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We establish several results concerning the expected general phenomenon that, given a multiplicative function $f:\mathbb{N}\to\mathbb{C}$, the values of $f(n)$ and $f(n+a)$ are "generally" independent unless $f$ is of a "special" form.…

We prove the Strengthened Hanna Neumann Conjecture, in its common graph theoretic formulation. Our original approach to this conjecture used cohomology of sheaves on graphs, although here we give a short combinatorial proof that we found in…

The Jacobian conjecture in dimension $n$ asserts that any polynomial endomorphism of $n$-dimensional affine space over a field of zero characteristic, with the Jacobian equal 1, is invertible. The Dixmier conjecture in rank $n$ asserts that…

We consider the alternating Riemann zeta function $\zeta^*(s)= \sum^{\infty} _{ n=1} \frac{(-1)^{n-1}}{n^s}$, which converges if $Re (s)>0 .$ By using Rouche's theorem, the Bolzano-Weierstrass theorem and by method of contradiction we…

Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…

Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number…

A new generalisation of Goldbach's conjecture (GGC) - also generalising that of Lemoine - is tested, introduced by the first author. It states that for every pair of positive integers $m_1, m_2$, every sufficiently large integer $n$…

We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups…

Newman's conjecture (proved by Rodgers and Tao in 2018) concerns a certain family of deformations $\{\xi_t(s)\}_{t \in \mathbb{R}}$ of the Riemann xi function for which there exists an associated constant $\Lambda \in \mathbb{R}$ (called…

Let $\mathcal{M}$ be a Banach bimodule over an associative Banach algebra $\mathcal{A}$, and let $F: \mathcal{A}\to \mathcal{M}$ be a linear mapping. Three main uses of the term \emph{generalized derivation} are identified in the available…

Assuming the Generalized Riemann Hypothesis(GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.

An heuristic proof of the Riemman conjecture is proposed. It is based on the old idea of Polya-Hilbert. A discrete/fractal derivative self adjoint operator whose spectrum may contain the nontrivial zeroes of the zeta function is presented.…

Let $Z_1,\, Z_2,\dots$ be independent and identically distributed complex random variables with common distribution $\mu$ and set $$ P_n(z) := (z - Z_1)\cdots (z - Z_n)\,. $$ Recently, Angst, Malicet and Poly proved that the critical points…

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from $\mathbb{F}_q^n$ to $\mathbb{F}_q$ to the set of affine functions from $\mathbb{F}_q^n$ to $\mathbb{F}_q$. We prove…

Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta-function in short intervals. We give three different formulations of these results. Assuming a…

A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\zeta_{k}$ and the method of the proof of the…

The paper proves a special case of a conjecture of Hellerstein, Shen and Williamson concerning non-real zeros of derivatives of real meromorphic functions.

In 1948, Feynman showed Dyson how the Lorentz force and Maxwell equations could be derived from commutation relations coordinates and velocities. Several authors noted that the derived equations are not Lorentz covariant and so are not the…

We study families of analytic and meromorphic functions with bounded generalized Schwarzian derivative $S_k(f)$. We show that these families are quasi-normal. Further, we investigate associated families, such as those formed by derivatives…

We prove the Kakeya set conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. This entails extending and combining the techniques of Arsovski [Ars21a] for $N=p^k$ and the author and Dvir [DD21] for…