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,
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,
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(Ta). 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[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
for all positive integersk 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 ck,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).
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 × nYoung tableaux given by the sequence
1, 1, 2, 42, 24024, 701149020, ... (sequence A039622 in the OEIS).
Other consequences
Denoting by pn the n-th prime number, let A result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0,
if n is sufficiently large.
The density hypothesis says that , where denote the number of zeros of with and , and it would follow from the Lindelöf hypothesis.[27][28]
More generally let then it is known that this bound roughly correspond to asymptotics for primes in short intervals of length .[29][30]
Ingham showed that in 1940,[31]Huxley that in 1971,[32] and Guth and Maynard that in 2024 (preprint)[33][34][35] and these coincide on , therefore the latest work of Guth and Maynard gives the closest known value to as we would expect from the Riemann hypothesis and improves the bound to or equivalently the asymptotics to .
In theory improvements to Baker, Harman, and Pintzestimates 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]
^Hardy, G. H.; Littlewood, J. E. (1923). "On Lindelöf's hypothesis concerning the Riemann zeta-function". Proc. R. Soc. A: 403–412.
^Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes". Acta Mathematica. 41: 119–196. doi:10.1007/BF02422942. ISSN0001-5962.
^Walfisz, Arnold (1924). "Zur Abschätzung von ζ(½ + it)". Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse: 155–158.
^Titchmarsh, E. C. (1932). "On van der Corput's method and the zeta-function of Riemann (III)". The Quarterly Journal of Mathematics. os-3 (1): 133–141. doi:10.1093/qmath/os-3.1.133. ISSN0033-5606.
^Phillips, Eric (1933). "The zeta-function of Riemann: further developments of van der Corput's method". The Quarterly Journal of Mathematics. os-4 (1): 209–225. doi:10.1093/qmath/os-4.1.209. ISSN0033-5606.
^Rankin, R. A. (1955). "Van der Corput's method and the theory of exponent pairs". The Quarterly Journal of Mathematics. 6 (1): 147–153. doi:10.1093/qmath/6.1.147. ISSN0033-5606.
^Kolesnik, Grigori (1982-01-01). "On the order of ζ (1/2+ it ) and Δ( R )". Pacific Journal of Mathematics. 98 (1): 107–122. doi:10.2140/pjm.1982.98.107. ISSN0030-8730.
^Kolesnik, G. (1985). "On the method of exponent pairs". Acta Arithmetica. 45 (2): 115–143. doi:10.4064/aa-45-2-115-143.
^Bombieri, E.; Iwaniec, H. (1986). "On the order of ζ (1/2+ it )". Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 13 (3): 449–472.
^Cramér, Harald (1936). "On the order of magnitude of the difference between consecutive prime numbers". Acta Arithmetica. 2 (1): 23–46. doi:10.4064/aa-2-1-23-46. ISSN0065-1036.
Huxley, M. N. (2002), "Integer points, exponential sums and the Riemann zeta function", Number theory for the millennium, II (Urbana, IL, 2000), A K Peters, pp. 275–290, MR1956254
Ingham, A. E. (1928), "Mean-Value Theorems in the Theory of the Riemann Zeta-Function", Proc. London Math. Soc., s2-27 (1): 273–300, doi:10.1112/plms/s2-27.1.273