Errata to An Invitation to Modern Number Theory

Additional problems available here.

• Chapter 1: 
  • page 4: d should be 2087 and not 2807 (although amazingly both values work!)
  • page 5: in equations (1.1), (1.2) and 2^k ≤ x: replace x with n.
  • page 9: Exercise 1.2.17: should refer to Lemma 6.4.9, not 6.4.3.
  • page 12: the left hand side of equation (1.18) should be N+k choose k, and not N+k choose k-1.
  • page 15: Definition 1.3.5: it should be blackboard N (for natural numbers).
  • page 16: after Definition 1.4.6 replace the empty set with the set containing the identity element.
  • page 16: Exercise 1.4.7: it should be a non-empty subset H.
  • page 17: In Definition 1.4.15 and Exercise 1.4.16, replace subgroup with group.
  • page 18: Exercise 1.4.18: replace subgroups of size p_1 or p_2 with p_1 - 1 and p_2 - 1.
  • page 18: Exercise 1.4.21: a is relatively prime to n.
     
• Chapter 2:
  • page 30: after (2.3) it should be the number of primes at most n (not at most x).
  • page 34: modify (2.16) and the text around it to: We write F(x) ~ G(x) if F(x) - G(x)  = o(F(x)), and say F and G are of the same
    order. Add to Exercise 2.2.4:  Let F(x) = x²/2 and G(x) = Σ_{n ≤ x} n. Prove F(x) ~ G(x).

• Chapter 3:
  • page 50: Definition 3.1.12: at the end it is order of f, not order of the f.
  • page 54: it should be residue and not reside.
  • page 56: insert a factor of (-1)ⁿ on the right hand side of the last line of (3.36).
  • page 56: (3.38): should be Lambda(n) and not Lambda(x) on the right hand side.
  • page 60: Exercise 3.2.19: the hint for the first statement should read (cosθ + 1)² and not (cosθ - 1)²; better yet, consider 2(cosθ + 1)².
  • page 79: Section 3.3.4 (Imprimitive Characters): The definition was slightly off, and should read: "By definition a character χ modulo m is periodic. If (n,m) > 1 then χ(n) = 0, and if (n,m) = 1 then |χ(n)| = 1 (and hence χ(n) \neq 0 for n relatively prime to m). If for n relatively prime to m the character χ has period equal to m, χ is a called a primitive character, otherwise it is imprimitive. For example, consider the character modulo 6 given by χ(1) = 1, χ(5) = -1; as 2 and 3 are not relatively prime to 6, χ(0) = χ(2) = χ(3) = χ(4) = 0. Let ψ be the character modulo 3 given by ψ(0) = 0, ψ(1) = 1 and ψ(2) = -1. For n relatively prime to 6, χ(n) = ψ(n). Thus χ is induced from a character modulo 3."
     
• Chapter 4:
  • page 95: Exercise 4.4.1 should read: Show that while (4.56) has integral solutions, (4.57) does not.
  • page 98: Exercise 4.4.13 should read: Let F(x₁,x₂) = x₁² + x₂². Prove the number of solutions N_F(p) of F(x₁,x₂) ≡ 0 mod p satisfies |N_F(p) - p| ≤ C(F) p^1/2 for some constant C(F). Note that although F(x₁,x₂) is not absolutely irreducible over C (it equals (x₁+ix₂)(x₁-ix₂)), it is over Q and Z.

• Chapter 5:
  • page 124: After definition 5.5.1, it should talk about applications of the approximation exponent being slightly greater than 4 having applications to n^2alpha and not n^k alpha.
  • page 131: Exercise 5.6.9: should read n^th digit.
  • page 133: after Exercise 5.7.4 it should be AB^+/-1 and not AB^+/-.
  • pages 134 - 135: in equations (5.81) through (5.84) a₀ should be replaced with |a₀|.
  • page 145: before Lemma 6.3.6 the reference should be to section 6.5, not section 6.3.6.

• Chapter 7:
  • page 158: In equation (7.2) the subscripts N₀+1 and N₀ should be replaced with -N₀+1 and -N₀.

• Chapter 8:
  • page 214: the variance and standard deviation in section 8.4.2 of X are both 1. This means the variance on page 215 should be 2N, not 2N/4. The first line of (8.73) should have (2N)^2N, not 2^2N.  Equation (8.74): It should be exp(-k² / N) and not exp(-2 k² / N). To see this, one should take the logarithm of the product in the denominator and Taylor expand. Equation (8.75) is correct. The fraction in (8.75) should be written as 2 / Sqrt(2π 2N); the exponential term is exp(-(2k)² / 2(2N)). This looks like a Gaussian with mean 0 and variance 2N except for the fact that we have a 2 in the numerator of the normalization constant. This 2 is because Prob(S_2N = m) is zero if m is odd, and thus if we spread this out to be over 2k and 2k+1, the factor of 2 is basically replaced with 1.
  • The exposition of the proof of the Central Limit Theorem on pages 214 - 215 has been expanded, and can be downloaded HERE.

