# Lindelöf hypothesis

In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see Lindelöf (1908)) 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 O notation). Since ε can be replaced by a smaller value, we can also write the conjecture as, 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) Convexity bound
1/6 0.1667 Hardy, Littlewood & ?
163/988 0.1650 Walfisz (1924)
27/164 0.1647 Titchmarsh (1932)
229/1392 0.164512 Phillips (1933)
0.164511 Rankin (1955)
19/116 0.1638 Titchmarsh (1942)
15/92 0.1631 Min (1949)
6/37 0.16217 Haneke (1962)
173/1067 0.16214 Kolesnik (1973)
35/216 0.16204 Kolesnik (1982)
139/858 0.16201 Kolesnik (1985)
32/205 0.1561 Huxley (2002, 2005)
53/342 0.1550 Bourgain (2017)
13/84 0.1548 Bourgain (2017)

## Relation to the Riemann hypothesis

Backlund (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 ck,j. This has been proved by Littlewood for k = 1 and by Heath-Brown (1979) for k = 2 (extending a result of Ingham (1926) 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).

## Other consequences

Denoting by pn 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.

## Notes and references

• Backlund, R. (1918–1919), "Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion", Ofversigt Finska Vetensk. Soc., 61 (9)
• Bourgain, Jean (2017), "Decoupling, exponential sums and the Riemann zeta function", Journal of the American Mathematical Society, 30 (1): 205–224, arXiv:1408.5794, doi:10.1090/jams/860, MR 3556291
• Conrey, J. B.; Farmer, D. W.; Keating, Jonathan P.; Rubinstein, M. O.; Snaith, N. C. (2005), "Integral moments of L-functions", Proceedings of the London Mathematical Society, Third Series, 91 (1): 33–104, arXiv:math/0206018, doi:10.1112/S0024611504015175, ISSN 0024-6115, MR 2149530
• Conrey, J. B.; Farmer, D. W.; Keating, Jonathan P.; Rubinstein, M. O.; Snaith, N. C. (2008), "Lower order terms in the full moment conjecture for the Riemann zeta function", Journal of Number Theory, 128 (6): 1516–1554, arXiv:math/0612843, doi:10.1016/j.jnt.2007.05.013, ISSN 0022-314X, MR 2419176
• Conrey, J. B.; Ghosh, A. (1998), "A conjecture for the sixth power moment of the Riemann zeta-function", International Mathematics Research Notices, 1998 (15): 775–780, doi:10.1155/S1073792898000476, ISSN 1073-7928, MR 1639551
• Edwards, H. M. (1974), Riemann's Zeta Function, New York: Dover Publications, ISBN 978-0-486-41740-0, MR 0466039
• Heath-Brown, D. R. (1979), "The fourth power moment of the Riemann zeta function", Proceedings of the London Mathematical Society, Third Series, 38 (3): 385–422, doi:10.1112/plms/s3-38.3.385, ISSN 0024-6115, MR 0532980
• 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, MR 1956254
• Huxley, M. N. (2005), "Exponential sums and the Riemann zeta function. V", Proceedings of the London Mathematical Society, Third Series, 90 (1): 1–41, doi:10.1112/S0024611504014959, ISSN 0024-6115, MR 2107036
• 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
• Ingham, A. E. (1940), "On the estimation of N(σ,T)", The Quarterly Journal of Mathematics. Oxford. Second Series, 11 (1): 291–292, Bibcode:1940QJMat..11..201I, doi:10.1093/qmath/os-11.1.201, ISSN 0033-5606, MR 0003649
• Karatsuba, Anatoly; Voronin, Sergei (1992), The Riemann zeta-function, de Gruyter Expositions in Mathematics, 5, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-013170-3, MR 1183467
• Keating, Jonathan P.; Snaith, N. C. (2000), "Random matrix theory and ζ(1/2+it)", Communications in Mathematical Physics, 214 (1): 57–89, Bibcode:2000CMaPh.214...57K, CiteSeerX 10.1.1.15.8362, doi:10.1007/s002200000261, ISSN 0010-3616, MR 1794265
• Lindelöf, Ernst (1908), "Quelques remarques sur la croissance de la fonction ζ(s)", Bull. Sci. Math., 32: 341–356
• Motohashi, Yõichi (1995), "A relation between the Riemann zeta-function and the hyperbolic Laplacian", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV, 22 (2): 299–313, ISSN 0391-173X, MR 1354909
• Motohashi, Yõichi (1995), "The Riemann zeta-function and the non-Euclidean Laplacian", Sugaku Expositions, 8 (1): 59–87, ISSN 0898-9583, MR 1335956
• Titchmarsh, Edward Charles (1986), The theory of the Riemann zeta-function (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853369-6, MR 0882550
• Voronin, S.M. (2001) [1994], "L/l058960", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4