Let $A$ and $\tilde{A}$ be Hilbert--Schmidt operators. For a constant $r>0$, let $i_+(r, A)$ be the number of the eigenvalues of $A$ taken with their multiplicities lying in the half-plane $\{z\in\mathbb{C}: \Re z>r\}$. We suggest the conditions that provide the equality $i_+(r, \tilde{A})=i_+(r, A)$.

We explore Fourier transforms of the reciprocal of the Riemann zeta function that have connections to the RH. A partial answer to a recently posed problem is explored by exploiting the fact that $\zeta(s)\neq0$ when $\Re(s)=1$.

Let $F \in \mathbb{R}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d > 1$ satisfying $(n - \dim V_{F}^*) > 4$, where $V_F^*$ is the singular locus of $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. Suppose there exists $\mathbf{x}_0 \in (0,1)^n \cap (V(F) \backslash V_F^*)$. Let $\mathbf{t} = (t_1, \ldots, t_n) \in \mathbb{R}^n$. Then for a smooth function $\varpi:\mathbb{R}^n \rightarrow \mathbb{R}$ with its support contained in a small neighbourhood of $\mathbf{x}_0$, we prove $$ \Big{|} \int_{0}^{\infty} \cdots \int_{0}^{\infty} \varpi(\mathbf{x}) x_1^{i t_1} \cdots x_n^{i t_n} e^{2 \pi i \tau F(\mathbf{x})} d \mathbf{x} \Big{|} \ll \min \{ 1, |\tau|^{-1} \}, $$ where the implicit constant is independent of $\tau$ and $\mathbf{t}$.

In our recent paper On sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$ (Functiones et Approximatio Commentarii Mathematici 57 (2017), 255--275), a factor $d^{2/3}$ was missing in Theorem 1. We show how to modify the proofs of the other two main results so that their conclusions are not affected by the extra factor.

We prove a joint universality theorem for two Dirichlet series associated with periodic arithmetical functions, resp. their analytic continutations. This result is applied to prove a uniqueness theorem which in particular shows that Hurwitz zeta-functions with a rational parameter do not share any complex value.

We establish a local-global principle for linear dependence of points in Mordell--Weil groups of abelian varieties over number fields. We give a complete characterization, in terms of a relation between the rank and the dimension, of abelian varieties with endomorphism ring equal to $\mathbb{Z}$ for which the principle holds. In the case of elliptic curves we prove the result in full generality, i.e., without the assumption on the endomorphism ring.

For a positive integer $n$, let $K_n$ be the $n$-th layer of the the cyclotomic $\mathbf{Z}_2$-extension of $\mathbf{Q}(\sqrt{5})$, which is the real quadratic field with the minimal discriminant. We prove that the class number of $K_5$ is 1.

In this paper, we study the critical orbit of a post-critically finite polynomial of the form $f_{c,d}(x) = x^d+c \in \mathbb{C}[x]$. We discover that in many cases the orbit elements satisfy some strong arithmetic properties. It is well known that the $c$ values for which $f_{c,d}$ has tail size $m\geq 1$ and period $n$ are the roots of a polynomial $G_d(m,n) \in \mathbb{Z}[x]$, and the irreducibility or not of $G_d(m,n)$ has been a great mystery. As a consequence of our work, for any prime $d$, we establish the irreducibility of these $G_d(m,n)$ polynomials for infinitely many pairs $(m,n)$. These appear to be the first known such infinite families of $(m,n)$. We also prove that all the iterates of $f_{c,d}$ are irreducible over $\mathbb{Q}(c)$ if $d$ is a prime and $f_{c,d}$ has a fixed point in its post-critical orbit.

Given a degree 1 function $F\in\mathcal{S}^{\sharp}$ and a real number $\alpha$, we consider the linear twist $F(s,\alpha)$, proving that it satisfies a functional equation reflecting $s$ into $1-s$, which can be seen as a Hurwitz-Lerch type of functional equation. We also derive some results on the distribution of the zeros of the linear twist.

Miller and F. Zhou have proved a balanced Voronoï summation formula for GL$_N$ over $\mathbb{Q}$, which allows one to control the dimensions of the Kloosterman sums appearing on either side of the Voronoï formula. In this note, we prove a balanced Voronoï formula over an arbitrary number field, starting with the Voronoï summation formula of A. Ichino and N. Templier over number fields, allowing one to extend recent results on spectral reciprocity laws to number fields, in special cases.

In [1], Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such that any set $E\subset \mathbb{F}_q^d$ with $|E|\gtrsim q^s$ determines $\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations for $k\leq d$. Because congruence classes can be identified with tuples of distances between distinct points when $k\leq d$, and because there are $\binom{k+1}{2}$ such pairs, this means any such $E$ determines a positive proportion of all congruence classes. In the $k>d$ case, fixing all pairs of distnaces leads to an overdetermined system, so $q^{\binom{k+1}{2}}$ is no longer the correct number of congruence classes. We determine the correct number, and prove that $|E|\gtrsim q^s$ still determines a positive proportion of all congruence classes, for the same $s$ as in the $k\leq d$ case.