• Chapter 11:
  • page 261: in (11.25) it should be f(y) inside the integral and not f(x).
  • page 262: In theorem 11.3.1 it should be T_N(x) and not T_Nf(x).
  • page 265: replace all the text from equation (11.41) to the end of the proof with: Therefore g_x0(x) is bounded everywhere, say by B. As
    g_x0 is a bounded function, it is square-integrable, and thus the Riemann-Lebesgue Lemma (see Exercise11.2.2) implies that its Fourier coefficients tend to zero. This completes the proof, as i ∫^1/2_1/2 g_x0(x) sin((2N+1)πx)dx = Im(∫^1/2_1/2 g_x0(x) exp(2π i (2N+1)x) dx); thus our integral is just the imaginary part of the 2N+1^st Fourier coefficient, which tends to zero as N →∞. Hence as N →∞, S_N(x₀) converges (pointwise) to f(x₀).
  • page 269: Right before Theorem 11.4.6 it should be |f(x)| <= C/|x|^a and not |f(x)| <= C/x^a.
  • page 270: Exercise 11.4.7: replace n^5 with n^6 (so we don't have to worry about n positive or negative).
  • page 270: Lemma 11.4.8: assume g is continuous.
  • page 270: Lemma 11.4.10: should be F is continuously differentiable, not F' is.  Some small typos, so read a bit carefully; for example dropped  the c_n before the f''' term towards the end, but doesn't matter as |c_n| < 1.
  • page 271: top of the page should be delta_0 < 1 and then later should be min(1, delta_0, min_{|n| >= N} delta_n).
    

• Chapter 12:
  • Section 12.1. We periodically continue the sequence y_n by setting y_{mN+n} = m + y_n; in particular, y_{N+1} = 1 + y_1. Problem 12.1.8 should read: Consider the spacings |y_2 - y_1|, ..., |y_N - y_{N-1}|, |y_{N+1} - y_N| where all the y_n's are as above. Show the average spacing is 1/N.
  • Section 12.3.1: Replace all (1/(2N+1)) Sum_{n=-N to N} with (1/N) Sum_{n=1 to N}. One can of course consider two-sided sequences, but the result is a bit stronger if we just consider one-sided sequences.
  • Exercise 12.6.3: This problem is for the k=1 case; i.e., we are considering the spacings between the n alpha mod 1.

   

• Chapter 13:
  • page 321: Should say we are approximate F_N on the major arc M_a,q. The integral should be over just the major arc M_a,q and not all the major arcs. In (13.74) replace Q³ with Q.
  • page 322: equation (13.78): should equal -1 otherwise (not 0).
  • page 324: equation (13.90): should equal N log N (not N).

• Chapter 14:
  • page 353: equation (14.111): c_p(N) instead of c_q(N).

• Chapter 15:
  • page 362: it should be Prob(A)dA and not Prob(A) in equation (15.5).
  • page 363: Exercise 15.1.5 should only be on even probability distributions, p(-x) = p(x).

• Chapter 16:
  • page 395: carriage return missing in equation (16.2.4) so hard to read: expression equals 1.

• Chapter 16:
  • page 395: carriage return missing in equation (16.2.4) so hard to read: expression equals 1.
  • page 398: in (16.38) it should be N^{1+m} and not N^{1+M}.

• Appendix A:
  • Exercise A.4.4: Clarification: While a_1, ..., a_{n+1} are distinct, it is possible to choose the same number twice when trying to find two numbers whose sum is divisible by 2n. If instead one required the two numbers to be distinct, then a_1, ..., a_{n+1} would have to be n+1 distinct numbers from {1,...,2n-1}.
  • Exercise A.5.9:  the problem on summing elements in the Cantor Set was not in the Monthly, but rather in the Mathematics Magazine (problem Q785, proposed by Jeffrey Shallit, Mathematics Magazine vol 64, no 5 (1991), page 351).

• Appendix B:
  • page 458: replace "this subgroup is disjoint from SO(2)" with "these matrices are disjoint from those in SO(2)".

• Appendix D:
  • page 475: should be Chapters 13 and 14 near the end of the page, not sections.