Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that uniform in the shift parameter. This applies to more general arithmetic functions such as of two squares, improving term representation number a sum prime and Fourier coefficients cusp forms, generalizing result Pitt.

For a smooth scheme X of pure dimension d over field k and an effective Cartier divisor D⊂X whose support is simple normal crossing divisor, we construct cycle class map from the Chow group zero-cycles with modulus to cohomology relative Milnor K-sheaf.

For a smooth scheme X of pure dimension d over field k and an effective Cartier divisor D⊂X whose support is simple normal crossing divisor, we construct cycle class map from the Chow group zero-cycles with modulus to cohomology relative Milnor K-sheaf.

Let s(·) denote the sum-of-proper-divisors function, that is, s(n) = ∑ d|n, d<n d. Erdős–Granville–Pomerance–Spiro conjectured that for any set A of asymptotic density zero, the preimage set s−1(A ) also has density zero. We prove a weak form of this conjecture: If (x) is any function tending to 0 as x→∞, and A is a set of integers of cardinality at most x 1 2 + , then the number of integers n ...

The main purpose of this paper is to prove an important generalization of the construction of the Incidence Divisor given in [BMg1] in the case of an ambient manifold. Let us first recall briefly the setting : let Z be a complex manifold and (X s) s∈S an analytic family of (closed) n−cycles in Z parametrized by a reduced complex space S. To a (n+1)−codimensional subspace Y in Z, which is assume...

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_...

The quotient digit selection in the SRT division algorithm is based on a few most significant bits of the remainder and divisor, where the remainder is usually represented in a redundant representation. The number of leading bits needed depends on the quotient radix and digit set, and is usually found by an extensive search, to assure that the next quotient digit can be chosen as valid for all ...

In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), highest common factor (hcf), or greatest commonmeasure (gcm), of two or more integers (when at least one of them is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.[1][2] This notion can be extended to polynomials, see ...

The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension g contains a Hodge structure of level g− 3 which we call the primal cohomology. The Hodge conjecture predicts that this is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version...