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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 $frac{9}{19}$.
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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.
In this paper, with the help of the theory of matrices and finite fields we generalize Zolotarevs theorem to an arbitrary finite dimensional vector space over $mathbb{F}_q$, where $mathbb{F}_q$ denotes the finite field with $q$ elements.
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.
Let $Gamma=PSL(2,Z[i])$ be the Picard group and $H^3$ be the three-dimensional hyperbolic space. We study the Prime Geodesic Theorem for the quotient $Gamma setminus H^3$, called the Picard manifold, obtaining an error term of size $O(X^{3/2+theta/2+epsilon})$, where $theta$ denotes a subconvexity exponent for quadratic Dirichlet $L$-functions defined over Gaussian integers.
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
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