Maximal prime gaps bounds

This paper presents research results, pertinent to the maximal prime gaps bounds. Four distinct bounds are presented: Upper bound, Infimum, Supremum and finally the Lower bound. Although the Upper and Lower bounds incur a relatively high estimation error cost, the functions representing them are quite simple. This ensures, that the computation of those bounds will be straightforward and efficient. The Lower bound is essential, to address the issue of the value of the lower bound implicit constant C, in the work of Ford et al (Ford, 2016). The concluding Corollary in this paper shows, that the value of the constant C does diverge, although very slowly. The constant C, will eventually take any arbitrary value, providing that a large enough N (for p <= N) is considered. The Infimum/Supremum bounds on the other hand are computationally very demanding. Their evaluation entails computations at an extreme level of precision. In return however, we obtain bounds, which provide an extremely close approximation of the maximal prime gaps. The Infimum/Supremum estimation error gradually increases over the range of p and attains at p = 18361375334787046697 approximately the value of 0.03.

Monopolistic competition with generalized additively separable preferences

We study monopolistic competition equilibria with free entry and social planner solutions under symmetric generalized additively separable preferences, which encompass known cases such as additive, homothetic, translog and other preferences. This setting can jointly produce competition and selection effects of entry, incomplete pass-through of cost changes and pricing to market. We discuss the inefficiencies of the market equilibrium under Gorman-Pollak preferences and show its optimality under implicit constant elasticity of substitution preferences. We propose a new specification of generalized translated power preferences, and discuss applications to trade and macroeconomics.

POSITIVE PROPORTION OF SHORT INTERVALS CONTAINING A PRESCRIBED NUMBER OF PRIMES

We prove that for every $m\geq 0$ there exists an $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}(m)>0$ such that if $0<\unicode[STIX]{x1D706}<\unicode[STIX]{x1D700}$ and $x$ is sufficiently large in terms of $m$ and $\unicode[STIX]{x1D706}$, then $$\begin{eqnarray}|\{n\leq x:|[n,n+\unicode[STIX]{x1D706}\log n]\cap \mathbb{P}|=m\}|\gg _{m,\unicode[STIX]{x1D706}}x.\end{eqnarray}$$ The value of $\unicode[STIX]{x1D700}(m)$ and the dependence of the implicit constant on $\unicode[STIX]{x1D706}$ and $m$ may be made explicit. This is an improvement of the author’s previous result. Moreover, we will show that a careful investigation of the proof, apart from some slight changes, can lead to analogous estimates when allowing the parameters $m$ and $\unicode[STIX]{x1D706}$ to vary as functions of $x$ or replacing the set $\mathbb{P}$ of all primes by primes belonging to certain specific subsets.

On a problem of Ivi\'c.

International audience Let $\gamma$ denote the imaginary parts of the nontrivial zeros of the Riemann zeta-function $\zeta(s)$. For sufficiently large $T$ and $\varepsilon>0$, Ivi\'c proved that $\sum_{T<\gamma\leq2T} \vert\zeta(\frac{1}{2}+i\gamma)\vert^2 <\!\!\!<_{\varepsilon} (T(\log T)^2\log\log T)^{3/2+\varepsilon},$ where the implicit constant depends only on $\varepsilon$. In this paper, this result is improved by (i) replacing $\vert\zeta(\frac{1}{2}+i\gamma)\vert^2$ by $\max\vert\zeta(s)\vert^2$, where the maximum is taken over all $s=\sigma+it$ in the rectangle $\frac{1}{2}-A/\log T\leq\sigma\leq2,\, \vert t-\gamma\vert\leq B(\log\log T)/\log T$ with some fixed positive constants $A, B,$ and (ii) replacing the upper bound by $T(\log T)^2\log\log T$. The method of proof differs completely from Ivi\'c's approach.

Indefinite quadratic polynomials

Letbe an indefinite quadratic form with real coefficients. A well-known result, due to Birch, Davenport and Ridout [1], [5] and [6], states that if n ≥21 then for any ε > 0 there is an integer vector x ≠O such thatRecently [3] we have quantified this result, obtaining a function g(n) such that g(n)→ ½ as n n→ ∞ and such that for any η > 0 and all large enough X there is an integer vector x satisfyingwhere |x| = max |xi|and the implicit constant in Vinogradov's ≪-notation is independent of X.