We investigate the value-distribution of Epstein zeta-functions ζ(s; Q), where Q is a positive definite quadratic form in n variables. We prove an asymptotic formula for the number of c-values, i.e., the roots of the equation ζ(s; Q) = c, where c is any fixed complex number. Moreover, we show that, in general, these c-values are asymmetrically distributed with respect to the critical line Re s =n/4. This complements previous results on the zero-distribution.[Proceedings of the Primeras Jornadas...

Let L be a finite Galois CM-extension of a totally real field K. We show that the validity of an appropriate special case of the Equivariant Tamagawa Number Conjecture leads to a natural construction for each odd prime p of explicit elements in the (non-commutative) Fitting invariants over ${\mathbb{Z}}_p\left[G\right]$ of a certain tame ray class group, and hence also in the analogous Fitting invariants of the p-primary part of the ideal class group of L. These elements involve the values at s=1 of the Artin L-series of characters...

1. Introduction. Let p be a prime number and ${\mathbb{Z}}_p$ the ring of p-adic integers. Let k be a finite extension of the rational number field ℚ, $k_{\infty }$ a ${\mathbb{Z}}_p$-extension of k, $k_n$ the nth layer of $k_{\infty }/k$, and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$. Iwasawa proved the following well-known theorem about the order $A_n$ of $A_n$: Theorem A (Iwasawa). Let $k_{\infty }/k$ be a ${\mathbb{Z}}_p$-extension and $A_n$ the p-Sylow subgroup of the ideal class group of $k_n$, where $k_n$ is the $n$th layer of $k_{\infty }/k$. Then there exist integers $\lambda =\lambda (k_{\infty }/k)\ge 0$, $\mu =\mu (k_{\infty }/k)\ge 0$, $\nu =\nu (k_{\infty }/k)$, and n₀ ≥ 0 such that $A_n=p^{\lambda n+\mu p^n+\nu }$ for...

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 $x⩾40c^{-5}{(1+b)}^{27/2}$.

Let $N$ be a sufficiently large integer. We prove that almost all sufficiently large even integers $n\in [N-6U,N+6U]$ can be represented as $$n=p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3,\left|p_1^2-{\displaystyle \frac{N}{6}}\right|\le U,\left|p_i^3-{\displaystyle \frac{N}{6}}\right|\le U,i=2,3,4,5,6,$$ where $U=N^{1-\delta +\epsilon }$ with $\delta \le 8/225$.