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 infinitesimal generator of the Lax-Phillips semigroup of the automorphic scattering system defined on the Poincaré upper half-plane for SL₂(ℤ). We show that its spectrum consists only of the poles of the resolvent of the generator, and coincides with the poles of the scattering matrix, counted with multiplicities. Using this we construct an operator whose eigenvalues, counted with algebraic multiplicities (i.e. dimensions of generalized eigenspaces), are precisely the non-trivial zeros...

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that ${\sum }_{n\le x}d²\left(n\right)=xP\left(logx\right)+E\left(x\right)$, where P(y) is a cubic polynomial in y and $E\left(x\right)=O\left(x^{3/5+\epsilon }\right)$, with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), $E\left(x\right)=O\left(x^{1/2+\epsilon }\right)$. In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵loglogx\right)$. In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove $E\left(x\right)=O\left(x^{1/2}\left(logx\right)⁵\right)$.

Let k ≥ 1 denote any positive rational integer. We give formulae for the sums $S_{odd}(k,f)={\sum }_{\chi (-1)=-1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 odd Dirichlet characters modulo f > 2) whenever k ≥ 1 is odd, and for the sums $S_{even}(k,f)={\sum }_{\chi (-1)=+1}\left|L(k,\chi )\right|²$ (where χ ranges over the ϕ(f)/2 even Dirichlet characters modulo f>2) whenever k ≥ 1 is even.