It is proved that the sequence $\left[n^c\right](n=1,2,\cdots )$ contains infinite squarefree integers whenever $1\<c\<\frac{61}{36}=1.6944\cdots$, which improves Rieger’s earlier range $1\<c\<1.5$.

For a number field $K$ with ring of integers ${𝒪}_K$, we prove an analogue over finite rings of the form ${𝒪}_K/{𝒫}^m$ of the fundamental theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where $𝒫$ is a big enough prime ideal of ${𝒪}_K$ and $m\>1$. In the appendix, F.Sato gives an application of the Theorems 1.1, 1.3 and the Theorems A, B, C in J.Denef and A.Gyoja [Character sums associated to prehomogeneous vector spaces, Compos. Math., 113(1998), 237–346] to the functional equation of $L$-functions...

We study the local properties of the time-dependent probability density function for the free quantum particle in a box, i.e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. √δ-families of initial functions are considered whose squared magnitudes approximate the periodic Dirac δ-function. The focus is on the set of rectilinear domains where the density has a special character, in particular,...

For a large odd integer N and a positive integer r, define b = (b₁,b₂,b₃) and $(N,r)=b\in \mathbb{N}³:1\le b_j\le r,(b_j,r)=1andb₁+b₂+b₃\equiv N\left(modr\right).$It is known that    $(N,r)=r²{\prod }_{p|r}p|N((p-1)(p-2)/p²){\prod }_{p|r}p\nmid N((p²-3p+3)/p²)$. Let ε > 0 be arbitrary and $R=N^{1/8-\epsilon }$. We prove that for all positive integers r ≤ R, with at most $O\left(Rlo{g}^{-A}N\right)$ exceptions, the Diophantine equation ⎧N = p₁+p₂+p₃, ⎨ $p_j\equiv b_j\left(modr\right),$ j = 1,2,3,$$⎩ with prime variables is solvable whenever b ∈ (N,r), where A > 0 is arbitrary.

1. Summary. In a sequence of three papers we study the circle problem and its generalization involving the logarithmic mean. Most of the deeper results in this area depend on estimates of exponential sums. For the circle problem itself Chen has carried out such estimates using three two-dimensional Weyl steps with complicated techniques. We make the same Weyl steps but our approach is simpler and clearer. Crucial is a good understanding of the Hessian determinant that appears and a simple...

1. Summary. In Part II we study arithmetic functions whose Dirichlet series satisfy a rather general type of functional equation. For the logarithmic Riesz mean of these functions we give a representation involving finite trigonometric sums. An essential tool here is the saddle point method. Estimation of the exponential sums in the special case of the circle problem will be the topic of Part III.

New bounds are given for the exponential sum$$\sum _{P\le p\<2P}^{}e\left(\alpha p^k\right)$$were $k\ge 5,p$ denotes a prime and $e\left(x\right)=exp\left(2\pi ix\right)$.

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\le 1/d$ of the $L$-functions of any degree $d\ge 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.