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Register a new userSummatory function of the number of prime factors
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Xianchang Meng
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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.
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The Stieltjes constants $gamma_k(a)$ appear in the regular part of the Laurent expansion for the Hurwitz zeta function $zeta(s,a)$. We present summatory results for these constants $gamma_k(a)$ in terms of fundamental mathematical constants such as the Catalan constant, and further relate them to products of rational functions of prime numbers. We provide examples of infinite series of differences of Stieltjes constants evaluating as volumes in hyperbolic $3$-space. We present a new series representation for the difference of the first Stieltjes constant at rational arguments. We obtain expressions for $zeta(1/2)L_{-p}(1/2)$, where for primes $p>7$, $L_{-p}(s)$ are certain $L$-series, and remarkably tight bounds for the value $zeta(1/2)$, $zeta(s)=zeta(s,1)$ being the Riemann zeta function.
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 $frac{9}{19}$.
Let $Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $fcolonmathbb{N}tomathbb{C}$ one has [ frac{1}{N}sum_{n=1}^N, f(Omega(n)+1)=frac{1}{N}sum_{n=1}^N, f(Omega(n))+mathrm{o}_{Ntoinfty}(1). ] This yields a new elementary proof of the Prime Number Theorem.
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.
The double sum sum_(s >= 1) sum_p 1/(p^s log p^s) = 2.00666645... over the inverse of the product of prime powers p^s and their logarithms, is computed to 24 decimal digits. The sum covers all primes p and all integer exponents s>=1. The calculational strategy is adopted from Cohens work which basically looks at the fraction as the underivative of the Prime Zeta Function, and then evaluates the integral by numerical methods.
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