Lindelöf hypothesis

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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, [math]\displaystyle{ \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]\displaystyle{ \zeta\!\left(\frac{1}{2} + it\right)\! = o(t^\varepsilon). }[/math]

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[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[citation needed]
35/216 0.16204 Kolesnik 1982[12]
139/858 0.16201 Kolesnik 1985[13]
32/205 0.1561 Huxley[14]
53/342 0.1550 Bourgain[15]
13/84 0.1548 Bourgain[16]

Relation to the Riemann hypothesis

Backlund[17] (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 [math]\displaystyle{ \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

[math]\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) }[/math]

for some constants ck,j . This has been proved by Littlewood for k = 1 and by Heath-Brown[18] for k = 2 (extending a result of Ingham[19] who found the leading term).

Country and Ghosh[20] suggested the value

[math]\displaystyle{ \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[21] 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 pn the n-th prime number, let [math]\displaystyle{ 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]\displaystyle{ 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:[22][23]

[math]\displaystyle{ g_n = O\!\left((\log p_n)^2\right)\!. }[/math]

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[24] and in the GL(1) and GL(2) case by Akshay Venkatesh and Philippe Michel[25] and in 2021 for the GL(n) case by Paul Nelson.[26][27]

See also

Notes and references

  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. Royal 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 (0): 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)" (in German). Acta Arithmetica 8 (4): 357–430. doi:10.4064/aa-8-4-357-430. ISSN 0065-1036. 
  12. ↑ 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. 
  13. ↑ Kolesnik, G. (1985). "On the method of exponent pairs". Acta Arithmetica 45 (2): 115-143. 
  14. ↑ (Huxley 2002), (Huxley 2005)
  15. ↑ (Bourgain 2017)
  16. ↑ (Bourgain 2017)
  17. ↑ (Backlund 1918–1919)
  18. ↑ (Heath-Brown 1979)
  19. ↑ (Ingham 1928)
  20. ↑ (Conrey Ghosh)
  21. ↑ (Keating Snaith)
  22. ↑ 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. 
  23. ↑ Banks, William; Ford, Kevin; Tao, Terence (2023). "Large prime gaps and probabilistic models". Inventiones mathematicae 233 (3): 1471–1518. doi:10.1007/s00222-023-01199-0. ISSN 0020-9910. 
  24. ↑ Bernstein, Joseph; Reznikov, Andre (2010-10-05). "Subconvexity bounds for triple L -functions and representation theory" (in en). Annals of Mathematics 172 (3): 1679–1718. doi:10.4007/annals.2010.172.1679. ISSN 0003-486X. http://annals.math.princeton.edu/2010/172-3/p05. 
  25. ↑ Michel, Philippe; Venkatesh, Akshay (2010). "The subconvexity problem for GL2". Publications Mathématiques de l'IHÉS 111 (1): 171–271. doi:10.1007/s10240-010-0025-8. 
  26. ↑ Nelson, Paul D. (2021-09-30). "Bounds for standard $L$-functions". arXiv:2109.15230 [math.NT].
  27. ↑ Hartnett, Kevin (2022-01-13). "Mathematicians Clear Hurdle in Quest to Decode Primes" (in en). https://www.quantamagazine.org/mathematicians-clear-hurdle-in-quest-to-decode-prime-numbers-20220113/. 



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