Analytic and Combinatorial Number Theory, NDMI045, summer term 2019/20
Lectures are on Friday 9:00 - 10:30 in the lecture room K334KA (seminary room of the Department of
Algebra) at the 3rd floor in the Karlin building.)
On exam and exam questions. Exam is in Czech or English and is preferably in contact form (oral with written
preparation) but distant form is possible on request, see SIS for terms.
1. Prove that natural numbers cannot be partitioned in arithmetic progressions with distinct common differences
and state and prove L. Euler's distinct x odd parts partition identity.
2. Prove the FTAlg (Fundamental Theorem of Algebra).
3. State and prove Dirichlet's asymptotics in the divisor problem.
4. Just state (without proof) the Cauchy theorem and the Theorem on residues, but explain all notions used.
5. Define the Bernoulli numbers B_r, prove the formula 1^k + 2^k + ... + (n-1)^k =
sum_{r=0}^k binom(k, r)*(B_r/(k+1-r))*n^{k+1-r} and state (without proof) the von Staudt--Clausen theorem on B_r.
6. State and prove (or at least give the main ideas of the proof of) the formula for the sum of the series
1^{-2k} + 2^{-2k} + 3^{-2k} + ...
7. State the functional equation for zeta(s) and give some main ideas of its proof.
8. Give the main ideas of Newman's proof of the PNT - state (without proof) Propositions 1-5 and Theorems 6 and 7.
1st lecture on February 28, 2020 (M. Klazar). Two examples
of results in analytic and combinatorial number
theory: partitions of the natural numbers in arithmetic progressions and L. Euler's distinct x odd parts identity. The FTAlg., beginning of the proof.
Lecture notes for the 1st lecture (contrary to what I write in the LN, unfortunately I will not have time to prove
the full asymptotics for p(n), but I will prove a weaker version instead).
2nd lecture on March 6, 2020 (M. Klazar). Finishing the proof of the FTAlg. Remarks on the two examples
in Lecture 1. Sums and integrals. Thm. 1 (approximating the sum of f(n), a < n <= b, for a monotonic
function f, by an integral), proof. Thm. 2 (Euler--MacLaurin summation formula for 1st derivatives), proof.
Lecture notes for the 2nd lecture
Starting March 11 classes in person are cancelled because of the coronavirus quarantine.
I will post my scanned handwritten lecture notes here on each Friday. The form of the exam will be determined later.
March 13, 2020 (M. Klazar). Asymptotics of the harmonic numbers H_n,
Theorem (Dirichlet's asymptotics in the divisor
problem), proof. Thm. 3 (Abel's summation), proof. An application:
deduction of the 2nd Mertens formula (asymptotics for sum_{p at most x}1/p) from the 1st one (asymptotics for sum_{p at most x}(log p)/p).
study text replacing the 3rd lecture.
March 20, 2020 (M. Klazar). Proof of the 1st Mertens formula. The Stieltjes integral.
Thm. 4 (General Euler--MacLaurin summation formula), without proof; the Bernoulli numbers appear.
study text replacing the 4th lecture.
Lecture notes Introduction to Number theory.
March 27, 2020 (M. Klazar). Another application of Abel's summation: if pi(x)/(x/log(x)) has a limit
for x going to infinity then it is 1. An alternative formula to the Euler - Maclaurin summation due to I. Pinelis, without proof.
Thm. 5 (the Poisson summation formula), without proof.
Applications of complex analysis - the Cauchy formula. Definition of the Cauchy formula.
study text replacing the 5th lecture.
April 3, 2020 (M. Klazar). We eventually derive the formula for zeta(2n) but need for it some
preparation. The Cauchy theorem, no proof (but see my
LN Mathematical
Analysis III, also in Czech version, the proof is in the 12th lecture).
Proposition: Int_{gamma}(z - z_0)^n dz = 2pi*i if n = -1 and = 0 for any other integer n,
for an admissible curve gamma going counter-clockwisely around z_0, no proof. Theorem on power series
expansions of holomorphic functions, no proof.
Theorem on rigidity of holomorphic functions, no proof. Meromorphic functions.
Theorem on residues, sketch of a proof.
study text replacing the 6th lecture.
April 10, 2020. No lecture - Good Friday. However, there will be lectures on May 1 and 8.
