Quantitative Riemann existence theorem over a number field

A Nonstandard Riemann Existence Theorem

We study elementary extensions of compact complex spaces and deduce that every complete type of dimension 1 is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay in [13].

Riemann ’ s zeta function and the prime number theorem

16 Riemann’s zeta function and the prime number theorem We now divert our attention from algebraic number theory to talk about zeta functions and L-functions. As we shall see, every global field has a zeta function that is intimately related to the distribution of its primes. We begin with the zeta function of the rational field Q, which we will use to prove the prime number theorem. We will ne...

A quantitative Khintchine–Groshev type theorem over a field of formal series

An asymptotic formula which holds almost everywhere is obtained for the number of solutions to the Diophantine inequalities ‖qA− p‖ < ψ(‖q‖), where A is an n ×m matrix (m > 1) over the field of formal Laurent series with coefficients from a finite field, and p and q are vectors of polynomials over the same finite field. AMS Subj. Classification: 11J83, 11J61

Revisiting Gauss's analogue of the prime number theorem for polynomials over a finite field

In 1901, von Koch showed that the Riemann Hypothesis is equivalent to the assertion that ∑ p≤x p prime 1 = ∫ x 2 dt log t + O( √ x log x). We describe an analogue of von Koch’s result for polynomials over a finite prime field Fp: For each natural number n, write n in base p, say n = a0 + a1 p + · · · + ak pk, and associate to n the polynomial a0 + a1T + · · · + akT k ∈ Fp[T ]. We let πp(X) deno...

Symmetry of Algebras over a Number Field