Lindelöf hypothesis explained

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,$$\backslash zeta\backslash !\backslash left(\backslash frac\; +\; it\backslash right)\backslash !\; =\; O(t^\backslash varepsilon)$$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,$$\backslash zeta\backslash !\backslash left(\backslash frac\; +\; it\backslash right)\backslash !\; =\; o(t^\backslash varepsilon).$$

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

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$$\backslash frac\; \backslash int\_0^T|\backslash zeta(1/2+it)|^\backslash ,dt\; =\; O(T^)$$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

\int ^{T|\zeta( 1 / 2+ it)|} _{0} ^{2k} dt = T\sum ^{k ^{2}} _{j =0} 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 for k = 2 (extending a result of Ingham who found the leading term).

Conrey and Ghosh suggested the value

(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 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, ... .

Other consequences

Denoting by p_n the n-th prime number, let

g _{n} = p _{n} - p _{n.} A result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ε > 0,$$g\_n\backslash ll\; p\_n^$$if n is sufficiently large.

A prime gap conjecture stronger than Ingham's result is Cramér's conjecture, which asserts that^{[16] [17]} $$g\_n\; =\; O\backslash !\backslash left((\backslash log\; p\_n)^2\backslash right).$$

The density hypothesis

The density hypothesis says that

N(\sigma,T)\le N ^{2(1-\sigma)+\epsilon} , where N(\sigma,T) denote the number of zeros \rho of \zeta(s) with ak{R}(s)\ge \sigma and | ak{I}(s)|\le T , and it would follow from the Lindelöf hypothesis.^{[18] [19]}

More generally let

N(\sigma,T)\le N ^{A(\sigma)(1-\sigma)+\epsilon} then it is known that this bound roughly correspond to asymptotics for primes in short intervals of length x ^{1-1/A(\sigma)} .^{[20] [21]}

Ingham showed that

A _{I(\sigma) = (3)/(2- \sigma)} in 1940,^[22] Huxley that A _{H(\sigma) = (3)/(3 \sigma -1)} in 1971,^[23] and Guth and Maynard that A _{GM} (\sigma) = (15)/(5 \sigma +3) in 2024 (preprint)^{[24] [25] [26]} and these coincide on \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 \sigma =1 / 2 as we would expect from the Riemann hypothesis and improves the bound to N(\sigma,T)\le N ^{(30)/(( 1- \sigma) + \epsilon 13)} or equivalently the asymptotics to 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^[27] and in the GL(1) and GL(2) case by Akshay Venkatesh and Philippe Michel^[28] and in 2021 for the GL(n) case by Paul Nelson.^{[29] [30]}

