Category: Analytic number theory

PNT for polynomials using the zeta function

In this blog post of mine I talk about the “polynomial zeta function” \(\zeta_q(s)\) and prove an analogue of Riemann hypothesis for it (indeed, this zeta has no zeroes at all). At the end I mention that this zeta function is, in fact, the “correct” zeta function for this purpose. In this blog post I am going to derive a formula analogous to the von Mangoldt’s explicit formula. This is a little project for me to see whether I’m able to work out all the details.

The proof is based on the exposition in Davenport’s Multiplicative Number Theory and requires some understanding of complex analysis.

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Proof of the Riemann hypothesis… for polynomials

This blog post consists of three parts. The first of them contains a somewhat nontechnical description of Riemann hypothesis. In the second one we discuss what the “correct” analogue of Riemann hypothesis is for polynomials over a finite field. Finally, in the last section, we prove the Riemann hypothesis for polynomials.

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Nonvanishing of L-functions via Dedekind zeta functions

The Dirichlet L-functions are an extremely important tool in studying primes in arithmetic progressions – their zeros “control” distribution of primes in arithmetic progressions in the same manner as zeros of Riemann zeta function control the overall distribution of primes. The first and the most elementary result involving these zeros, which is the key result in most proofs of Dirichlet’s theorem, is that there is never a zero at point \( s=1\). This post will present a proof of this fact using results from algebraic number theory.

This post is based solely on the content of Marcus’s Number Fields. The prerequisities for it are basic results about ideals in number fields and a minute amount of complex analysis. No background in analytic number theory is necessary.

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