L’article donne des réponses optimales ou presque optimales aux questions suivantes, qui remontent à Stieltjes, Landau et Bohr, et concernent des séries de Dirichlet $A_j=\sum _{n=1}^{\infty }a(j,n)n^{-s}$$(j=1,2...$

On étudie sommairement la distribution des valeurs de $L^{\text{'}}(1,\chi )$ ($\chi$ : caractère de Dirichlet primitif réel) et on constate qu’on a en général $L^{\text{'}}(1,\chi )\<{\pi }^2/6$; on démontre par ailleurs que si $L^{\text{'}}(1,\chi )\<({\pi }^2/6)-ϵ$, alors$L(1,\chi )\>c\left(ϵ\right)/logk$ ($k$ : conducteur de $\chi$; $c\left(ϵ\right)$: constante positive effectivement calculable.

Mertens’ product formula asserts that$$\prod _{p\le x}\left(1-\frac{1}{p}\right)logx\to e^{-\gamma }$$as $x\to \infty$. Calculation shows that the right side of the formula exceeds the left side for $2\le x\le {10}^8$. It was suggested by Rosser and Schoenfeld that, by analogy with Littlewood’s result on $\pi \left(x\right)-\mathrm{li}x$, this and a complementary inequality might change their sense for sufficiently large values of $x$. We show this to be the case.

Let $J_k\left(n\right):=n^k{\prod }_{p\mid n}(1-p^{-k})$ (the $k$-th Jordan totient function, and for $k=1$ the Euler phi function), and consider the associated error term$$E_k\left(x\right):=\sum _{n\le x}{J}_k\left(n\right)-\frac{x^{k+1}}{(k+1)\zeta (k+1)}.$$When $k\ge 2$, both $i_k:=E_k\left(x\right)x^{-k}$ and $s_k:=lim\; sup{E}_k\left(x\right)x^{-k}$ are finite, and we are interested in estimating these quantities. We may consider instead$$Ik:=\underset{n\in \mathbb{N},n\to \infty }{lim\; inf}$$d 1 (d)dk ( 12 - { nd} ), since from [AS] $i_k=I_k-{\left(\zeta (k+1)\right)}^{-}1$ and from the present paper $s_k=-i_k$. We show that $I_k$ belongs to an interval of the form$$\left(\frac{1}{2\zeta \left(k\right)}-\frac{1}{(k-1)N^{k-1}},\frac{1}{2\zeta \left(k\right)}\right),$$where $N=N\left(k\right)\to \infty$ as $k\to \infty$. From a more practical point of view we describe an algorithm capable of yielding arbitrary good approximations of $I_k$. We apply this algorithm...

In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is √n at prime arguments.