We discuss evaluating fractional Stieltjes constants $\gamma_{\alpha}(a)$, arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions $\zeta^{(\alpha)}(s,a)$. We give an upper bound for the absolute value of $C_\alpha(a)=\gamma_\alpha(a)-\log^\alpha(a)/a$ and an asymptotic formula $\widetilde{C}_{\alpha}(a)$ for $C_{\alpha}(a)$ that yields a good approximation even for most small values of $\alpha$. We bound $|\widetilde{C}_{\alpha}(a)|$ and based on this we conjecture a tighter bound for $|C_\alpha(a)|$.

We establish the precise analogue of a large sieve inequality due to L. Zhao with special characters to prime square moduli for Gaussian integers.

Recently, Zhang and Lv find an asymptotic formula for the fourth power mean of the general $3$-dimensional Kloosterman sum mod $p$. In this article we prove an improvement of their asymptotic formula.

Let $a_1,\cdots,a_6$ be non-zero integers satisfying $(a_i,a_j)=1, 1\leq i \lt j \leq 6$ and $b$ be any integer. For the Diophantine equation $a_1p_1+a_2p_2^3+\cdots+a_6p_6^3=b$ we prove that (i) if all $a_1,\cdots,a_6$ are positive and $b\gg \max \{|a_j|\}^{34+\varepsilon}$, then the equation is soluble in primes $p_j$, and (ii) if $a_1,\cdots,a_6$ are not all of the same sign, then the equation has prime solutions satisfying $\max \{ p_1,p_2^3,\cdots,p_6^3 \}\ll |b|+\max \{|a_j|\}^{33+\varepsilon}$, where the implied constants depend only on $\varepsilon$.

This paper is a corrigendum to the article ``On the ideal theorem for number fields''. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.

Uniformly for small $q$ and $(a,q)=1$, we obtain an estimate for the weighted number of ways a sufficiently large integer can be represented as the sum of a prime congruent to $a$ modulo $q$ and a square-free integer. Our method is based on the notion of local model developed by Ramaré and may be viewed as an abstract circle method.

The dual spaces $d(p), 1 \lt p \lt \infty ,$ of the discrete Cesàro (Banach) spaces $\mathrm{ces} (q)$, $1 \lt q \lt \infty ,$ were studied by G. Bennett, A. Jagers and others. These (reflexive) dual Banach spaces induce the non-normable Fréchet spaces $d (p+) := \bigcap_{r \gt p} d (r), $ for $1 \leq p \lt \infty ,$ and the (LB)-spaces $d (p-) := \bigcup_{ 1 \lt r \lt p } d (r), $ for $ 1 \lt p \leq \infty ,$ recently introduced and investigated in [11]. Here a detailed study is made of various aspects, such as the spectrum, continuity, compactness, mean ergodicity and supercyclicity of the Cesàro operator, multiplication operators and inclusion operators when they act on (and between) such spaces.