A remark on vorticity maximal function

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

Historical Remark on Ramanujan′s Tau Function

It is shown that Ramanujan could have proved a special case of his conjecture that his tau function is multiplicative without any recourse to modularity results.

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

A Remark on Partial Sums Involving the Mobius Function

Let 〈P〉 ⊂ N be a multiplicative subsemigroup of the natural numbers N= {1, 2, 3, . . .} generated by an arbitrary set P of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈P〉:n≤x (μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈P (1− (1/p)) (the case where P is all the primes is a we...

ON THE MAXIMAL SPECTRUM OF A MODULE

Let $R$ be a commutative ring with identity. The purpose of this paper is to introduce and study two classes of modules over $R$, called $mbox{Max}$-injective and $mbox{Max}$-strongly top modules and explore some of their basic properties. Our concern is to extend some properties of $X$-injective and strongly top modules to these classes of modules and obtain some related results.