Lindelöf hypothesis

In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf<ref>see Template:Harvtxt</ref> about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis. It says that for any ε > 0, <math display="block">\zeta\!\left(\frac{1}{2} + it\right)\! = O(t^\varepsilon)</math> as t tends to infinity (see big O notation). Since ε can be replaced by a smaller value, the conjecture can be restated as follows: for any positive ε, <math display="block">\zeta\!\left(\frac{1}{2} + it\right)\! = o(t^\varepsilon).</math>

The μ functionEdit

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:

μ(1/2) ≤	μ(1/2) ≤	Author
1/4	0.25	Lindelöf<ref>Template:Harvtxt</ref>	Convexity bound
1/6	0.1667	Hardy & Littlewood<ref name="Hardy & Littlewood 1923 pp. 403–412">Template:Cite journal</ref><ref name="Hardy Littlewood 1916 pp. 119–196">Template:Cite journal</ref>
163/988	0.1650	Walfisz 1924<ref name="Walfisz 1924 pp. 115-143">Template:Cite journal</ref>
27/164	0.1647	Titchmarsh 1932<ref name="Titchmarsh 1932 pp. 133–141">Template:Cite journal</ref>
229/1392	0.164512	Phillips 1933<ref name="Phillips 1933 pp. 209–225">Template:Cite journal</ref>
	0.164511	Rankin 1955<ref name="Rankin 1955 pp. 147–153">Template:Cite journal</ref>
19/116	0.1638	Titchmarsh 1942<ref name="Titchmarsh 1942 pp. 11–17">Template:Cite journal</ref>
15/92	0.1631	Min 1949<ref name="Min 1949 pp. 448–472">Template:Cite journal</ref>
6/37	0.16217	Haneke 1962<ref name="Haneke 1963 pp. 357–430">Template:Cite journal</ref>
173/1067	0.16214	Kolesnik 1973<ref name="Kolesnik 1973 pp. 7-30">Template:Cite journal</ref>
35/216	0.16204	Kolesnik 1982<ref name="Kolesnik 1982 pp. 107–122">Template:Cite journal</ref>
139/858	0.16201	Kolesnik 1985<ref name="Kolesnik 1985 pp. 115-143">Template:Cite journal</ref>
9/56	0.1608	Bombieri & Iwaniec 1986<ref>Template:Cite journal</ref>
32/205	0.1561	Huxley<ref>Template:Harvtxt, Template:Harvtxt</ref>
53/342	0.1550	Bourgain<ref>Template:Harvtxt</ref>
13/84	0.1548	Bourgain<ref>Template:Harvtxt</ref>

Relation to the Riemann hypothesisEdit

Backlund<ref>Template:Harvtxt</ref> (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.

Means of powers (or moments) of the zeta functionEdit

The Lindelöf hypothesis is equivalent to the statement that <math display="block">\frac{1}{T} \int_0^T|\zeta(1/2+it)|^{2k}\,dt = O(T^{\varepsilon})</math> 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

for some constants c_{k,Template:Hairspj}. This has been proved by Littlewood for k = 1 and by Heath-Brown<ref>Template:Harvtxt</ref> for k = 2 (extending a result of Ingham<ref>Template:Harvtxt</ref> who found the leading term).

Conrey and Ghosh<ref>Template:Harvtxt</ref> suggested the value

<math>\frac{42}{9!}\prod_ p \left((1-p^{-1})^4(1+4p^{-1}+p^{-2})\right)</math>

for the leading coefficient when k is 6, and Keating and Snaith<ref>Template:Harvtxt</ref> 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 × n Young tableaux given by the sequence

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

Other consequencesEdit

Denoting by p_n the n-th prime number, let <math>g_n = p_{n + 1} - p_n.\ </math> A result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0, <math display="block">g_n\ll p_n^{1/2+\varepsilon}</math> if n is sufficiently large.

A prime gap conjecture stronger than Ingham's result is Cramér's conjecture, which asserts that<ref name="Cramér1936">Template:Cite journal</ref><ref name="Banks Ford Tao 2023 pp. 1471–1518">Template:Cite journal</ref> <math display="block"> g_n = O\!\left((\log p_n)^2\right).</math>

The density hypothesisEdit

File:DensityHypothesis.png

The known zero-free region roughly speaking corresponds to the bottom right corner of the image, and the Riemann hypothesis would push the entire diagram down to the x-axis <math>A_{RH}(\sigma>1/2)=0</math>. At the other extreme, the upper boundary <math>A_{DH}(1-\sigma)=2(1-1/2)=1</math> of this diagram corresponds to the trivial bound coming from the Riemann-von Mangoldt formula.(Various other estimates do exist<ref>Template:Cite arXiv</ref>)

The density hypothesis says that <math>N(\sigma,T)\le N^{2(1-\sigma)+\epsilon}</math>, where <math>N(\sigma,T)</math> denote the number of zeros <math>\rho</math> of <math>\zeta(s)</math>with <math>\mathfrak{R}(s)\ge \sigma</math> and <math>|\mathfrak{I}(s)|\le T</math>, and it would follow from the Lindelöf hypothesis.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

More generally let <math>N(\sigma,T)\le N^{A(\sigma)(1-\sigma)+\epsilon}</math> then it is known that this bound roughly correspond to asymptotics for primes in short intervals of length <math>x^{1-1/A(\sigma)}</math>.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Ingham showed that <math>A_I(\sigma)=\frac{3}{2-\sigma}</math> in 1940,<ref>Template:Cite journal</ref> Huxley that <math>A_H(\sigma)=\frac{3}{3\sigma-1}</math> in 1971,<ref>Template:Cite journal</ref> and Guth and Maynard that <math>A_{GM}(\sigma)=\frac{15}{5\sigma+3}</math> in 2024 (preprint)<ref>Template:Cite arXiv</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and these coincide on <math>\sigma_{I,GM}=7/10<\sigma_{H,GM}=8/10<\sigma_{I,H}=3/4</math>, therefore the latest work of Guth and Maynard gives the closest known value to <math>\sigma=1/2</math> as we would expect from the Riemann hypothesis and improves the bound to <math>N(\sigma,T)\le N^{\frac{30}{13}(1-\sigma)+\epsilon}</math> or equivalently the asymptotics to <math>x^{17/30}</math>.

In theory improvements to Baker, Harman, and Pintz estimates for the Legendre conjecture and better Siegel zeros free regions could also be expected among others.

L-functionsEdit

The Riemann zeta function belongs to a more general family of functions called L-functions. In 2010, new methods to obtain sub-convexity estimates for L-functions in the PGL(2) case were given by Joseph Bernstein and Andre Reznikov<ref>Template:Cite journal</ref> and in the GL(1) and GL(2) case by Akshay Venkatesh and Philippe Michel<ref name="SubconvexityGL2">Template:Cite journal</ref> and in 2021 for the GL(n) case by Paul Nelson.<ref>Template:Cite arXiv</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Notes and referencesEdit

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