In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same method, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials. Moreover, we establish a connection between these results and the generalized Dedekind sums and Hardy--Berndt sums.

We have recently proved several explicit versions of the prime ideal theorem under GRH. Here we further explore the method, in order to deduce its strongest consequence for the case where $x$ diverges.

Let $\mathbb{F}_p$ be a finite field of size $p$ where $p$ is an odd prime. Let $f(x)\in\mathbb{F}_p[x]$ be a~polynomial of positive degree $k$ that is not a $d$-th power in $\mathbb{F}_p[x]$ for all $d\mid p-1$. Furthermore, we require that $f(x)$ and $x$ are coprime. The main purpose of this paper is to give an estimate of the number of pairs $(\xi,\xi^\alpha f(\xi))$ such that both $\xi$ and $\xi^\alpha f(\xi)$ are primitive roots of $p$ where $\alpha$ is a given integer. This answers a question of Han and Zhang.

In this note, we prove that the tower given by Bassa, Garcia and Stichtenoth in [4] is a subtower of the one given by Anbar, Beelen and Nguyen in [2]. This completes the study initiated in [16,2] to relate all known towers over cubic finite fields meeting Zink's bound with each other.

Hardy and Littlewood conjectured that every sufficiently large integer is either a square or the sum of a prime and a square. Let $E(X)$ be the number of positive integers up to $X\ge4$ for which this property does not hold. We prove \[E(X)\ll X^{1/2}(\log X)^A(\log\log X)^4\] with $A=3/2$ under the Generalized Riemann Hypothesis. This is a small improvement on the previous remarks of Mikawa (1993) and Perelli-Zaccagnini (1995) which claim $A=4,3$ respectively.

We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem, cp. [10] to this setting, we apply the polynomial large sieve inequality that has been proved in [7] and which includes recent progress in Vinogradov's mean value theorem due to Parsell \emph{et al.} in [9]. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.

Let $p=2\ell^f+1$ be a prime number with $f \geq 2$ and an odd prime number $\ell$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ with $[K_t : \mathbb{Q}]=2\ell^t$. Denote by $h_{p,t}^-$ the relative class number of $K_t$, and by $h_{p,t}^+$ the class number of the maximal real subfield $K_t^+$. It is known that the ratio $h_{p,f}^-/h_{p,f-1}^-$ is odd (and hence so is $h_{p,f}^+/h_{p,f-1}^+$) whenever $2$ is a primitive root modulo $\ell^2$. We show that $h_{p,f}^+/h_{p,f-1}^+$ is odd under a somewhat milder assumption on $\ell$ and that the ratio $h_{p,f-1}^-/h_{p,f-2}^-$ is always odd when $\ell=3$.

We compute the first moment of twisted Hecke $L$-functions of weight $2$ and prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.

Let $K$ be a number field and suppose $L/K$ is a finite Galois extension. We establish a bound for the least prime ideal occurring in the Chebotarev Density Theorem. Namely, for every conjugacy class $C$ of $\mathrm{Gal}(L/K)$, there exists a prime ideal $\mathfrak{p}$ of $K$ unramified in $L$, for which its Artin symbol $\big[ \frac{L/K}{\mathfrak{p}} \big] = C$, for which its norm $N^K_{\mathbb{Q}}\mathfrak{p}$ is a rational prime, and which satisfies \[ N^K_{\mathbb{Q}} \mathfrak{p} \ll d_L^{40}, \] where $d_L = |\mathrm{disc}(L/\mathbb{Q})|$. All implicit constants are effective and absolute.