Riemann hypothesis

For the musical term, see Riemannian theory.

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 ⁠1/2⁠. Many consider it to be the most important unsolved problem in pure mathematics.^[1] It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.

Unsolved problem in mathematics:

Do all non-trivial zeroes of the Riemann zeta function have a real part of one half?

(more unsolved problems in mathematics)

This plot of Riemann's zeta (ζ) function (here with argument z) shows trivial zeros where ζ(z) = 0, a pole where ζ(z) = ${\displaystyle \infty }$, the critical line of nontrivial zeros with Re(z) = 1/2 and slopes of absolute values.

The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:

The real part of every nontrivial zero of the Riemann zeta function is ⁠1/2⁠.

Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers ⁠1/2⁠ + i t, where t is a real number and i is the imaginary unit.

Riemann zeta function

The Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergent infinite series

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }$

Leonhard Euler considered this series in the 1730s for real values of s, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product

${\displaystyle \zeta (s)=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}}={\frac {1}{1-2^{-s}}}\cdot {\frac {1}{1-3^{-s}}}\cdot {\frac {1}{1-5^{-s}}}\cdot {\frac {1}{1-7^{-s}}}\cdots }$

where the infinite product extends over all prime numbers p.^[2]

The Riemann hypothesis discusses zeros outside the region of convergence of this series and Euler product. To make sense of the hypothesis, it is necessary to analytically continue the function to obtain a form that is valid for all complex s. Because the zeta function is meromorphic, all choices of how to perform this analytic continuation will lead to the same result, by the identity theorem. A first step in this continuation observes that the series for the zeta function and the Dirichlet eta function satisfy the relation

${\displaystyle \left(1-{\frac {2}{2^{s}}}\right)\zeta (s)=\eta (s)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-\cdots ,}$

within the region of convergence for both series. But the eta function series on the right converges not just when the real part of s is greater than one, but more generally whenever s has positive real part. Thus the zeta function can be redefined as ${\displaystyle \eta (s)/(1-2/2^{s})}$, extending it from Re(s) > 1 to a larger domain: Re(s) > 0, except for the points where ${\displaystyle 1-2/2^{s}}$ is zero. These are the points ${\displaystyle s=1+2\pi in/\log 2}$ where ${\displaystyle n}$ can be any nonzero integer; the zeta function can be extended to these values too by taking limits (see Dirichlet eta function § Landau's problem with ζ(s) = η(s)/0 and solutions), giving a finite value for all values of s with positive real part except the simple pole at s = 1.

In the strip 0 < Re(s) < 1 this extension of the zeta function satisfies the functional equation

${\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).}$

One may then define ζ(s) for all remaining nonzero complex numbers s (Re(s) ≤ 0 and s ≠ 0) by applying this equation outside the strip, and letting ζ(s) equal the right side of the equation whenever s has non-positive real part (and s ≠ 0).

If s is a negative even integer, then ζ(s) = 0, because the factor sin(πs/2) vanishes; these are the zeta function's trivial zeros. (If s is a positive even integer this argument does not apply because the zeros of the sine function are canceled by the poles of the gamma function as it takes negative integer arguments.)

The value ζ(0) = −1/2 is not determined by the functional equation, but is the limiting value of ζ(s) as s approaches zero. The functional equation also implies that the zeta function has no zeros with negative real part other than the trivial zeros, so all nontrivial zeros lie in the critical strip where s has real part between 0 and 1.

• 
  Riemann zeta function along the critical line with Re(s) = 1/2. Real values are shown on the horizontal axis and imaginary values are on the vertical axis. Re(ζ(1/2 + it)), Im(ζ(1/2 + it)) is plotted with t ranging between −30 and 30.^[3]
• Animation showing in 3D the Riemann zeta function critical strip (blue, where s has real part between 0 and 1), critical line (red, for real part of s equals 0.5) and zeroes (cross between red and orange): [x,y,z] = [Re(ζ(r + it)), Im(ζ(r + it)), t] with 0.1 ≤ r ≤ 0.9 and 1 ≤ t ≤ 51
• 
  The real part (red) and imaginary part (blue) of the Riemann zeta function ζ(s) along the critical line in the complex plane with real part Re(s) = 1/2. The first nontrivial zeros, where ζ(s) equals zero, occur where both curves touch the horizontal x-axis, for complex numbers with imaginary parts Im(s) equaling ±14.135, ±21.022 and ±25.011.

Origin

... es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habe indess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für den nächsten Zweck meiner Untersuchung entbehrlich schien.

... it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the immediate objective of my investigation.

— Riemann's statement of the Riemann hypothesis, from (Riemann 1859). (He was discussing a variant of the zeta function, modified in a way that the real line be mapped to the critical line.)

At the death of Riemann, a note was found among his papers, saying "These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself].

