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We give a new formula for the Chebotarev densities of Frobenius elements in Galois groups. This formula is given in terms of smallest prime factors $p_{mathrm{min}}(n)$ of integers $ngeq2$. More precisely, let $C$ be a conjugacy class of the Galois group of some finite Galois extension $K$ of $mathbb{Q}$. Then we prove that $$-lim_{Xrightarrowinfty}sum_{substack{2leq nleq X[1pt]left[frac{K/mathbb{Q}}{p_{mathrm{min}}(n)}right]=C}}frac{mu(n)}{n}=frac{#C}{#G}.$$ This theorem is a generalization of a result of Alladi from 1977 that asserts that largest prime divisors $p_{mathrm{max}}(n)$ are equidistributed in arithmetic progressions modulo an integer $k$, which occurs when $K$ is a cyclotomic field $mathbb{Q}(zeta_k)$.
We prove an analogue of Kroneckers second limit formula for a continuous family of indefinite zeta functions. Indefinite zeta functions were introduced in the authors previous paper as Mellin transforms of indefinite theta functions, as defined by Zw
Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In t
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $mathbb N$ as limiting values of $q$-series as $qto zeta$ a root of un
In this article, we prove that a general version of Alladis formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom $A$ or Axiom $A^{#}$. As applications, we apply our main results to certain semigroups coming from algeb
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.