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In higher dimension things become more complicated. For a finite abelian group $G$ define $\mathfrak{s}(G)$ to be the least integer $N$, such that every sequence $x_1, \ldots, x_N$ of elements of $G$ …

The standard appproach is via Baker's method of linear forms in logarithms. We have $x_n+\sqrt{3}y_n=(2+\sqrt{3})^n$, thus $2x_n=(2+\sqrt{3})^n+(2-\sqrt{3})^n$. Now assume that $x_n=7^m$, and consider …

Whether for a finite set $\mathcal{R}$ of roots the approximation $$ \pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho) $$ is "on average" better …

The best known bounds seem to be due to Nicely [``A new error analysis for Brun's constant,'' Virginia J. Sci. 52 (2001), no. 1, 45–55), who showed that Brun's constant is $$ 1.9021605823 \pm 0.000000 …

As far as I know there are two approaches to Goldbach type problems, the circle method and sieve methods. In the sequel I will restrict myself to the circle method, hoping that someone else writes som …

No. There are various results which give counterexamples. For example, Rankin's construction of large prime gaps boils down to the fact that if $p_1, \ldots, p_k$ denote all prime numbers below $x$, …

There exist surprising counterexamples. Elsholtz and Dietmann found the following: If $p\equiv 7\pmod{8}$ is prime, then the equation $x^2+y^2+z^4=p^2$ has no non-trivial solution. You might argue tha …

In the 1960's Turán wrote several papers on a function-theoretic sieve. He managed to express the number of prime twins in terms of roots of $L$-series. He began like you did by expressing $\Lambda$ a …

Let $p_k$ be the $k$-th prime number, and pick a sequence of primes $q_k$, such that $q_k\sim p_k^{3/2}$. Let $G$ be the arithmetic semigroup consisting of all integers not divisible by one of the $q_ …

If $a/b$ is not too large, you can compute the probability using arguments as in the computation leading to the asymptotics for smooth numbers. In theory, you can compute for all $\beta, \gamma$ a rea …

In "The equation $\omega(n)=\omega(n+1)$" (Mathematika 50, 99-101, 2003) it was shown that the equation $\omega(n)=\omega(n+1)$ has infinitely many solutions. Pintz has a series of results on consecut …

Wolke (On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247, MathSciNet MR0291121 (45 #215)) has worked on this question. His motivation was Gallagher's approach to the …

A non-trivial lower bound can be obtained using linear forms in $p$-adic logarithms. Suppose that $\left\{\left(\frac{4}{3}\right)^n\right\}$ is small. Clearly it is a rational number with denominator …

I am looking for a reference for the following statement: For every integer $d$, there exist an integer $k$, such that for all polynomials $P_1, \ldots, P_k$ of degree $d$ there exist integers $N, q, …

The right context of your question is probably the area of Beurling primes. An arithmetic semigroup is a semigroup $(G,\cdot)$ together with a norm $\|\cdot\|:G\rightarrow[1,\infty)$, such that $\|gh\ …