math.NT — Number Theory
Contributed by Changxia Lin
Let Ξ(t)=ξ(1/2+it) denote the Riemann ξ-function in the
critical-line parametrisation, let Ξ(t)=Σ_{k≥0}(-1)^k c_k t^{2k} be
its even Taylor expansion at the origin, and let
D(n,r)=det[c_{n+i-j}]_{i,j=0}^{r-1} be the associated Toeplitz minors.
By the Schoenberg--Edrei characterisation of Pólya frequency
sequences, positivity of D(n,r) for all n≥0 and all r≥1 is equivalent
to the Riemann hypothesis (RH); for each fixed r the inequalities
D(n,r)≥0 supply a necessary condition for RH, and for r=2 they were
proved unconditionally by Csordas, Norfolk and Varga, so that the
second level of the resulting hierarchy is not discriminating. We show
that this failure of discrimination persists far beyond level 2, and
we determine the range over which it does so: every level r≤43 is
non-discriminating.
Parametrising the zeros of the generating function G(z)=Σ_{k≥0}c_k z^k
by β_j=-1/z_j, the dual Jacobi--Trudi identity yields the exact
identity D(n,r)=c_0^r s_{(r^n)}(β), where s_{(r^n)} denotes the Schur
function of the rectangular partition (r^n) in the alphabet β. A zero
of ζ off the critical line contributes to β a conjugate pair
ρ e^{±iθ} whose argument θ coincides with the vertical angle of the
corresponding violating pair, whereas zeros on the critical line
contribute positive real variables. For an alphabet of cardinality two
we prove the closed form
s_{(a,b)}(ρ e^{iθ}, ρ e^{-iθ}) = ρ^{a+b} sin((a+1-b)θ)/sin θ
= ρ^{a+b} U_{a-b}(cos θ),
where U_k is the Chebyshev polynomial of the second kind; the sign is
therefore governed by the criterion U_k(cos θ)>0 if and only if
(k+1)θ<π.
Through the coproduct expansion s_λ(X⊔Y)=Σ_μ s_μ(X)s_{λ/μ}(Y), the
Littlewood--Richardson expansion s_{λ/μ}=Σ_ν c^λ_{μν}s_ν of each skew
factor, and the positivity of Schur functions on positive real
alphabets, the sign analysis reduces to two-variable constituents.
Since c^λ_{μν}>0 forces ν⊆λ=(r^n), the index k=ν_1-ν_2 of each
constituent is bounded by the number of columns r and is independent
of the shift n. All constituents then have the same sign, so no
cancellation occurs. The same bound persists for an alphabet
containing several conjugate pairs: iterating the coproduct over the
pairs and using the fact that the entry 1 occurs at most once in each
column of a semistandard tableau bounds every constituent index by
ν_1≤r, independently of the number of pairs. This yields the
following result.
Blind Zone Theorem. Let the alphabet contain any number of violating
pairs, each subtending a vertical angle at most θ_max, and suppose
θ_max<π/(r+1). Then every minor D(n,r') with r'≤r is strictly
positive, notwithstanding the presence of non-real zeros.
The critical strip together with γ_1=14.134725141... forces
θ≤θ_max=2 arctan(1/(2γ_1))=0.07071826... radians, whence
π/θ_max=44.424...; consequently every order r≤43 is
non-discriminating: for all n≥0 and all r≤43 the minor D(n,r) is
positive if RH fails and non-negative if RH holds, so no minor of
order r≤43 can certify either the truth or the falsity of RH. The
bound is on the order r and not on the shift n; the corresponding
n-based assertion is false, and the mechanism responsible is exhibited
in the remarks section. The threshold is established for an arbitrary
number of violating pairs; the case of several pairs is settled in the
coproduct section and requires no separation condition on the moduli
and no bound on the number of pairs. No claim of a proof of the
Riemann hypothesis is made.
Primarily human-written textHuman understanding: all parts1 Riemann zeta function 2 xi-function 3 Turan inequalities 4 Toeplitz minors