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

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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:

μ(1/2) ≤ μ(1/2) ≤ Author
1/4 0.25 Lindelöf[2] Convexity bound
1/6 0.1667 Hardy & Littlewood[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[16]
53/342 0.1550 Bourgain[17]
13/84 0.1548 Bourgain[18]

Relation to the Riemann hypothesis

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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

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The Lindelöf hypothesis is equivalent to the statement that   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 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).

Conrey and Ghosh[22] suggested the value

 

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

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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.

A prime gap conjecture stronger than Ingham's result is Cramér's conjecture, which asserts that[24][25]  

The density hypothesis

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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  . At the other extreme, the upper boundary   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  , 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 Pintz estimates for the Legendre conjecture and better Siegel zeros free regions could also be expected among others.

L-functions

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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

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Notes and references

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  1. ^ see Lindelöf (1908)
  2. ^ Lindelöf (1908)
  3. ^ Hardy, G. H.; Littlewood, J. E. (1923). "On Lindelöf's hypothesis concerning the Riemann zeta-function". Proc. R. Soc. A: 403–412.
  4. ^ 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. ISSN 0001-5962.
  5. ^ Walfisz, Arnold (1924). "Zur Abschätzung von ζ(½ + it)". Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse: 155–158.
  6. ^ 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. ISSN 0033-5606.
  7. ^ 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. ISSN 0033-5606.
  8. ^ 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. ISSN 0033-5606.
  9. ^ Titchmarsh, E. C. (1942). "On the order of ζ(½+ it )". The Quarterly Journal of Mathematics. os-13 (1): 11–17. doi:10.1093/qmath/os-13.1.11. ISSN 0033-5606.
  10. ^ Min, Szu-Hoa (1949). "On the order of 𝜁(1/2+𝑖𝑡)". Transactions of the American Mathematical Society. 65 (3): 448–472. doi:10.1090/S0002-9947-1949-0030996-6. ISSN 0002-9947.
  11. ^ Haneke, W. (1963). "Verschärfung der Abschätzung von ξ(½+it)". Acta Arithmetica (in German). 8 (4): 357–430. doi:10.4064/aa-8-4-357-430. ISSN 0065-1036.
  12. ^ Kolesnik, G. A. (1973). "On the estimation of some trigonometric sums". Acta Arithmetica (in Russian). 25 (1): 7–30. ISSN 0065-1036. Retrieved 2024-02-05.
  13. ^ 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. ISSN 0030-8730.
  14. ^ Kolesnik, G. (1985). "On the method of exponent pairs". Acta Arithmetica. 45 (2): 115–143. doi:10.4064/aa-45-2-115-143.
  15. ^ 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.
  16. ^ Huxley (2002), Huxley (2005)
  17. ^ Bourgain (2017)
  18. ^ Bourgain (2017)
  19. ^ Backlund (1918–1919)
  20. ^ Heath-Brown (1979)
  21. ^ Ingham (1928)
  22. ^ Conrey & Ghosh (1998)
  23. ^ Keating & Snaith (2000)
  24. ^ 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. ISSN 0065-1036.
  25. ^ Banks, William; Ford, Kevin; Tao, Terence (2023). "Large prime gaps and probabilistic models". Inventiones Mathematicae. 233 (3): 1471–1518. arXiv:1908.08613. doi:10.1007/s00222-023-01199-0. ISSN 0020-9910.
  26. ^ Trudgian, Timothy S.; Yang, Andrew (2023). "Toward optimal exponent pairs". arXiv:2306.05599 [math.NT].
  27. ^ "25a". aimath.org. Retrieved 2024-07-16.
  28. ^ "Density hypothesis - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2024-07-16.
  29. ^ "New Bounds for Large Values of Dirichlet Polynomials, Part 1 - Videos | Institute for Advanced Study". www.ias.edu. 2024-06-04. Retrieved 2024-07-16.
  30. ^ "New Bounds for Large Values of Dirichlet Polynomials, Part 2 - Videos | Institute for Advanced Study". www.ias.edu. 2024-06-04. Retrieved 2024-07-16.
  31. ^ Ingham, A. E. (1940). "ON THE ESTIMATION OF N (σ, T )". The Quarterly Journal of Mathematics. os-11 (1): 201–202. doi:10.1093/qmath/os-11.1.201. ISSN 0033-5606.
  32. ^ Huxley, M. N. (1971). "On the Difference between Consecutive Primes". Inventiones Mathematicae. 15 (2): 164–170. doi:10.1007/BF01418933. ISSN 0020-9910.
  33. ^ Guth, Larry; Maynard, James (2024). "New large value estimates for Dirichlet polynomials". arXiv:2405.20552 [math.NT].
  34. ^ Bischoff, Manon. "The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved". Scientific American. Retrieved 2024-07-16.
  35. ^ Cepelewicz, Jordana (2024-07-15). "'Sensational' Proof Delivers New Insights Into Prime Numbers". Quanta Magazine. Retrieved 2024-07-16.
  36. ^ Bernstein, Joseph; Reznikov, Andre (2010-10-05). "Subconvexity bounds for triple L -functions and representation theory". Annals of Mathematics. 172 (3): 1679–1718. arXiv:math/0608555. doi:10.4007/annals.2010.172.1679. ISSN 0003-486X. S2CID 14745024.
  37. ^ Michel, Philippe; Venkatesh, Akshay (2010). "The subconvexity problem for GL2". Publications Mathématiques de l'IHÉS. 111 (1): 171–271. arXiv:0903.3591. CiteSeerX 10.1.1.750.8950. doi:10.1007/s10240-010-0025-8. S2CID 14155294.
  38. ^ Nelson, Paul D. (2021-09-30). "Bounds for standard $L$-functions". arXiv:2109.15230 [math.NT].
  39. ^ Hartnett, Kevin (2022-01-13). "Mathematicians Clear Hurdle in Quest to Decode Primes". Quanta Magazine. Retrieved 2022-02-17.