Related papers: Ordinality and Riemann Hypothesis I
We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…
We extend Riemann's rearrangement theorem on conditionally convergent series of real numbers to multiple instead of simple sums.
We prove a general local existence theorem for nonlinear systems of Cauchy-Riemann operator of any order in one complex variable with initial values at a given point, which is a counterpart of local existence of ODE.
In this paper a new integral for the remainder of $\pi(x)$ is obtained. It is proved that there is an infinite set of the formulae containing miscellaneous parts of this integral.
We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd…
The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the…
In this note, we give a simple proof that the Riemann Hypothesis is unprovable in any reasonable axiom system.
The Riemann hypothesis is proved by quantum-extending the zeta Riemann function to a quantum mapping between quantum $1$-spheres with quantum algebra $A=\mathbb{C}$, in the sense of A. Pr\'astaro \cite{PRAS01, PRAS02}. Algebraic topologic…
We prove that a good average order on the Goldbach generating function implies that the real parts of the non-trivial zeros of the Riemann zeta function are strictly less than 1. This together with existing results establishes an…
A proposed solution to the Riemann Hypothesis
A new parametric integral is obtained as a consequence of the Riemann hypothesis. An asymptotic multiplicability is the main property of this integral.
In this article, it is proved that the non-trivial zeros of the Riemann zeta function must lie on the critical line, known as the Riemann hypothesis.
All possible products of all elements of an odd order finite group are considered. A set of all such products is called as a K-set. A hypothesis of K-set coincidence of any group of an odd order with its commutant is proposed and the…
We provide conditions under which the union of two first-order theories has the amalgamation property.
We extend Painlev\'e's determinateness theorem to the case of first order ordinary differential equations in the complex domain with known terms allowed be multivalued in the dependent variable as well; multivaluedness is supposed to be…
The theorem presented in this paper allows the creation of large prime numbers (of order up to o(n^2)) given a table of all primes up to n.
The basic purpose of the present paper is the full solutions of the inverse problem (i.e. a finding of necessary and sufficient conditions) for the operator with complex periodic coefficients.
A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order.…
We consider a natural generalisation of the Painlev\'e property and use it to identify the known integrable cases of the Lane-Emden equation with a real positive index. We classify certain first-order ordinary differential equations with…
Let $p_{1}$, ..., $p_{k}$ be the first $k$ odd primes in succession. Let $n$ be an even integer such that $n > p_{k}$. We conjecture that if none of $n - p_{1}$, ..., $n - p_{k}$ are prime, then at least one of them has a prime factor which…