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Heat Expansion and Zeta

Alain Connes

Abstract

We compute the full asymptotic expansion of the heat kernel Trace$(\exp(-tD^2))$ where $D$ is, assuming RH, the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. The coefficients of the expansion are explicit expressions involving Bernoulli and Euler numbers. We relate the divergent terms with the heat kernel expansion of the Dirac square root of the prolate wave operator investigated in our joint work with Henri Moscovici.

Heat Expansion and Zeta

Abstract

We compute the full asymptotic expansion of the heat kernel Trace where is, assuming RH, the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. The coefficients of the expansion are explicit expressions involving Bernoulli and Euler numbers. We relate the divergent terms with the heat kernel expansion of the Dirac square root of the prolate wave operator investigated in our joint work with Henri Moscovici.
Paper Structure (5 sections, 4 theorems, 75 equations, 2 figures)

This paper contains 5 sections, 4 theorems, 75 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Explicit formula
  3. Archimedean contribution
  4. Contribution of the primes
  5. Counting function

Key Result

Theorem 1.1

Assume RH and let $D$ be the self-adjoint operator whose spectrum is formed of the imaginary parts of non-trivial zeros of the Riemann zeta function. One then has the asymptotic expansion for $t\to 0$ where $a_0=-\frac{1}{4}$ and for $k>0$, using Bernouilli numbers $B_j$ and Euler numbers $E(k)$,

Figures (2)

  • Figure 1: Graph of ${\rm Tr}(\exp(-D^2/a))$.
  • Figure 2: Graph of discrepancy.

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 5.1