Short Note on the Riemann Hypothesis

Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)<e^{\gamma }\times n \times\log\log n$ holds for all natural numbers $n>5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma\approx0.57721$ is the Euler-Mascheroni constant. Let $q_{1}=2,q_{2}=3,\ldots,q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n>5040$ such that Robin inequality does not hold and we prove that $n^{\left(1-\frac{0.6253}{\log q_{m}}\right)}<N_{m}$, where $N_{m}=\prod_{i =1}^{m}q_{i}$ is the primorial number of order $m$ and $q_{m}$ is the largest prime divisor of $n$. In addition, we show that $q_{m}$ will not have an upper bound by some positive value for these counterexamples and therefore, the value of $q_{m}$ tends to infinity as $n$ goes to infinity.

Short Note on the Riemann Hypothesis

Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)<e^{\gamma }\times n \times\log\log n$ holds for all natural numbers $n>5040$, where $\sigma(n)$ is the sum-of-divisors function of $n$ and $\gamma\approx0.57721$ is the Euler-Mascheroni constant. Let $q_{1}=2,q_{2}=3,\ldots,q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m}q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n>5040$ such that Robin inequality does not hold and we prove that $n^{\left(1-\frac{0.6253}{\log q_{m}}\right)}<N_{m}$, where $N_{m}=\prod_{i =1}^{m}q_{i}$ is the primorial number of order $m$ and $q_{m}$ is the largest prime divisor of $n$. In addition, we show that $q_{m}$ will not have an upper bound by some positive value for these counterexamples and therefore, the value of $q_{m}$ tends to infinity as $n$ goes to infinity.

On the least common multiple of polynomial sequences at prime arguments

Cilleruelo conjectured that if [Formula: see text] is an irreducible polynomial of degree [Formula: see text] then, [Formula: see text] In this paper, we investigate the analog of prime arguments, namely, [Formula: see text] where [Formula: see text] denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of [Formula: see text]

MONOMIAL AND MONOLITHIC CHARACTERS OF FINITE SOLVABLE GROUPS

Let G be a finite solvable group and let p be a prime divisor of $|G|$ . We prove that if every monomial monolithic character degree of G is divisible by p, then G has a normal p-complement and, if p is relatively prime to every monomial monolithic character degree of G, then G has a normal Sylow p-subgroup. We also classify all finite solvable groups having a unique imprimitive monolithic character.

A geometric approach to Quillen’s conjecture

We introduce admissible collections for a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of | G | \lvert G\rvert . Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex, Ann. of Math. (2) 137 (1993), 3, 473–529], our techniques are simpler.

On some punctured codes of ℤq-Simplex codes

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

Korselt rational bases of prime powers

Let N be a positive integer, be a subset of ℚ and . N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α2p − α1 divides α2N − α1 for every prime divisor p of N. By the Korselt set of N over , we mean the set of all such that N is an α-Korselt number. In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of is studied where we prove that is empty if and only if l = 2. Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].