Let $a(n)$ and $b(n)$ be arithmetic sequences, and $$A(s)=\sum_{n\ge1}a(n)n^{-s}, \qquad B(s)=\sum_{n\ge1}b(n)n^{-s},$$ be the two Dirichlet series related by a certain functional equation. Let $m$ be the \emph{analytic degree} of the functional equation. For $x>0$ and a positive integer $N$, Friedlander and Iwaniec (2005) define the sharply truncated nonlinear dual sum $$\mathcal B_{\ell,D}(x,N) := \sum_{\substack{n\in\mathbb N\\ n\le N}} b(n)n^{-\beta_m} \cos\left( 2\pi m\left(\frac{nx}{D}\right)^{1/m} +\frac{\pi\ell}{4} \right),$$ where $D\ge1$ is the conductor, $\beta_m:=\frac{m+1}{2m}$, and $\ell=m-3-2k$ is determined by the archimedean weight $k$ of the functional equation. Their Conjecture 1 predicts that, for every $\varepsilon>0$, $$\mathcal B_{\ell,D}(x,N) \ll_{\varepsilon,\boldsymbol\kappa} (DNx)^\varepsilon,$$ uniformly in the variables $x$ and $N$, with the degree, conductor, and archimedean datum fixed. We give counterexamples to this prediction with $$A(s)=B(s)=\zeta(s)^m,\; m\geq 4$$ where $$\zeta(s):=\sum_{n\ge1}n^{-s} \qquad(\operatorname{Re}s>1)$$ is the Riemann zeta function.
For integers $r\ge2$ and $n\ge1$, let $$ F_{r,n}(q)=\prod_{k=1}^{n}\frac{1-q^{rk}}{1-q^k}. $$ The coefficient of $q^j$ in \(F_{r,n}(q)\) counts partitions of $j$ into parts at most $n$, each occurring at most $r-1$ times. Hughes proved that $F_{2,n}(q)=\prod_{k=1}^{n}(1+q^k)$ is unimodal for every $n\ge 1$. This result...
For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusi...
Omkar Baraskar, Prashant Gokhale, Sarvagya Jain et al.· 0 citations
Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ and let $f\colon\mathbb R^n\to\mathbb R^k$ be a polynomial mapping of degree at most $d$. We show that \[ \mu(f\in A) \le C\bigl(\lambda_k(A)\bigr)^{\frac{1}{k(d-1)+1}} \] for every Borel set $A\subset\mathbb R^k$ whenever the image measure $\mu\circ f^{-1...
Let $(a_n)_{n\geqslant 0}$ be a sequence of integers. Its dual sequence $(a_n^*)_{n\geqslant 0}$ is defined by \begin{equation*} a_n^* := \sum_{k=0}^{n} \binom{n}{k}(-1)^k a_k. \end{equation*} Let $p>3$ be a prime. In this paper we mainly investigate congruences modulo $p^2$ involving central binomial coefficients and...
Let $\mathcal N$ be a von Neumann algebra equipped with a normal faithful semifinite trace, and let $\gamma$ be a trace-preserving automorphism of $\mathcal N$. We consider the ergodic averages along the prime numbers \[ A_N(x) := \frac1{|P_N|} \sum_{q\in P_N}\gamma^q(x), \qquad P_N:=\{q\leq N:q\ \text{is prime}\}. \]...
Fix an integer $K\ge2$, and let $C_K^n =\{(e^{\frac{2\pi ij}{K}})_{j=0}^{K-1}\}^n$ be the product of cyclic groups of order $K$. For a Fourier character $\chi_\alpha$, let $s(\alpha)$ be the number of active coordinates. We give a self-contained proposed proof that the dimension-free Bohnenblust--Hille constants govern...
Joseph Slote, Alexander Volberg· 2 citations
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