SCHWEITZER 2016

For which complex numbers $\alpha$ does there exist a completely multiplicative, complex-valued arithmetic function $f$ such that $$\sum_{n<x}f(n)=\alpha x+O(1)\,\,?$$

Let $K=(V,E)$ be a finite, simple, complete graph. Let $d$ be a positive integer. Let $\phi:E\to \mathbb{R}^d$ be a map from the edge set to Euclidean space, such that the preimage of any point in the range defines a connected graph on the entire vertex set $V$ , and the points assigned to the edges of any triangle in $K$ are collinear. Show that the range of $\phi$ is contained in a line.

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$ , there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$ .

Prove that there exists a sequence $a(1),a(2),\dots,a(n),\dots$ of real numbers such that $$a(n+m)\le a(n)+a(m)+\frac{n+m}{\log (n+m)}$$ for all integers $m,n\ge 1$ , and such that the set $\{a(n)/n:n\ge 1\}$ is everywhere dense on the real line.

Remark. A theorem of de Bruijn and Erdős states that if the inequality above holds with $f(n + m)$ in place of the last term on the right-hand side, where $f(n)\ge 0$ is nondecreasing and $\sum_{n=2}^\infty f(n)/n^2<\infty$ , then $a(n)/n$ converges or tends to $(-\infty)$ .

Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$ , $n\in\mathbb{Z}$ , there exists an $x\in\mathbb{R}$ with $$\limsup_{K\to \infty} \frac{\#\{k\le K\,: k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n$$ for all $n\in\mathbb{Z}$ , where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$ -fold iterate of $f$ ?

Let $\Gamma(s)$ denote Euler's gamma function. Construct an even entire function $F(s)$ that does not vanish everywhere, for which the quotient $F(s)/\Gamma(s)$ is bounded on the right halfplane $\{\Re(s)>0\}$ .

Show that the unit sphere bundle of the $r$ -fold direct sum of the tautological (universal) complex line bundle over the space $\mathbb{C}P^{\infty}$ is homotopically equivalent to $\mathbb{C}P^{r-1}$ .

For which integers $n>1$ does there exist a rectangle that can be subdivided into $n$ pairwise noncongruent rectangles similar to the original rectangle?

For $p_0,\dots,p_d\in\mathbb{R}^d$ , let $$S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}.$$ Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$ , and choose $p_0,\dots,p_d$ independently with distribution $\pi$ . Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$ .

Let $X$ and $Y$ be independent, identically distributed random points on the unit sphere in $\mathbb{R}^3$ . For which distribution of $X$ will the expectation of the (Euclidean) distance of $X$ and $Y$ be maximal?