April 17, 2020. I am sorry for the delay, the study text will be posted here tomorrow on April 21.
Here it is: study text replacing the 7th lecture. Euler's ``proof'' that 1 + 1/4 + 1/9 + ... = pi^2/6.
Bernoulli numbers B_k.
The theorem of von Staudt and Clausen on Bernoulli numbers: B_{2n} + the sum of all 1/p s. t. p - 1 divides 2n is always
an integer, proof next time.
April 24, 2020. study text replacing the 8th lecture.
Beginning of the proof of the von Staudt--Clausen theorem. Proposition:
S_k(n) := 1^k + 2^k + ... + (n-1)^k = sum_{r=0}^k binom(k, r)*(B_r/(k+1-r))*n^{k+1-r}, proof. The p-adic norm |...|_p.
Proposition: the properties of |...|_p, no proof. Remarks on p-adic Analysis.
Claim 1: lim_{n-->0}S_k(n)/n = B_k (in |...|_p), proof. Claim 2: If p is a prime
and k a natural number, then S_k(p) = -1 (mod p) if p-1 divides k and = 0 (mod p) else, proof of the 1st case (conclusion next time).
May 1, 2020. See May 8.
May 8, 2020. study text replacing the 9th lecture (updated on May 12). Proof of the 2nd case.
Claim 3: If k>0 is an even integer and p is prime then |B_k - p^{-1}S_k(p)|_p is at most 1, proof. Conclusion of the
proof of the theorem of von Staudt and Clausen, follows from Claims 2 and 3. Theorem: If k is a natural number then
sum_{n=1}^{infinity}n^{-2k} = (-1)^{k+1}*2^{2k-1}*B_{2k}*pi^{2k}/(2k)!, proof by the theorem on residues. study text replacing
the 10th lecture (updated on May 12). Theorem (properties of zeta(s)): zeta(s) has a meromorphic extension to the
punctured complex plane C - {1}, with Res(zeta(s), 1) = 1, and satisfies there the functional equation zeta(s) = 2^s*pi^{s-1}*sin(pi*s/2)*Gamma(1-s)*zeta(1-s)
where Gamma(s) is the Euler gamma function, proof (with the help of Theorem 2 of lecture 2).
May 15, 2020. study text replacing the 11th lecture (updated on May 20).
The PNT (Prime Number Theorem): pi(x) ~ x/log(x) for x --> +infinity. Proof will be given in this and the next lecture.
Proposition 1: theta(x) = sum_{p at most x} log(p) = O(x), proof.
Proposition 2: For x --> +infinity, the PNT holds iff theta(x) = x + o(x), proof.
Proposition 3: If the integral int_1^{+infinity}(theta(x) - x)x^{-2}dx converges then theta(x) = x + o(x), proof.
Proposition 4: If F(s) = sum_p(log(p)p^{-s}) for Re(s)>1, then int_0^{+infinity}(theta(e^t)e^{-t} - 1)e^{-zt}dt =
(z+1)^{-1}F(z+1) - z^{-1} for Re(z)>0, proof.
Proposition 5: The function (z+1)^{-1}F(z+1) - z^{-1} (from the previous proposition) has a holomorphic extension
from Re(z)>0 to the closed halfplane Re(z)>=0, proof in the next lecture.
Theorem 6 (Wiener and Ikehara, 1932): If the real function f defined on [0, +infinity) is (i) bounded, (ii)
integrable over bounded intervals, and (iii) such that its Laplace transform g(z) = int_0^{+infinity}f(t)e^{-zt}dt has a holomorphic extension from Re(z)>0 to
the closed halfplane Re(z)>=0, then g(0) = int_0^{+infinity}f(t)dt (in particular this integral converges), proof in the next lecture.
Deduction of the PNT from Propositions 1-5 and Theorem 6.
May 22, 2020. study text replacing the 12th lecture (updated on May 20 (sic!)).
Theorem 7 (on the zeros of zeta(s)): zeta(s) is nonzero on the closed halfplane Re(s)>=1, proof.
Proof of Proposition 5: logarithmic derivative of zeta(s) gives a good formula for F(s) - (s-1)^{-1}.
Proof of Theorem 6: D.J. Newman's proof by the Cauchy formula, based on Newman's trick (introduction of the
integration kernel G(z)).
May 2020