Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An electrostatic depiction of the validity of the Riemann Hypothesis and
  a formula for the N-th zero at large N   We construct a vector field E from the real and imaginary parts of an entire
function xi (z) which arises in the quantum statistical mechanics of
relativistic gases when the spatial dimension d is analytically continued into
the complex z plane. This function is built from the Gamma and Riemann zeta
functions and is known to satisfy the functional identity xi (z) = xi (1-z). E
satisfies the conditions for a static electric field. The structure of E in the
critical strip is determined by its behavior near the Riemann zeros on the
critical line Re (z) = 1/2, where each zero can be assigned a + or - vorticity
of a related pseudo-magnetic field. Using these properties, we show that a
hypothetical Riemann zero that is off the critical line leads to a frustration
of this "electric" field.
  We formulate our argument more precisely in terms of the potential Phi
satisfying E = - gradient Phi, and construct Phi explicitly. One outcome of our
analysis is a formula for the n-th zero on the critical line for large n
expressed as the solution of a simple transcendental equation. Riemann's
counting formula for the number of zeros on the entire critical strip can be
derived from this formula. Our result is much stronger than Riemann's counting
formula, since it provides an estimate of the n-th zero along the critical
line.
  This provides a simple way to estimate very high zeros to very good accuracy,
and we estimate the 10^{10^6}-th one.

5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
3
The categories of this paper are: physics
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex