In this paper, we consider the $k$-th Riesz mean for the coefficients of the Rankin--Selberg $L$-function $L_{f\times f}(s)$ related to the Godement--Jacquet $L$-function with respect to $SL(n,\mathbb{Z})$. We establish an asymptotic formula for the $k$-th Riesz mean with an improved range and a better error term. As a result, we get an asymptotic relation for the partial sum of the coefficients of $L_{f \times f}(s)$.

We study the étale cohomology of generalized Suzuki curves in characteristic two as representations of their automorphism groups. We calculate the number of rational points and the zeta functions of the curves. As an application, we obtain several maximal curves.

In this paper, we show that a weak form of Greenberg's conjecture for the cyclotomic $\mathbb{Z}_{2}$-extension of $\mathbb{Q}(\sqrt{p})$ is true, where $p$ is an odd prime number satisfying certain arithmetic conditions. As an application, we obtain an upper bound of the Iwasawa $\lambda$-invariant of the cyclotomic $\mathbb{Z}_{2}$-extension of $\mathbb{Q}(\sqrt{p})$ under some additional conditions on $\mathbb{Q}(\sqrt{p})$.

Let $p$ be a prime and $K$ be an arbitrary field which can be embedded into both $\mathbb{R}$ and $\mathbb{Q}_p$.We propose an algorithm that generates $p$-adic continued fraction expansions for elements in $K$ that converge in $\mathbb{Q}_p$.This algorithm produces finite continued fraction expansions for rational numbers.In the case of $p=2$ and if $K$ is a quadratic field, the continued fraction expansions for irrational elements in $K$ generated by this algorithm converge in $\mathbb{R}$, and they are eventually periodic.For an element $\alpha$ in $K$, let $p_n/q_n$ denote the $n$-th convergent.There exist constants $u_1$ and $u_2$ in ${\mathbb{R}}_{>0}$ with $u_1 + u_2 = 2$, and constants $C_1$ and $C_2$ in ${\mathbb{R}}_{>0}$such that $|\alpha - p_n/q_n| < C_1/|q_n|^{u_1}$ and $|\alpha - p_n/q_n|_2 < C_2/|q_n|^{u_2}$. Here, $|\cdot|_2$ represents the $2$-adic distance. For prime numbers $p > 2$, we present numerical experiences.

Since 1772, when Euler first described two methods of obtaining two pairs of biquadrates with equal sums, several methods of solving the diophantine equation $x^4+y^4=z^4+w^4$ have been published. All these methods yield parametric solutions in terms of homogeneous bivariate polynomials of odd degrees. In this paper we describe a method that yields three parametric solutions of the aforesaid diophantine equation in terms of homogeneous bivariate polynomials of even degrees, namely degrees $74$, $88$ and $132$ respectively.

Let $S := \{p_1,\ldots ,p_{\ell}\}$ be a finite set of primes and $\mathcal{U}_S$ be the set of all rational integers whose prime factors are all in $S$. Let $(U_n)_{n\geq 0}$ be a non-degenerate linear recurrence sequence with order at least two. In this paper, we extend the results of [1, 3]. In particular, for fixed positive integers $\la_1, \la_2,\ldots, \la_d$, we provide a finiteness result for the solutions of the Diophantine equation $\la_{1}U_{n_1} + \cdots + \la_{d}U_{n_d}= z_1 +\cdots +z_r,$ in non-negative integers $n_1, \ldots, n_d$ and rational integers $z_1, \ldots, z_r$, where $n_1 > n_2 > \dots > n_d,$ and $z_1, \ldots, z_r\in \mathcalU_S$.

Gonek, Graham, and Lee showed the Riemann Hypothesis (RH) can be reformulatedin terms of estimates for twisted sums $\sum_{n\le x}\Lambda(n) n^{-iy}$,where $\Lambda$ is the von Mangoldt function. Using an approachbased on the Perron formula but with a modified contour, we strengthentheir results. We also establish two generalizations,where $\Lambda$ in the twisted sumsis replaced by the generalized von Mangoldt function $$\Lambda_k(n):=\sum_{d\mid n}\mu(d)\Big(\log\frac{n}{d}\Big)^k$$ or the $k$-fold convolution $$\Lambda^k:=\mathop{\underbrace{\Lambda\star\cdots\star\Lambda}}\limits_{k\text{~copies}}.$$ Our results show that the validity of RH is governed by not only the distribution of prime numbers, but also by the distributionof almost-primes, i.e., natural numbers that have no more than$k$ distinct prime divisors.

Due to their singularities, multiple zeta functions behave sensitively at non-positive integer points. In this article, we focus on the asymptotic behavior at the origin $(0,\dots, 0)$ and unveil the generating series of the asymptotic coefficients as a generalization of the classical Gregory coefficients. This enables us to reveal the underlying symmetry of the asymptotic coefficients. Additionally, we extend the relationship between the asymptotic coefficients and the Gregory coefficients to include Hurwitz multiple zeta functions.

In this paper, for any number field $K$ generated by a root $\alpha$ of a monic irreducible trinomial $F(x)=x^{11}+ax^3+b \in \mathbbZ[x]$ and for every rational prime $p$, we characterize when $p$ divides the index of $K$. We also describe the prime power decomposition of the index $i(K)$. In such a way we give a partial answer of Problem $22$ of Narkiewicz ([20]) for this family of number fields. As an application of our results, if $i(K)\neq1$, then $K$ is not monogenic. We illustrate our results by some computational examples.

For $\alpha>1$, set $\beta=1/(\alpha-1)$. We show that, for every $1<\alpha<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^\alpha}\rfloor - \lfloor{m^\alpha}\rfloor$ is equal to $\beta\alpha^{-\beta}\zeta(\beta)d^{\beta-1} + o(d^{\beta-1})$ as $d\to\infty$, where $\zeta$ denotes the Riemann zeta function. We use this result to derive an asymptotic formula for the number of triplets $(l,m,n)$ of positive integers such that $l<x$ and $\lfloor{l^\alpha}\rfloor + \lfloor{m^\alpha}\rfloor = \lfloor{n^\alpha}\rfloor$. Furthermore, we prove that the additive energy of the sequence $(\lfloor{n^\alpha}\rfloor)_{n=1}^N$, i.e., the number of quadruples $(n_1,n_2,n_3,n_4)$ of positive integers with $\lfloor{n_1^\alpha}\rfloor+\lfloor{n_2^\alpha}\rfloor=\lfloor{n_3^\alpha}\rfloor+\lfloor{n_4^\alpha}\rfloor$ and $n_1,n_2,n_3,n_4\le N$, is equal to $O_\alpha(N^{4-\alpha})$ when $1<\alpha\le4/3$.

Let $\zeta(s)$ be the Riemann zeta function and $\rho=\beta+i\gamma$ its nontrivial zero. In 1984 S.M. Gonek started to consider the mean value $\sum_{0<\gamma<T}\zeta(\rho+\delta)\zeta(1-\rho+øverline{\delta})$ for imaginary $\delta$. Here we obtain asymptotic formulas for this mean value for general complex $\delta$.

We prove a reciprocity formula that relates a spectral average of products of triple product integrals involving automorphic forms of weights $0$ and $1/2$ to the classical Rankin-Selberg integrals for automorphicforms of weight $0$.