Related papers: On the conditional bounds for Siegel zeros

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite…

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do…

In these notes, we refine Mitsui's Prime Number Theorem from 1957, which for a number field $K$ predicts how many prime elements there are in bounded convex sets in $K \otimes_{\mathbf Q} \mathbf R$, by incorporating potential Siegel zeros…

In this paper, we prove the lower bound of the unconditional large gap is 3.5555 which improves the obtained value 3.079 in the literature. Next, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly…

A celebrated conjecture of Hardy and Littlewood provides with an asymptotic formula for the counting function of the twin primes. We give an unconditional proof of such a formula by means of a finite Ramanujan expansion of the counting…

With the uniform positions we prove theorems of Landau and Hardy-Littlwood type for Goldbach, Chen, Lemoine-Levy and other binary partitions of positive integers. We also pose some new conjectures.

Let $L/K$ be a Galois extension of number fields. We prove two lower bounds on the maximum of the degrees of the irreducible complex representations of ${\rm Gal}(L/K)$, the sharper of which is conditional on the Artin Conjecture and the…

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the…

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…

The paper investigates various $p$-adic versions of Littlewood's conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff…

We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known constant…

The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl…

Using the concepts of Hyperbolic Classification of Natural Numbers, Essential Regions and Goldbach Conjecture Function we prove that the existence of a proof of the Goldbach Conjecture in First-Order Arithmetic would imply the existence of…

We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| $\le$ y. We prove the lower bound part of the conjecture for…

Let A_1,...,A_n be finite subsets of a field F, and let f(x_1,...,x_n)=x_1^k+...+x_n^k+g(x_1,...,x_n)\in F[x_1,...,x_n] with deg g<k. We obtain a lower bound for the cardinality of {f(x_1,...,x_n): x_1\in A_1,...,x_n\in A_n, and x_i\not=x_j…

The Equidistribution Conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general…

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the…

In this paper, a simple explanation for the Goldbach Conjecture is given. We have shown that the probability of violating the conjecture not only for the prime numbers, but also for any subset of natural numbers whose distribution is…

Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients…

Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. We prove the conjecture for type G_2, and…