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Analytic criteria equivalent to the Riemann hypothesis

Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve the divisor function σ(n).)

The Riesz criterion was given by Riesz (1916), to the effect that the bound

holds for all ε > 0 if and only if the Riemann hypothesis holds. See also the Hardy–Littlewood criterion.

Nyman (1950) proved that the Riemann hypothesis is true if and only if the space of functions of the form

where ρ (z) is the fractional part of z, 0 ≤ θ ≤ 1, and

is dense in the Hilbert space L (0,1) of square-integrable functions on the unit interval. Beurling (1955) extended this by showing that the zeta function has no zeros with real part greater than 1/ p if and only if this function space is dense in L (0,1). This Nyman-Beurling criterion was strengthened by Baez-Duarte to the case where θ ν ∈ { 1 / k } k ≥ 1 {\displaystyle \theta _{\nu }\in \{1/k\}_{k\geq 1}}.

Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation

has no non-trivial bounded solutions φ {\displaystyle \varphi } for 1 / 2 < σ < 1 {\displaystyle 1/2<\sigma <1}.

Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is Li's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.

Speiser (1934) proved that the Riemann hypothesis is equivalent to the statement that ζ ′ (s) {\displaystyle \zeta '(s)}, the derivative of ζ (s) {\displaystyle \zeta (s)}, has no zeros in the strip

That ζ (s) {\displaystyle \zeta (s)} has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.

The Farey sequence provides two equivalences, due to Jerome Franel and Edmund Landau in 1924.

The de Bruijn–Newman constant denoted by Λ and named after Nicolaas Govert de Bruijn and Charles M. Newman, is definedas the unique real number such that the function

that is parametrised by a real parameter λ, has a complex variable z and is defined using a super-exponentially decaying function

has only real zeros if and only if λ ≥ Λ.Since the Riemann hypothesis is equivalent to the claim that all the zeroes of H (0, z) are real, the Riemann hypothesis is equivalent to the conjecture that Λ ≤ 0 {\displaystyle \Lambda \leq 0}. Brad Rodgers and Terence Tao discovered the equivalence is actually Λ = 0 {\displaystyle \Lambda =0} by proving zero to be the lower bound of the constant. Proving zero is also the upper bound would therefore prove the Riemann hypothesis. As of April 2020 the upper bound is Λ ≤ 0.2 {\displaystyle \Lambda \leq 0.2}.