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Discrete Fourier Transform and $L$-functions

D. Liu

TL;DR

This paper shows a converse to Dirichlet's theorem by generalizing Guinand's zero-indicator to Dirichlet $L$-functions. It constructs a Dirichlet polynomial $D_x(z)=\sum_{n\le x}\frac{\Lambda(n)}{n^{1/2-iz}}$ via Mellin inversion and relates it to $-\frac{L'}{L}$ through contour integration, producing a zero-indicator identity linking prime powers to zeros $\rho$. It develops a Discrete Fourier Transform decomposition across residue classes modulo $q$, with a DFT matrix $M$ satisfying $M M^{*}=\varphi(q) I$, and includes compensation terms for non-primitive characters. Numerical experiments illustrate the predicted spikes at zeros and the DC/compensation structure, validating the transform-based view of the explicit formulas.

Abstract

We give the converse to Dirichlet's theorem on primes in arithmetic progressions by generalizing an old result of Guinand.

Discrete Fourier Transform and $L$-functions

TL;DR

This paper shows a converse to Dirichlet's theorem by generalizing Guinand's zero-indicator to Dirichlet -functions. It constructs a Dirichlet polynomial via Mellin inversion and relates it to through contour integration, producing a zero-indicator identity linking prime powers to zeros . It develops a Discrete Fourier Transform decomposition across residue classes modulo , with a DFT matrix satisfying , and includes compensation terms for non-primitive characters. Numerical experiments illustrate the predicted spikes at zeros and the DC/compensation structure, validating the transform-based view of the explicit formulas.

Abstract

We give the converse to Dirichlet's theorem on primes in arithmetic progressions by generalizing an old result of Guinand.

Paper Structure

This paper contains 7 sections, 7 theorems, 63 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Proof
  3. Proof of \ref{['tmZt']}
  4. Proof of \ref{['tmDr']}
  5. Decomposition
  6. Imaginary Part
  7. Numerical Experiment

Key Result

theorem 1.1

Assume RH, let $\delta_{\zeta}\mleft( y \mright) = k_{\gamma}$ if $y = \gamma$ for a zero $1 / 2 + i \gamma$ of order $k_{\gamma}$, and $0$ otherwise. Let $x > e$ and $\varepsilon_{x} = 3 \log x / x$, then for $y \in \mathbb{R}$ as $x \to + \infty$,

Figures (5)

  • Figure 1: Zero indicator of $\zeta\mleft( s \mright)$ around origin.
  • Figure 2: Zero indicator of $\zeta\mleft( s \mright)$ higher up.
  • Figure 3: Zero indicator of $L\mleft( s, \chi \mright)$.
  • Figure 4: Quadratic decomposition.
  • Figure 5: Real vs. Imaginary parts of \ref{['ld']} and \ref{['indZt']}.

Theorems & Definitions (7)

  • theorem 1.1
  • theorem 1.2
  • corollary 1
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4