proof of the Dirichlet Theorem 68 Chapter 4. p-adic numbers 71 x 4.1. Valuation fields 71 1. Basic properties 71 2. Valuations over rationals 74 3. The replenishment of a valuation field 76 x 4.2. Construction and properties of p-adic fields 79 1. Ring of p-adic integers and its properties 80 2. The field of p-adic rationals is the replenishment of rationals in p-adic metric 81 3. Applications 84 x 4.3. Problems 85 Bibliography 86 Glossary 87 Index 88 CHAPTER 1 Algebraic and transcendental numbers x 1.1. Field of algebraic numbers. Ring of algebraic integers 1. Preliminary information. Let us recall some basic notions from Abstract Algebra. Throughout we use the following notations: P is the set of all prime numbers; N is the set of positive integers (the set of natural numbers); Z is the set of all integers; Q is the set of all rational numbers; R is the set of all real numbers; C is the set of all complex numbers, C = R + ιR, ι2 = −1. Given a field F , symbol F [x] denotes the ring of polynomials in variable n x with coefficients in F . If f(x) = a0 + a1x + ··· + anx 2 F [x], ai 2 F , is chosen so that an 6= 0, then n is called the degree of f(x), it is denoted by deg f, while an 2 F is called the leading coefficient of f(x), and if an = 1 then f(x) is called monic. If f(x) = 0 (all coefficients are equal to zero) then the degree of f(x) is said to be −∞. If f(x); g(x) 2 F [x], g(x) 6= 0, then there exist unique q(x); r(x) 2 F [x] such that f(x) = g(x)q(x) + r(x); deg r < deg g: (1.1) These polynomials (quotient q(x) and remainder r(x)) can be found by the well-known division algorithm. If r(x) = 0 then we write g j f (g divides f). One may easily note the similarity between division algorithms in the ring of integers Z and in the ring of polynomials F [x]. Indeed, these are par- ticular examples of Euclidean rings, and there are many common features and problems that can be solved in similar ways for integers and polynomi- als. In particular, the greatest common divisor (gcd) d of two polynomials f; g 2 F [x] is defined as a monic common divisor which is divided by every 4 x 1.1. Field of algebraic numbers 5 other common divisor, i.e., d = gcd(f; g) if and only if d j f, d j g, and for every h 2 F [x] with h j f and h j g it follows that h divides d. To find gcd of f and g, one may use the Euclidean algorithm based on the following observation: If f and g are related by (1.1) then gcd(f; g) = gcd(g; r). Moreover, if d = gcd(f; g) then there exist p(x); s(x) 2 F [x] such that f(x)p(x) + g(x)s(x) = d(x): Exercise 1.1. Let f1; : : : ; fn 2 F [x] be a finite family of polynomials over a field F . Prove that there exists a unique monic greatest common divisor of f1; : : : ; fn. Suppose R is a commutative ring with an identity (e.g., R = Z or R = F [x] as above). A subset I ⊆ R is called an ideal of R if a ± b 2 I for every a; b 2 I, and ax 2 I for every a 2 I, x 2 R. For example, the set of all even integers is an ideal of Z; the set ff(x) 2 F [x] j f(α) = 0g is an ideal of F [x], where α is an element of some extension field of F . Since an intersection of any family of ideals is again an ideal, for every set M ⊆ R there exists minimal ideal of R which contains M, it is denoted by (M). It is easy to note that X (M) = xiai j xi 2 R; ai 2 M : i An ideal I of R is said to be principal if there exists a 2 R such that I = (a), where (a) stands for (fag). Recall that a commutative ring R is called an integral domain (or simply a domain) if ab = 0 implies a = 0 or b = 0 for all a; b 2 R. In particular, Z and F [x] are integral domains. An integral domain R such that every ideal of R is principal is called a principal ideal domain. Exercise 1.2. Prove that Z and F [x] (where F is a field) are principal ideal domains. In particular, if f(x); g(x) 2 F [x] then (ff; gg) = (gcd(f; g)): If R is a domain, then we can consider the field of fractions of R. In order to construct it we start with the Cartesian product R × (R n f0g) of R a (here each pair (a; b) corresponds to fraction b ). Now define an equivalence relation (a1; b1) ∼ (a2; b2) () a1b2 = a2b1. Let Q be the set of equivalence classes of R × (R n f0g) under this equivalence. Define the addition and multiplication on representatives by (a1; b1) + (a2; b2) = (a1b2 + a2b1; b1b2); (a1; b1) · (a2; b2) = (a1a2; b1b2): x 1.1. Field of algebraic numbers 6 We leave for the reader to prove that all operations defined are correct and that Q is a field under these operations. Let I be an ideal of R. Then R is split into a disjoint union of congruence classes a+I = fa+x j x 2 Ig, a 2 R, and the set of all these classes (denoted by R=I) is a ring with respect to natural operations (a + I) + (b + I) = (a + b) + I; (a + I)(b + I) = ab + I: The ring R=I obtained is called a factor ring of R over I. For example, Z=(n) = Zn, the ring of remainders modulo n. A proper ideal I of R is maximal if there are no proper ideals J of R such that I ⊂ J. For example, if R = Z then (n) is maximal if and only if n = ±p, where p is a prime natural number; if R = F [x] then (f) is maximal if and only if the polynomial f is irreducible over F . Note that if I is a maximal ideal of a commutative ring R then R=I is a field. Indeed, if a + I 6= 0 (i.e., a2 = I) then J = fxa + b j x 2 R; b 2 Ig is an ideal of R such that I ⊂ J. Therefore, J = R, and thus all equations of the form (a + I)X = c + I, c 2 R, have solutions in R=I. Exercise 1.3. Prove that R[x]=(x2 + x + 1) is a field isomorphic to the field C of complex numbers. 2. Minimal polynomial. A complex number α 2 C is algebraic if there exists a nonzero polynomial f(x) 2 Q[x] such that f(α) = 0. A non-algebraic complex number is said to be transcendental. An algebraic number α is called an algebraic integer if there exists a monic polynomial f(x) 2 Z[x] such that f(α) = 0. Every rational number α 2 Q ⊂ C is obviously an algebraic one. More- over, as we will see later, a rational number is an algebraic integer if and only if it is an integer. p p 1 3 Exercise 1.4. (1) Prove that 2 and + ι are algebraic in- 2 2 tegers.
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