The estimations of lower order $\left(R\right)$ $\lambda$ in terms of the sequences $\left\{a_n\right\}$ and $\left\{{\lambda }_n\right\}$ for an entire Dirichlet series $f\left(s\right)={\sum }_{n=1}^{\infty }{a}_n{e}^{s\lambda n}$, have been obtained, namely : $$\begin{array}{cc}\hfill \lambda & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{{\lambda }_{n_p}log{\lambda }_{n_{p-1}}}{log|a_{n_p}{|}^{-1}}\hfill \\ & =\underset{\left\{{\lambda }_{n_p}\right\}}{max}lim\underset{p\to \infty }{inf}\frac{({\lambda }_{n_p}-{\lambda }_{n_{p-1}})log{\lambda }_{n_{p-1}}}{log\left|a_{n_{p-1}}\right|a_{n_p}|}.\hfill \end{array}$$ One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

In this paper, we give certain upper bounds for the 2k-th moments, k ≥ 1/2, of derivatives of Dirichlet L-functions at s = 1/2 under the assumption of the Generalized Riemann Hypothesis.

We prove that for any real $\theta$ there are infinitely many values of $s=\sigma +it$ with $\sigma \to 1+$ and $t\to +\infty$ such that $$\Re \{exp\left(i\theta \right)logL(s,\chi )\}\ge log\frac{logloglogt}{loglogloglogt}+O\left(1\right).$$ The proof relies on an effective version of Kronecker’s approximation theorem.

Under the Generalized Riemann Hypothesis, it is proved that for any $k\ge 200$ there is $N_k\>0$ depending on $k$ only such that every even integer $\ge N_k$ is a sum of two odd primes and $k$ powers of $2$.

For a positive integer $n$ we write $\phi \left(n\right)$ for the Euler function of $n$. In this note, we show that if $b>1$ is a fixed positive integer, then the equation $$\phi \left(x\frac{b^n-1}{b-1}\right)=y\frac{b^m-1}{b-1},\text{where}x,y\in \{1,...,b-1\},$$ has only finitely many positive integer solutions $(x,y,m,n)$.