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Register a new userUniform explicit Stewarts theorem on prime factors of linear recurrences
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Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of ErdH{o}s and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewarts result.
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We show that for all large enough $x$ the interval $[x,x+x^{1/2}log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals $[x,x+x^{1/2+epsilon}].$ We also incorporate some ideas from Harmans book `Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomaki and Radziwi{l}{l}(2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of $log x$ when applying Harmans sieve method.
Let $Lambda(n)$ be the von Mangoldt function, and let $[t]$ be the integral part of real number $t$. In this note, we prove that for any $varepsilon>0$ the asymptotic formula $$ sum_{nle x} LambdaBig(Big[frac{x}{n}Big]Big) = xsum_{dge 1} frac{Lambda(d)}{d(d+1)} + O_{varepsilon}big(x^{9/19+varepsilon}big) qquad (xtoinfty)$$ holds. This improves a recent result of Bordell`es, which requires $frac{97}{203}$ in place of $frac{9}{19}$.
We consider the summatory function of the number of prime factors for integers $leq x$ over arithmetic progressions. Numerical experiments suggest that some arithmetic progressions consist more number of prime factors than others. Greg Martin conjectured that the difference of the summatory functions should attain a constant sign for all sufficiently large $x$. In this paper, we provide strong evidence for Greg Martins conjecture. Moreover, we derive a general theorem for arithmetic functions from the Selberg class.
In this paper we use a theorem first proved by S.W.Golomb and a famous inequality by J.B. Rosser and L.Schoenfeld in order to prove that there exists an exact formula for $pi(n)$ which holds infinitely often.
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