Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

Representation of an integer as a prime plus a product of two small factors

Abstract: In this paper we consider the following problem, which seems to have been brought to light fairly recently by M. Car. Can every sufficiently large integer n be expressed as n = p + ab with p prime and 1 ≤ a , b ≤ n ½ ? Certainly one should expect this to be possible. Taking b = 1, for example, p will be restricted to the range n − n ½ ≤ p n , and this interval is conjectured to contain a prime, for large enough n . Alternatively, providing that n is not a square, we expect n = p + a 2 to be solvable for sufficiently large n . However, although the statement that n = p + ab , with a , b ≤ n ½ , is far weaker than either of the aforementioned conjectures, it is nevertheless rather tricky to show that solutions must in fact exist.

On the divisor function and the Riemann zeta-function in short intervals

Abstract: We obtain, for $T^\epsilon \le U=U(T)\le T^{1/2-\epsilon}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(\Delta(t+U) - \Delta(t))^2 dt, $$ where $\Delta(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. Upper bounds of the form $O_\epsilon(T^{1+\epsilon}U^2)$ for the above integrals with biquadrates instead of square are shown to hold for $T^{3/8} \le U =U(T) \ll T^{1/2}$. The connection between the moments of $E(t+U) - E(t)$ and $|\zeta(1/2+it)|$ is also given. Generalizations to some other number-theoretic error terms are discussed.

Division method and division device

Abstract: PROBLEM TO BE SOLVED: To provide a division device for which arithmetic accuracy is improved and a circuit scale is reduced. SOLUTION: A divisor is divided into a mantissa part and an exponent part, division is performed at the mantissa part and then only the bit shift of a power of two is performed by a bit shift circuit 5 for the calculation of the exponent part. By contriving the division of the divisor, the calculation of the mantissa part is performed by making the divisor be in common as much as possible in a division ROM 1 and a linear arithmetic interpolation circuit 2 as well.

On the distribution of eigenspaces in classical groups over finite rings and the Cohen-Lenstra heuristic

Abstract: In 2006 Jeffrey Achter proved that the distribution of divisor class groups of degree 0 of function fields with a fixed genus and the distribution of eigenspaces in symplectic similitude groups are closely related to each other Gunter Malle proposed that there should be a similar correspondence between the distribution of class groups of number fields and the distribution of eigenspaces in ceratin matrix groups Motivated by these results and suggestions we study the distribution of eigenspaces corresponding to the eigenvalue one in some special subgroups of the general linear group over factor rings of rings of integers of number fields and derive some conjectural statements about the distribution of \(p\)-parts of class groups of number fields over a base field \(K_{0}\) Where our main interest lies in the case that \(K_{0}\) contains the \(p\)th roots of unity, because in this situation the \(p\)-parts of class groups seem to behave in an other way like predicted by the popular conjectures of Henri Cohen and Jacques Martinet In 2010 based on computational data Malle has succeeded in formulating a conjecture in the spirit of Cohen and Martinet for this case Here using our investigations about the distribution in matrixgroups we generalize the conjecture of Malle to a more abstract level and establish a theoretical backup for these statements

On the cohomology of a simple normal crossings divisor

Abstract: We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $\Delta(D)$ with coefficients in a presheaf of vector spaces. This presheaf consists precisely of the corresponding cohomology data on the components of $D$ and on their intersections. We use this formula to give a Hodge decomposition for SNC divisors and investigate the toric setting. We also conjecture the existence of such a formula for effective non-reduced divisors with SNC support, and show that this would imply the vanishing of the higher simplicial cohomologies of the dual complex associated to a resolution of an isolated rational singularity.