Related papers: On the conditional bounds for Siegel zeros
We present four generalized small cancellation conditions for finite presentations and solve the word- and conjugacy problem in each case. Our conditions $W$ and $W^*$ contain the non-metric small cancellation cases C(6), C(4)T(4), C(3)T(6)…
In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in short intervals, and also on the…
Furstenberg has associated to every topological group $G$ a universal boundary $\partial(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\partial(G,H)$. In the case…
We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly…
The Hardy--Littlewood prime $k$-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made…
We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function $\zeta(s)$ if and only if there is a positive proportion…
Recent work of Borwein, Choi, and the second author examined a collection of polynomials closely related to the Goldbach conjecture: the polynomial $F_N$ is divisible by the $N$th cyclotomic polynomial if and only if there is no…
In 2016, Ellenberg and Gijswijt established a new upper bound on the size of subsets of $\mathbb{F}^n_q$ with no three-term arithmetic progression. This problem has received much mathematical attention, particularly in the case $q = 3$,…
We show that the Riesz mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the…
We give a simple proof of a recent result by J. Schleischitz dealing with a counterexample to the uniform Littlewood conjecture. Our construction is based on simple properties of Fibonacci numbers.
Work of Clifford, Munn and Ponizovski{\u\i} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations…
We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…
Bombieri and Zannier established lower and upper bounds for the limit infimum of the Weil height in fields of totally p-adic numbers and generalizations thereof. In this paper, we use potential theoretic techniques to generalize the upper…
We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions which is constraint-preserving and sufficiently general to include…
We prove near-optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application we prove near-optimal upper bounds for the average of…
As a sequel to the paper [9], we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward-backward parabolic equations with diffusion fluxes violating Fourier's inequality.
This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in…
Let $$\gamma^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $\gamma^*<\gamma_0\leq 1$, $c_0=1/\gamma_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann…
We correct a small gap found in the authors' paper 'On bounds for the effective differential Nullstellensatz' (J Algebra 449:1-21, 2016). This gap is due to an inequality that does not generally hold. However, under one additional…
We apply Tatuzawa's version of Siegel's theorem to derive two lower bounds on the size of the principal genus of positive definite binary quadratic forms.