On the Riemann Hypothesis in Function-Fields

On the Riemann Hypothesis for function fields

We prove a variant of Connes’s trace formula and show how it can be used to give a new proof of the Riemann hypothesis for L-functions with Größencharacter for function fields.

Riemann Hypothesis for function fields

1 1 Preliminaries 1 1.1 Function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Primes and Divisors . . . . . . . . . . . . . . . . . . . . 2 1.2.2 The Picard Group . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Riemann-Roch . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Notation . . . . ...

Riemann hypothesis from the Dedekind psi function

Let P be the set of all primes and ψ(n) = n ∏ n∈P,p|n(1 + 1/p) be the Dedekind psi function. We prove that, under the assumption that the n-th prime pn is always larger than the first Chebyshev function θ(pn), the Riemann hypothesis is satisfied if and only if f(n) = ψ(n)/n − e log logn < 0 for all integers n > n0 = 30 (D), where γ ≈ 0.577 is Euler’s constant. This inequality is equivalent to R...

On The Riemann Hypothesis For The Characteristic p Zeta Function

Comment on the Riemann Hypothesis

The Riemann hypothesis is identified with zeros of N = 4 supersymmetric gauge theory four-point amplitude. The zeros of the ζ(s) function are identified with th complex dimension of the spacetime, or the dimension of the toroidal compactification. A sequence of dimensions are identified in order to map the zeros of the amplitude to the Riemann hypothesis.