Lindelöf hypothesis

In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf^[1] 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, $${\displaystyle \zeta \!\left({\frac {1}{2}}+it\right)\!=O(t^{\varepsilon })}$$ 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, $${\displaystyle \zeta \!\left({\frac {1}{2}}+it\right)\!=o(t^{\varepsilon }).}$$

The μ function

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 1908[2]	Convexity bound
1/6	0.1667	Hardy & Littlewood 1916, 1923[3][4]
163/988	0.1650	Walfisz 1924[5]
27/164	0.1647	Titchmarsh 1932[6]
229/1392	0.164512	Phillips 1933[7]
	0.164511	Rankin 1955[8]
19/116	0.1638	Titchmarsh 1942[9]
15/92	0.1631	Min 1949[10]
6/37	0.16217	Haneke 1962[11]
173/1067	0.16214	Kolesnik 1973[12]
35/216	0.16204	Kolesnik 1982[13]
139/858	0.16201	Kolesnik 1985[14]
9/56	0.1608	Bombieri & Iwaniec 1986[15]
32/205	0.1561	Huxley 2002, 2005[16]
53/342	0.1550	Bourgain 2017[17]
13/84	0.1548	Bourgain 2017[18]

Relation to the Riemann hypothesis

Backlund^[19] (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 function

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^[20] for k = 2 (extending a result of Ingham^[21] who found the leading term).

Conrey and Ghosh^[22] 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 and Snaith^[23] 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 consequences

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

A prime gap conjecture stronger than Ingham's result is Cramér's conjecture, which asserts that^[24][25] $${\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right).}$$

The density hypothesis

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 ${\displaystyle A_{RH}(\sigma >1/2)=0}$. At the other extreme, the upper boundary ${\displaystyle A_{DH}(1-\sigma )=2(1-1/2)=1}$ of this diagram corresponds to the trivial bound coming from the Riemann-von Mangoldt formula.(Various other estimates do exist^[26])

The density hypothesis says that ${\displaystyle N(\sigma ,T)\leq N^{2(1-\sigma )+\epsilon }}$, where ${\displaystyle N(\sigma ,T)}$ denote the number of zeros ${\displaystyle \rho }$ of ${\displaystyle \zeta (s)}$with ${\displaystyle {\mathfrak {R}}(s)\geq \sigma }$ and ${\displaystyle |{\mathfrak {I}}(s)|\leq T}$, and it would follow from the Lindelöf hypothesis.^[27][28]

More generally let ${\displaystyle N(\sigma ,T)\leq N^{A(\sigma )(1-\sigma )+\epsilon }}$ then it is known that this bound roughly correspond to asymptotics for primes in short intervals of length ${\displaystyle x^{1-1/A(\sigma )}}$.^[29][30]

Ingham showed that ${\displaystyle A_{I}(\sigma )={\frac {3}{2-\sigma }}}$ in 1940,^[31] Huxley that ${\displaystyle A_{H}(\sigma )={\frac {3}{3\sigma -1}}}$ in 1971,^[32] and Guth and Maynard that ${\displaystyle A_{GM}(\sigma )={\frac {15}{5\sigma +3}}}$ in 2024 (preprint)^[33][34][35] and these coincide on ${\displaystyle \sigma _{I,GM}=7/10<\sigma _{H,GM}=8/10<\sigma _{I,H}=3/4}$, therefore the latest work of Guth and Maynard gives the closest known value to ${\displaystyle \sigma =1/2}$ as we would expect from the Riemann hypothesis and improves the bound to ${\displaystyle N(\sigma ,T)\leq N^{{\frac {30}{13}}(1-\sigma )+\epsilon }}$ or equivalently the asymptotics to ${\displaystyle x^{17/30}}$.

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-functions

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^[36] and in the GL(1) and GL(2) case by Akshay Venkatesh and Philippe Michel^[37] and in 2021 for the GL(n) case by Paul Nelson.^[38][39]

See also

• Z function#The Lindelöf hypothesis