See also

• Z function#The Lindelöf hypothesis

Notes and references

• • • • • • 2001 pbk reprint

Notes and References

1. see
2. Hardy . G. H. . Littlewood . J. E.. 1923 . On Lindelöf's hypothesis concerning the Riemann zeta-function . Proc. R. Soc. A . 403–412.
3. Hardy . G. H. . Littlewood . J. E. . Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes . Acta Mathematica . 41 . 1916 . 0001-5962 . 10.1007/BF02422942 . 119–196.
4. Walfisz . Arnold . 1924 . Zur Abschätzung von ζ(½ + it) . Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse . 155–158.
5. Titchmarsh . E. C. . On van der Corput's method and the zeta-function of Riemann (III) . The Quarterly Journal of Mathematics . os-3 . 1 . 1932 . 0033-5606 . 10.1093/qmath/os-3.1.133 . 133–141.
6. Phillips . Eric . The zeta-function of Riemann: further developments of van der Corput's method . The Quarterly Journal of Mathematics . os-4 . 1 . 1933 . 0033-5606 . 10.1093/qmath/os-4.1.209 . 209–225.
7. Rankin . R. A. . Van der Corput's method and the theory of exponent pairs . The Quarterly Journal of Mathematics . 6 . 1 . 1955 . 0033-5606 . 10.1093/qmath/6.1.147 . 147–153.
8. Titchmarsh . E. C. . On the order of ζ(½+ it) . The Quarterly Journal of Mathematics . os-13 . 1 . 1942 . 0033-5606 . 10.1093/qmath/os-13.1.11 . 11–17.
9. Min . Szu-Hoa . On the order of (1/2+) . Transactions of the American Mathematical Society . 65 . 3 . 1949 . 0002-9947 . 10.1090/S0002-9947-1949-0030996-6 . 448–472.
10. Haneke . W. . Verschärfung der Abschätzung von ξ(½+it) . Acta Arithmetica . 8 . 4 . 1963 . German . 0065-1036 . 10.4064/aa-8-4-357-430 . 357–430.
11. Kolesnik . G. A. . On the estimation of some trigonometric sums . Acta Arithmetica . 25 . 1 . Russian . 1973 . 0065-1036 . 7–30 . 2024-02-05.
12. Kolesnik . Grigori . On the order of ζ (1/2+ it) and Δ(R) . Pacific Journal of Mathematics . 98 . 1 . 1982-01-01 . 0030-8730 . 10.2140/pjm.1982.98.107 . 107–122.
13. Kolesnik . G. . 1985 . On the method of exponent pairs . Acta Arithmetica . 45 . 2 . 115–143. 10.4064/aa-45-2-115-143 .
14. Bombieri . E. . Iwaniec . H. . On the order of ζ (1/2+ it) . Annali della Scuola Normale Superiore di Pisa - Classe di Scienze . 1986 . 13 . 3 . 449–472.
15. ,
16. Cramér . Harald . 1936 . On the order of magnitude of the difference between consecutive prime numbers . Acta Arithmetica . 2 . 1 . 23–46 . 10.4064/aa-2-1-23-46 . 0065-1036.
17. Banks . William . Ford . Kevin . Tao . Terence . 2023 . Large prime gaps and probabilistic models . Inventiones Mathematicae . 233 . 3 . 1471–1518 . 1908.08613 . 10.1007/s00222-023-01199-0 . 0020-9910.
18. Web site: 25a . 2024-07-16 . aimath.org.
19. Web site: Density hypothesis - Encyclopedia of Mathematics . 2024-07-16 . encyclopediaofmath.org.
20. Web site: 2024-06-04 . New Bounds for Large Values of Dirichlet Polynomials, Part 1 - Videos Institute for Advanced Study . 2024-07-16 . www.ias.edu . en.
21. Web site: 2024-06-04 . New Bounds for Large Values of Dirichlet Polynomials, Part 2 - Videos Institute for Advanced Study . 2024-07-16 . www.ias.edu . en.
22. Ingham . A. E. . 1940 . ON THE ESTIMATION OF N (σ, T) . The Quarterly Journal of Mathematics . en . os-11 . 1 . 201–202 . 10.1093/qmath/os-11.1.201 . 0033-5606.
23. Huxley . M. N. . 1971 . On the Difference between Consecutive Primes. . Inventiones Mathematicae . 15 . 2 . 164–170 . 10.1007/BF01418933 . 0020-9910.
24. Guth . Larry . Maynard . James . 2024 . New large value estimates for Dirichlet polynomials . math.NT . 2405.20552.
25. Web site: Bischoff . Manon . The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved . 2024-07-16 . Scientific American . en.
26. Web site: Cepelewicz . Jordana . 2024-07-15 . 'Sensational' Proof Delivers New Insights Into Prime Numbers . 2024-07-16 . Quanta Magazine . en.
27. Bernstein. Joseph. Reznikov. Andre. 2010-10-05. Subconvexity bounds for triple L -functions and representation theory. Annals of Mathematics. en. 172. 3. 1679–1718. 10.4007/annals.2010.172.1679. 14745024. 0003-486X. free. math/0608555.
28. Michel. Philippe. Philippe Michel (number theorist). Venkatesh. Akshay. 2010. The subconvexity problem for GL₂. Publications Mathématiques de l'IHÉS. 111. 1. 171–271. 0903.3591. 10.1.1.750.8950. 10.1007/s10240-010-0025-8. 14155294.
29. Nelson. Paul D.. 2021-09-30. Bounds for standard $L$-functions. math.NT. 2109.15230.
30. Web site: Hartnett. Kevin. 2022-01-13. Mathematicians Clear Hurdle in Quest to Decode Primes. 2022-02-17. Quanta Magazine. en.