There are many proposed partial prime number formulas, however, no formula can generate all prime numbers. Here we show three formulas which can obtain the entire prime numbers set from the positive integers, based on the Möbius function plus the “omega” function, or the Omega function, or the divisor function. The history of searching for a prime number formula goes back to the ancient Egyptia...

Critical to the understanding of a graph are its chromatic number and whether or not it is perfect. Here we prove when Γ(Zn), the zero-divisor graph of Zn, is perfect and show an alternative method to [D] for determining the chromatic number in those cases. We go on to determine the chromatic number for Γ(Zp[x]/〈x 〉) where p is prime and show that an isomorphism exists between this graph and Γ(...

We consider the generalized Jacobian J̃0(N) of a modular curveX0(N)with respect to a reduced divisor given by the sum of all cusps on it. WhenN is a power of a prime≥ 5, we exhibit that the group of rational torsion points J̃0(N)(Q)Tor tends to be much smaller than the classical Jacobian. 2010 Mathematics Subject Classification: Primary 14H40; Secondary 11G16, 11F03, 14G35.

Let H be a Krull monoid with finite class group such that each class contains a prime divisor (e.g., the multiplicative monoid of the ring of algebraic integers of some number field). It is shown that it can be determined whether the class group is of the form Z/nZ ⊕ Z/nZ, for n ≥ 3, just by considering the system of sets of lengths of H.

Using the techniques of automata theory, we give another proof of the function field analogue of the Mahler-Manin conjecture and prove transcendence results for the power series associated to higher divisor functions σk(n) = ∑ d|n d k. Let p be a prime number, and let k be an algebraic closure of Fp. Let q be a variable and consider a4, a6 ∈ k[[q]] defined by

We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: proof recent estimate by Drappeau Topacogullari arithmetical correlations; theorem Erdős-Wintner with support equal to the level set an additive function at shifted argument; law iterated logarithm distribution prime factors integers weighted $...

‎Let $R$ be commutative ring with identity and $M$ be an $R$-module‎. ‎The zero divisor graph of $M$ is denoted $Gamma{(M)}$‎. ‎In this study‎, ‎we are going to generalize the zero divisor graph $Gamma(M)$ to submodule-based zero divisor graph $Gamma(M‎, ‎N)$ by replacing elements whose product is zero with elements whose product is in some submodules $N$ of $M$‎. ‎The main objective of this pa...

Let {pj} ∞ j=1 denote the set of prime numbers in increasing order, let ΩN ⊂ N denote the set of positive integers with no prime factor larger than pN and let PN denote the probability measure on ΩN which gives to each n ∈ ΩN a probability proportional to 1 n . This measure is in fact the distribution of the random integer IN ∈ ΩN defined by IN = ∏N j=1 p Xpj j , where {Xpj } ∞ j=1 are independ...