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.
For a cubic algebraic extension $K$ of $mathbb{Q}$, the behavior of the ideal counting function is considered in this paper. Let $a_{K}(n)$ be the number of integral ideals of the field $K$ with norm $n$. An asymptotic formula is given for the sum $$
sumlimits_{n_{1}^2+n_{2}^2leq x}a_{K}(n_{1}^2+n_{2}^2). $$
Let $k,pin mathbb{N}$ with $p$ prime and let $finmathbb{Z}[x_1,x_2]$ be a bivariate polynomial with degree $d$ and all coefficients of absolute value at most $p^k$. Suppose also that $f$ is variable separated, i.e., $f=g_1+g_2$ for $g_iinmathbb{Z}[x_
i]$. We give the first algorithm, with complexity sub-linear in $p$, to count the number of roots of $f$ over $mathbb{Z}$ mod $p^k$ for arbitrary $k$: Our Las Vegas randomized algorithm works in time $(dklog p)^{O(1)}sqrt{p}$, and admits a quantum version for smooth curves working in time $(dlog p)^{O(1)}k$. Save for some subtleties concerning non-isolated singularities, our techniques generalize to counting roots of polynomials in $mathbb{Z}[x_1,ldots,x_n]$ over $mathbb{Z}$ mod $p^k$. Our techniques are a first step toward efficient point counting for varieties over Galois rings (which is relevant to error correcting codes over higher-dimensional varieties), and also imply new speed-ups for computing Igusa zeta functions of curves. The latter zeta functions are fundamental in arithmetic geometry.
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 conject
ured 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.
Rank-2 Drinfeld modules are a function-field analogue of elliptic curves, and the purpose of this paper is to investigate similarities and differences between rank-2 Drinfeld modules and elliptic curves in terms of supersingularity. Specifically, we
provide an explicit formula of a supersingular polynomial for rank-2 Drinfeld modules and prove several basic properties. As an application, we give a numerical example of an asymptotically optimal tower of Drinfeld modular curves.
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.