On the difference between consecutive primes

Inventiones mathematicae -     

On the several differences between primes

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,4(5 × 10¹⁰) = 118905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computedB ₄(5×10¹⁰)=1.197054473029 andB ₄=1.197054±7×10^?6. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5×10¹⁰, yielding the counting function π_2,6(5 × 10¹⁰) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computedB ₆(5×10¹⁰)=0.93087506039231 andB ₆=1.135835±1.2×10^?6.     

On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

On the sum of distinct primes or squares of primes

In 1965 Erd?s introduced $f_2(s)$: $f_2(s)$ is the smallest integer such that every $l>f_2(s)$ is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, $f_2(s)?p_2+p_3+?+p_{s+1}+3106$, and the set of s with the equality has the density 1.     

Algorithm for a sum of reciprocals

An algorithm is described for the approximate calculation of a collection of sums of the form _k= _j–1 ⁿ c_j/(_j+_k), 1kn, where 0<_j. The working time of the algorithm is 0(n(t+ log n)(t+log n)) if _k calculated to within 2^–t; here the function (l) denotes the time of multiplication of twoZ-bit numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 3–6, 1984.In conclusion, the author expresses thanks to A. O. Slisenko and Yu. A. Kuznetsov, who drew the attention of the author to the problem considered.     

On admissible constellations of consecutive primes

Admissible constellations of primes are patterns which, like the twin primes, no simple divisibility relation would prevent from being repeated indefinitely in the series of primes. All admissible constellations, formed ofconsecutive primes, beginning with a prime <1000, are established, and some properties of such constellations in general are conjectured.Dedicated to Peter Naur on the occasion of his 60th birthday     

On the sum of the reciprocals of cycle lengths in sparse graphs

For a graphG let ℒ(G)=Σ{1/k contains a cycle of lengthk}. Erdős and Hajnal [1] introduced the real functionf(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviouslyf(1)=0. We provef (k+1/k)≧(300k logk)⁻¹ for all sufficiently largek, showing that sparse graphs of large girth must contain many cycles of different lengths.     

On the mod-Gaussian convergence of a sum over primes

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates ${\mathrm{Im }}\log \zeta (1/2+it)$ . This Dirichlet polynomial is sufficiently long to deduce Selberg’s central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that ${\mathrm{Im }}\log \zeta (1/2+it)$ satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.     

On the rational approximation of the sum of the reciprocals of the Fermat numbers

Let $\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}$ denote the generating function of the ruler function, and $\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}$ ; note that the special value $\mathcal{F}(1/2)$ is the sum of the reciprocals of the Fermat numbers $F_{n}:=2^{2^{n}}+1$ . The functions $\mathcal{F}(z)$ and $\mathcal{G}(z)$ as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers $\mathcal {F}(\alpha)$ and $\mathcal{G}(\alpha)$ are transcendental for all algebraic numbers α which satisfy 0<α<1. For a sequence u, denote the Hankel matrix $H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}$ . Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |α?p/q|<q ^?μ has infinitely many solutions (p,q)∈?×?. In this paper, we first prove that the determinants of $H_{n}^{1}(\mathbf {g})$ and $H_{n}^{1}(\mathbf{f})$ are nonzero for every n?1. We then use this result to prove that for b?2 the irrationality exponents $\mu(\mathcal{F}(1/b))$ and $\mu(\mathcal{G}(1/b))$ are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.     

On the sum of two primes and k powers of two

Let X be a large integer. We prove that, for any fixed positiveinteger k, a suitable asymptotic formula for the number of representationsof an even integer N [1,X] as the sum of two primes and k powersof 2 holds with at most exceptions.     

The exceptional set for the distribution of primes between consecutive powers

A well known conjecture about the distribution of primes asserts that between two consecutive squares there is always at least one prime number. The proof of this conjecture is quite out of reach at present, even under the assumption of the Riemann Hypothesis. This paper is concerned with the distribution of prime numbers between two consecutive powers of integers, as a natural generalization of the afore-mentioned conjecture.