Let $b>a>0$. We prove the following asymptotic formula $$\sum _{n⩾0}|\{x/(n+a)\}-\{x/(n+b)\}|=\frac{2}{\pi }\zeta (3/2)\sqrt{cx}+O\left(c^{2/9}{x}^{4/9}\right),$$ with $c=b-a$, uniformly for $x⩾40c^{-5}{(1+b)}^{27/2}$.

In the paper the asymptotics for Dirichlet polynomials associated to certain cusp forms are obtained.

A formula for the mean value of multiplicative functions associated to certain cusp forms is obtained. The paper is a continuation of [4].

If the counting function N(x) of integers of a Beurling generalized number system satisfies both ${\int }_1^{\infty }{x}^{-2}|N\left(x\right)-Ax|dx<\infty$ and $x^{-1}\left(logx\right)(N\left(x\right)-Ax)=O\left(1\right)$, then the counting function π(x) of the primes of this system is known to satisfy the Chebyshev bound π(x) ≪ x/logx. Let f(x) increase to infinity arbitrarily slowly. We give a construction showing that ${\int }_1^{\infty }|N\left(x\right)-Ax|x^{-2}dx<\infty$ and $x^{-1}\left(logx\right)(N\left(x\right)-Ax)=O\left(f\left(x\right)\right)$ do not imply the Chebyshev bound.

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.