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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 …

Goldston, Graham Pintz and Yildirim ( https://arxiv.org/abs/0803.2636 ) showed that among three linear forms $\ell_1, \ell_2, \ell_3$, which satisfy the obvious local conditions, there are two, say $\ …

Studying the distribution of patterns of the Moebius function falls into an easy part, which deals with the distribution of zeroes, and a difficult part, which deals with the distribution of signs. Th …

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, …

Checking whether an integer $n$ has a divisor in a given interval is essentially equivalent to factorization. Factoring is essentially equivalent to finding the smallest prime factor. So suppose that …

Pyber showed that the number of groups of order $n$ is $\leq n^{\frac{2}{27}\nu(n)^3+C\nu(n)^{3/2}}$, where $\nu$ is the highest power of a prime dividing $n$ and $C$ is an absolute constant. On the o …

The assertion is generally believed to be true. In fact, much more is conjectured: For every $\epsilon>0$ we have $\left|\zeta(\frac{1}{2}+it)-\sum_{n\leq t^\epsilon} n^{-\frac{1}{2}-it}\right|= \math …

It is not true anymore that a proof of this conjecture would lead to significant simplifications. Peterfalvi proved in 1984 a weaker version of this conjecture, which suffices to get rid of the chapte …