Lindelöf hypothesis

If σ is real, then μ(σ) is defined to be the infimum of all real numbers a such that ζ(σ + iT) = O(T ^a). It is trivial to check that μ(σ) = 0 for σ > 1, and the functional equation of the zeta function implies that μ(σ) = μ(1 − σ) − σ + 1/2. The Phragmén–Lindelöf theorem implies that μ is a convex function. The Lindelöf hypothesis states μ(1/2) = 0, which together with the above properties of μ implies that μ(σ) is 0 for σ ≥ 1/2 and 1/2 − σ for σ ≤ 1/2.

Lindelöf's convexity result together with μ(1) = 0 and μ(0) = 1/2 implies that 0 ≤ μ(1/2) ≤ 1/4. The upper bound of 1/4 was lowered by Hardy and Littlewood to 1/6 by applying Weyl's method of estimating exponential sums to the approximate functional equation. It has since been lowered to slightly less than 1/6 by several authors using long and technical proofs, as in the following table:

Backlund harvtxt error: no target: CITEREFBacklund (help) (1918–1919) showed that the Lindelöf hypothesis is equivalent to the following statement about the zeros of the zeta function: for every ε > 0, the number of zeros with real part at least 1/2 + ε and imaginary part between T and T + 1 is o(log(T)) as T tends to infinity. The Riemann hypothesis implies that there are no zeros at all in this region and so implies the Lindelöf hypothesis. The number of zeros with imaginary part between T and T + 1 is known to be O(log(T)), so the Lindelöf hypothesis seems only slightly stronger than what has already been proved, but in spite of this it has resisted all attempts to prove it.

The Lindelöf hypothesis is equivalent to the statement that

${\displaystyle {\frac {1}{T}}\int _{0}^{T}|\zeta (1/2+it)|^{2k}\,dt=O(T^{\varepsilon })}$

for all positive integers k and all positive real numbers ε. This has been proved for k = 1 or 2, but the case k = 3 seems much harder and is still an open problem.

There is a much more precise conjecture about the asymptotic behavior of the integral: it is believed that

${\displaystyle \int _{0}^{T}|\zeta (1/2+it)|^{2k}\,dt=T\sum _{j=0}^{k^{2}}c_{k,j}\log(T)^{k^{2}-j}+o(T)}$

for some constants c_k,j. This has been proved by Littlewood for k = 1 and by Heath-Brown (1979) for k = 2 (extending a result of Ingham (1926) harvtxt error: no target: CITEREFIngham1926 (help) who found the leading term).

Conrey & Ghosh (1998) suggested the value

${\displaystyle {\frac {42}{9!}}\prod _{p}\left((1-p^{-1})^{4}(1+4p^{-1}+p^{-2})\right)}$

for the leading coefficient when k is 6, and Keating & Snaith (2000) used random matrix theory to suggest some conjectures for the values of the coefficients for higher k. The leading coefficients are conjectured to be the product of an elementary factor, a certain product over primes, and the number of n by n Young tableaux given by the sequence

1, 1, 2, 42, 24024, 701149020, … (sequence A039622 in the OEIS).

Denoting by p_n the n-th prime number, a result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0,

${\displaystyle p_{n+1}-p_{n}\ll p_{n}^{1/2+\varepsilon }\,}$

if n is sufficiently large. However, this result is much worse than that of the large prime gap conjecture.

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