Lindelöf Hypothesis
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Lindelöf hypothesis is a
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...
by Finnish mathematician Ernst Leonard Lindelöf about the rate of growth of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
on the critical line. This hypothesis is implied by the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
. It says that for any ''ε'' > 0, \zeta\!\left(\frac + it\right)\! = O(t^\varepsilon) as ''t'' tends to infinity (see
big O notation Big ''O'' notation is a mathematical notation that describes the asymptotic analysis, limiting behavior of a function (mathematics), function when the Argument of a function, argument tends towards a particular value or infinity. Big O is a memb ...
). Since ''ε'' can be replaced by a smaller value, the conjecture can be restated as follows: for any positive ''ε'', \zeta\!\left(\frac + it\right)\! = o(t^\varepsilon).


The μ function

If σ is real, then ''μ''(σ) is defined to be the
infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
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 In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
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 In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of a function, graph of the function lies above or on the graph between the two points. Equivalently, a function is conve ...
. 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 Hardy may refer to: People * Hardy (surname) * Hardy (given name) * Hardy (singer), American singer-songwriter Places Antarctica * Mount Hardy, Enderby Land * Hardy Cove, Greenwich Island * Hardy Rocks, Biscoe Islands Australia * Hardy, ...
and Littlewood to 1/6 by applying
Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
's method of estimating
exponential sum In mathematics, an exponential sum may be a finite Fourier series (i.e. a trigonometric polynomial), or other finite sum formed using the exponential function, usually expressed by means of the function :e(x) = \exp(2\pi ix).\, Therefore, a typi ...
s 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:


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 In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
at least 1/2 + ''ε'' and
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
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 \frac \int_0^T, \zeta(1/2+it), ^\,dt = O(T^) for all positive
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s ''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 In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
. There is a much more precise conjecture about the asymptotic behavior of the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
: it is believed that : \int_0^T, \zeta(1/2+it), ^ \, dt = T\sum_^c_\log(T)^ + 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 :\frac\prod_ p \left((1-p^)^4(1+4p^+p^)\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 In mathematics, a Young tableau (; plural: tableaux) is a combinatorial object useful in representation theory and Schubert calculus. It provides a convenient way to describe the group representations of the symmetric and general linear groups ...
given by the
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
:1, 1, 2, 42, 24024, 701149020, ... .


Other consequences

Denoting by ''p''''n'' the ''n''-th prime number, let g_n = p_ - p_n.\ A result by Albert Ingham shows that the Lindelöf hypothesis implies that, for any ''ε'' > 0, g_n\ll p_n^ if ''n'' is
sufficiently large In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it does not have the said property across all its ordered instances, but will after some instances have ...
. A
prime gap A prime gap is the difference between two successive prime numbers. The ''n''-th prime gap, denoted ''g'n'' or ''g''(''p'n'') is the difference between the (''n'' + 1)-st and the ''n''-th prime numbers, i.e., :g_n = p_ - p_n. ...
conjecture stronger than Ingham's result is
Cramér's conjecture In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and ...
, which asserts that g_n = O\!\left((\log p_n)^2\right).


The density hypothesis

The density hypothesis says that N(\sigma,T)\le N^, where N(\sigma,T) denote the number of zeros \rho of \zeta(s)with \mathfrak(s)\ge \sigma and , \mathfrak(s), \le T, and it would follow from the Lindelöf hypothesis. More generally let N(\sigma,T)\le N^ then it is known that this bound roughly correspond to asymptotics for primes in short intervals of length x^. Ingham showed that A_I(\sigma)=\frac in 1940, Huxley that A_H(\sigma)=\frac in 1971, and Guth and Maynard that A_(\sigma)=\frac in 2024 (preprint) and these coincide on \sigma_=7/10<\sigma_=8/10<\sigma_=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^ or equivalently the asymptotics to x^. In theory improvements to Baker, Harman, and Pintz
estimates In the Westminster system of government, the ''Estimates'' are an outline of government spending for the following fiscal year presented by the Cabinet (government), cabinet to parliament. The Estimates are drawn up by bureaucrats in the finance ...
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-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...
s. 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 and in the GL(1) and GL(2) case by Akshay Venkatesh and Philippe Michel and in 2021 for the GL(''n'') case by Paul Nelson.


See also

* Z function#The Lindelöf hypothesis


Notes and references

* * * * *
2001 pbk reprint
* * * * * * * * * * * * {{DEFAULTSORT:Lindelof hypothesis Conjectures Zeta and L-functions Unsolved problems in number theory Analytic number theory