Göttingen Junior Number Theory Seminar
Winter 2013/2014
08.01.2014 - Julia Brandes: On generalised Möbius Functions
Abstract: I will give an easy introduction to generalized Möbius functions and how they can be used to describe the principle of inclusion and exclusion analytically. I found them in a paper (Giancarlo Rota (1964), on the foundations of combinatorial Theory, I: Theory of Möbius Functions) and I think they are a very neat idea. I might also sketch what I think I'll need them for.
15.01.2014 - Berke Topacogullari: A quadratic divisor problem
Abstract: N/A
22.01.2014 - Lasse Grimmelt: The number of divisors on average
Abstract: Prove of an asymptotic formula of:
$$r_1(x):=#\{n∈ ℕ: n d(n)≤x\}$$
and
$$r_2(x):=#\{n∈ ℕ:n/{d(n)}≤x\}$$
Idea of Proof:
Combining an Idea of Bateman in "The distribution of values of the Euler function" and a lightly generalized Selberg-Delange method. For $r_2(x)$ a generalization of Perron's formula for series over the rational numbers is used.
29.01.2014 - Fabian Dehnert: Variance for $k$-free numbers in arithmetic progressions
Abstract: We sketch the proof of the variance for k-free numbers in arithmetic progressions based on Vaughan (2005), i.e.
let
$$Q_k(x;q,a) = ∑↙{n≤ x,n ≡ a (q)} μ_k(n)$$
and
$$g(q,a)= ∑ ↙{m=1,(m^k,q)| a}↖∞ {μ(m)(m^k,q)}/{m^k q},$$
then the Variance is defined as
$$V(x,Q)= ∑ ↙{q≤ Q} ∑ ↙{a=1}↖q (Q_k(x;q,a)- x g(q,a) )^2$$
and a asymptotic is obtained via the circle method.
More precisely,
$$V(x,Q) = c_k x^{1/k} Q^{2-1/k} + O( x^{1/(2k)}Q^{2-1/(2k)} \exp (-c_k^{*} {(\log2x/Q)^{3/5}}/{(\log \log 3x/Q)^{1/5}} ) ).$$
Furthermore if f is a suitable polynomial of degree d we give result on the sparse variance
$$V_f(x,y) =∑ ↙{y_0(f) ≤ u ≤ y} f'(u) ∑ ↙{l=1}↖{f(u)} ( Q_k(x;u,l) - x g(u,l))^2$$
05.02.2014 - Alexander Adam: A sequential Riesz-like criterion for the Riemann hypothesis
Abstract: We will show the steps of the proof of the Baez-Duarte criterion in a rigorous manner. Some knowledge about elementary special functions is nice too have. (gamma, beta function, Pochhammer symbol).
The criterion itself is not hard to understand. It imposes a condition on a pole-free half-space of the inverse Riemann zeta function. In this talk we will generalize the Baez-Duarte criterion to arbitrary Dirichlet series in one variable
19.02.2014 - Tomos Parry: A variance for primes in arithmetic progressions
Abstract: One approximates the number of primes up to x in an arithmetic progression $a+nq, (a,q)=1$, using the classical approximation $x φ (q)$. The error term is then $O (x \log x)$ and it is hard to improve on this. On average, however, much more can be said, and therefore people study the variance - the average over all the progressions with moduli up to some $Q≤ x$ of the error term squared. We then have the Barban-Davenport-Halberstam Theorem, which gives an upper bound for the variance, and then we have Montgomery's and also Hooley's refinements of it to an asymptotic formula. I will talk about a paper of Vaughan where he uses a new approximation instead of the classical one, and studies the variance with this new approximation. The main term is then no bigger than the main term for the classical variance and, moreover, the error term is better as $Q$ approaches $x$.
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