We provide systematic evaluations, in terms of binary quadratic representations of $4p$, for the $p$-th Fourier coefficients of each member $f$ of an infinite class $\mathcal{C}$ of CM eigenforms. As an application, previously conjectured evaluations of three algebro-geometric character sums can now be formulated explicitly without reference to eigenforms. There are several non-CM newforms that appear to share some properties with the eigenforms in $\mathcal{C}$, and we pose some conjectures about their Fourier coefficients.

Let $A$ be a finite multiset of integers. If $B$ be a multiset such that $A$ and $B$ are $t$-complementing multisets of integers, then $B$ is periodic. We obtain the Biro-type upper bound for the smallest such period of $B$: Let $\varepsilon>0$. We assume that $diam(A)\ge n_0(\varepsilon)$ and that $\sum_{a\in A}w_A(a)\leq (diam(A)+1)^{c}$, where $c$ is any constant such that $c< 100\log2-2$. Then $B$ is periodic with period $\log k\leq (diam(A)+1)^{\frac{1}{3}+\varepsilon}$.

We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function over the zeros of $\zeta(s)$ that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.

We propose a simple measure of how slowly a bounded real sequence clusters. This measure, called \emph{separation}, is the infimum, over all finite segments of the sequence with at least two terms, of the ratio of the least distance between the terms in the segment to the general supremum of such a distance for a segment of that length. An example of a highly separated sequence is given. To create a more separated sequence, we modify van der Corput's construction, replacing the powers of a base by the even-numbered terms of the Fibonacci sequence. The result coincides initially with the sequence built stepwise by maximizing separation for each extra term. We conjecture that these sequences are the same and of maximal separation.

For a finite set $S$ of local primes of a countable Hilbertian field $K$ and for $\sigma_1,\ldots,\sigma_e\in\Gal(K)$ we denote the field of totally $S$-adic numbers by $\K_{tot,S}$, the fixed field of $\sigma_1,\ldots,\sigma_e$ in $\K_{tot,S}$ by $\K_{tot,S}({\mathbf \sigma})$, and the maximal Galois extension of $K$ in $\KtotS({\mathbf \sigma})$ by $\KtotS[{\mathbf \sigma}]$. We prove that for almost all ${\mathbf \sigma}\in\Gal(K)^e$ the absolute Galois group of $\K_{tot,S}[{\mathbf \sigma}]$ is isomorphic to the free product of $\hat{F}_\omega$ and a free product of local factors over $S$.

In classical prime number theory, several relations are considered to be equivalent to the Prime Number Theorem. For Beurling generalized numbers, some auxiliary conditions may be needed to deduce one relation from another one. We show conditions under which the Beurling analog of the sharp version of Mertens' sum formula does or does not hold.

In order to prove a result on fourth moments of modular L-functions, Duke derived an identity of the divisor function over the rational Hamiltonian quaternion algebra. Recently Kim and the author generalized Duke's divisor function, the identity and other related results from level two to general prime level. In this note, we consider such identities in general.

For a positive integer $n$, let $\sigma(n):= \sum_{d \in \mathb{N}, d|n} d$. The explicit evaluation of such arithmetic sums as $\sum_{(a,b,c) \in \ABIFnn^3, a+2b+4c=n} \sigma(a)\sigma(b) \sigma(c)$ and $\sum_{(a,b) \in \ABIFnn^2, a+2b=n} a \sigma(a)\sigma(b)$ is carried out for all positive integers $n$.

We give an explicit version of a result due to D. Burgess. Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $\chi$ takes on a particular value is less than $\left\{\frac{\pi e\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.