Muharem Avdispahić

On the Classes ΛBV and V [ ν]

We prove inclusion relations between Watermans and Chanturiyas classes and point to some corollaries thereof. The situation which occurs in connection with Zygmunds theorem for Watermans classes is clarified.

On the determination of the jump of a function by its Fourier series

We shall suppose that I=[0, 2rc] and that f is 2~-periodic. The relationship of this notion of bounded variation to another interesting generalization given by Z. A. ~anturijas modulus of variation [4], was discussed by the author in [1] and [2]. The existing inclusion relations yield that certain theorems on the absolute convergence of Fourier series of functions of Waterman classes {n~}BV, 0 < ~ < 1, are equivalent to the corresponding statements for ~anturijas V[n~], 0 < y < 1, and further to the older ones for Wiener classes of bounded p-variation, [2]. The case c~= 1; i.e. 2 ,=n, when we also speak of harmonic bounded variation (ttBV) is, however, a distinguished one. It appears that some important classical theorems, valid for the Jordan class BV, find in HBV their natural setting. D. Waterman has proved in [9] that the Fourier series of functions of this class converge everywhere and converge uniformly on closed intervals of continuity. (See also ~anturija [5]. Note that his class V[v(n)] with Zv(k)/k~< co is contained in HBV.) Further, the classes L and HBV are complementary in the sense that for a product of fCL and gEHBV the Parseval formula holds (Waterman, [11]). Both of these results are best possible in that they are not valid for any strictly larger class ABV. In this note we shall show this is also the case with the equation

ON THE ERROR TERM IN THE PRIME GEODESIC THEOREM

Taking the integrated Chebyshev-type counting function of the appropriate order, we improve the error term in Parks prime geodesic theorem for hyperbolic manifolds with cusps. The obtained estimate coincides with the best known result in the Riemann surfaces case.

A Note on Weil’s Explicit Formula

Taking Weil’s point on adelic nature of explicit formulas, we extend the class of test functions to which his formula in Barner‐Burnol version holds.

On Koyama’s refinement of the prime geodesic theorem

We give a new proof of the best presently known error term in the prime geodesic theorem for compact Riemann surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived yielding to a further reduction of the error term outside a set of finite logarithmic measure.

Zeta and normal zeta functions for a subclass of space groups

We calculate zeta and normal zeta functions of space groups with the point group isomorphic to the cyclic group of order 2. The obtained results are applied to determine the number of subgroups, resp. normal subgroups, of a given index for each of these groups.

On the prime geodesic theorem for hyperbolic 3 -manifolds

Through the Selberg zeta approach, we reduce the exponent in the error term of the prime geodesic theorem for cocompact Kleinian groups or Bianchi groups from Sarnaks 53 to 32. At the cost of excluding a set of finite logarithmic measure, the bound is further improved to 139.

Strong boundedness, strong convergence and generalized variation

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all

A derivative on the field of p-adic numbers

{L^{p}}