We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.

The aim of this work is to prove new results on a class of digital functions with special emphasis on shifted primes as arguments. Our method lies on the estimate of exponential sums of the form $\sum_{n\leq x}\Lambda (n)\exp(2i\pi f(n+c_n)+\beta n)$ where $f$ a digital function, $\mathbf{c}=(c_n)$ is an almost-periodic sequence in $ \mathbb{Z}$ and $\beta $ is a real parameter, which extend the works of Mauduit-Rivat \cite{mr1} and Martin-Mauduit-Rivat \cite{mmr} to the case of the shifted prime numbers satisfying a digital constraint.

If $ P(z)=a_nz^n+\sum_{j=\mu}^{n}a_{n-j}z^{n-j}$, $1\leq\mu\leq n$, is a polynomial of degree $n$ having all its zeros in $|z|\le k$, $k\leq 1$, then it was recently claimed by K. K. Dewan, Naresh Singh, Abdullah Mir [\textit{Extensions of some polynomial inequalities to the polar derivative}, {J. Math. Anal. Appl.} \textbf{352} (2009), 807--815] that for every real or complex number $\alpha$, with $|\alpha|\geq k^\mu$, \begin{align*} \underset{\left|z\right|=1}{\max}\left|D_\alpha P(z)\right|&\geq \frac{n\left(|\alpha|-k^\mu\right)}{1+k^\mu}\underset{\left|z\right|=1}{\max}\left|P(z)\right|+\frac{n\left(|\alpha|+1\right)}{k^{n-\mu}\left(1+k^\mu\right)}m\\ &\quad +n\left(\dfrac{k^\mu-A_\mu}{1+k^\mu}\right)\underset{\left|z\right|=1}{\max}|P(z)|+\dfrac{n(A_\mu-k^\mu)}{k^n(1+k^\mu)}m \end{align*} where $m=\min_{|z|=k}|P(z)|$, $D_\alpha P(z)$ is a polar derivative of $P(z)$ with respect to the point $\alpha\in\mathbb{C}$ and $A_\mu$ is given by (1.11). The proof of this result is not correct. In this paper, we present certain more refined results which not only provides a correct proof of above inequality as a special case but also yields a refinement of above and other related result.

We show that twisted monomial Gauss sums modulo prime powers can be evaluated explicitly once the power is sufficiently large.

In this paper, we shall investigate several properties of integrals defined by $\int_1^{\infty}t^{-\theta}P(t)\log^jtdt$ with a complex variable $\theta$ and a non-negative integer $j$, where $P(x)$ is the error term in the circle problem of Gauss. We shall also study the analytic continuation of several types of the Dirichlet series related with the circle problem, and study a proof of the functional equation of the Dedekind zeta-function associated with the Gaussian number field ${\mathbb{Q}}(\sqrt{-1})$.

Let $E/F$ be a non-abelian Galois extension of number fields of degree $q^{3}$. We give some expressions for the order of the Sylow $p$-subgroup of tame kernel of $E$ and some of its subfields containing $F$, where $p$ is a prime, $q$ is an odd prime, $p\neq q$. As applications, we give some results about the orders of the Sylow $p$-subgroups of tame kernels when $E/mathbb{Q}(\zeta_{3})$ is a Galois extension of number fields with non-abelian Galois group of order $27$.

We study the problem of the existence of a true exceptional set for an asymptotic formula, that is a minimal set --- up to finite modifications --- such that the asymptotic formula holds outside such a set. We give an analytic and a descriptive set theoretic characterisations for the existence of a true exceptional set, which we then apply by showing the non-existence of a true exceptional set in some well known situations. We prove in fact that, both from a category and a measure theoretic points of view, most asymptotic formulas do not have a true exceptional set.

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm [20] gave an upper bound for modular forms of a given weight and level. This was adapted by Ram Murty [16] and Ghitza [5] to the case of two eigenforms of the same level but having potentially different weights. We consider their expansions modulo a prime ideal, presenting a new bound. In the process of analysing this bound, we generalise a result of Bach and Sorenson [2], who provide a practical upper bound for the least prime in an arithmetic progression.

Under the assumption of the Riemann Hypothesis, we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.

Let $h \geq 2$ be an integer. We say that a set $\mathcal{A}$ of positive integers is an asymptotic basis of order $h$ if every large enough positive integer can be represented as the sum of $h$ terms from $\mathcal{A}$. A set of positive integers $\mathcal{A}$ is called a Sidon set if all the sums $a+b$ with $a,b \in \mathcal{A}$, $a \leq b$ are distinct. In this paper we prove the existence of Sidon set $\mathcal{A}$ which is an asymptotic basis of order $4$ by using probabilistic methods.

In this article, we give a complete description of the characteristic polynomials of simple supersingular abelian varieties over finite fields. We list them for the dimensions up to $7$