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition

Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes π(x) less than or equal to a given number x, which he published in his 1859 paper "On the Number of Primes Less Than a Given Magnitude". His formula was given in terms of the related function

${\displaystyle \Pi (x)=\pi (x)+{\tfrac {1}{2}}\pi (x^{1/2})+{\tfrac {1}{3}}\pi (x^{1/3})+{\tfrac {1}{4}}\pi (x^{1/4})+{\tfrac {1}{5}}\pi (x^{1/5})+{\tfrac {1}{6}}\pi (x^{1/6})+\cdots }$

which counts the primes and prime powers up to x, counting a prime power pⁿ as 1⁄n. The number of primes can be recovered from this function by using the Möbius inversion formula,

${\displaystyle {\begin{aligned}\pi (x)&=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi (x^{1/n})\\&=\Pi (x)-{\frac {1}{2}}\Pi (x^{1/2})-{\frac {1}{3}}\Pi (x^{1/3})-{\frac {1}{5}}\Pi (x^{1/5})+{\frac {1}{6}}\Pi (x^{1/6})-\cdots ,\end{aligned}}}$

where μ is the Möbius function. Riemann's formula is then

${\displaystyle \Pi _{0}(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\log 2+\int _{x}^{\infty }{\frac {dt}{t(t^{2}-1)\log t}}}$

where the sum is over the nontrivial zeros of the zeta function and where Π₀ is a slightly modified version of Π that replaces its value at its points of discontinuity by the average of its upper and lower limits:

${\displaystyle \Pi _{0}(x)=\lim _{\varepsilon \to 0}{\frac {\Pi (x-\varepsilon )+\Pi (x+\varepsilon )}{2}}.}$

The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral

${\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\log t}}.}$

The terms li(x^ρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e. they should be considered as Ei(ρ log x). The other terms also correspond to zeros: the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. For some graphs of the sums of the first few terms of this series see Riesel & Göhl (1970) or Zagier (1977).

This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributed about the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checked that a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemann hypothesis.

The result has caught the imagination of most mathematicians because it is so unexpected, connecting two seemingly unrelated areas in mathematics; namely, number theory, which is the study of the discrete, and complex analysis, which deals with continuous processes.

— (Burton 2006, p. 376)

Consequences

The practical uses of the Riemann hypothesis include many propositions known to be true under the Riemann hypothesis, and some that can be shown to be equivalent to the Riemann hypothesis.

Distribution of prime numbers

Riemann's explicit formula for the number of primes less than a given number states that, in terms of a sum over the zeros of the Riemann zeta function, the magnitude of the oscillations of primes around their expected position is controlled by the real parts of the zeros of the zeta function. In particular, the error term in the prime number theorem is closely related to the position of the zeros. For example, if β is the upper bound of the real parts of the zeros, then^[4] ${\displaystyle \pi (x)-\operatorname {li} (x)=O\left(x^{\beta }\log x\right)}$, where ${\displaystyle \pi (x)}$ is the prime-counting function, ${\displaystyle \operatorname {li} (x)}$ is the logarithmic integral function, ${\displaystyle \log(x)}$ is the natural logarithm of x, and big O notation is used here. It is already known that 1/2 ≤ β ≤ 1.^[5]

Corrections to an estimate of the prime-counting function using zeros of the zeta function. The magnitude of the correction term is determined by the real part of the zero being added in the correction.

Von Koch (1901) proved that the Riemann hypothesis implies the "best possible" bound for the error of the prime number theorem. A precise version of von Koch's result, due to Schoenfeld (1976), says that the Riemann hypothesis implies

${\displaystyle |\pi (x)-\operatorname {li} (x)|<{\frac {1}{8\pi }}{\sqrt {x}}\log(x),\qquad {\text{for all }}x\geq 2657,}$

Schoenfeld (1976) also showed that the Riemann hypothesis implies

${\displaystyle |\psi (x)-x|<{\frac {1}{8\pi }}{\sqrt {x}}\log ^{2}x,\qquad {\text{for all }}x\geq 73.2,}$

where ${\displaystyle \psi (x)}$ is Chebyshev's second function.

Dudek (2014) proved that the Riemann hypothesis implies that for all ${\displaystyle x\geq 2}$ there is a prime ${\displaystyle p}$ satisfying

${\displaystyle x-{\frac {4}{\pi }}{\sqrt {x}}\log x<p\leq x}$.

The constant 4/π may be reduced to (1 + ε) provided that x is taken to be sufficiently large. This is an explicit version of a theorem of Cramér.

Growth of arithmetic functions

The Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to the primes counting function above.

One example involves the Möbius function μ. The statement that the equation

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}$

is valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to the Riemann hypothesis. From this we can also conclude that if the Mertens function is defined by

${\displaystyle M(x)=\sum _{n\leq x}\mu (n)}$

then the claim that

${\displaystyle M(x)=O\left(x^{{\frac {1}{2}}+\varepsilon }\right)}$

for every positive ε is equivalent to the Riemann hypothesis (J. E. Littlewood, 1912; see for instance: paragraph 14.25 in Titchmarsh (1986)). The determinant of the order n Redheffer matrix is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. Littlewood's result has been improved several times since then, by Edmund Landau,^[6] Edward Charles Titchmarsh,^[7] Helmut Maier and Hugh Montgomery,^[8] and Kannan Soundararajan.^[9] Soundararajan's result is that, conditional on the Riemann hypothesis,

${\displaystyle M(x)=O\left(x^{1/2}\exp \left((\log x)^{1/2}(\log \log x)^{14}\right)\right).}$

The Riemann hypothesis puts a rather tight bound on the growth of M, since Odlyzko & te Riele (1985) disproved the slightly stronger Mertens conjecture

${\displaystyle |M(x)|\leq {\sqrt {x}}.}$

Another closely related result is due to Björner (2011), that the Riemann hypothesis is equivalent to the statement that the Euler characteristic of the simplicial complex determined by the lattice of integers under divisibility is ${\displaystyle o(n^{1/2+\epsilon })}$ for all ${\displaystyle \epsilon >0}$ (see incidence algebra).

The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ(n). A typical example is Robin's theorem,^[10] which states that if σ(n) is the sigma function, given by

${\displaystyle \sigma (n)=\sum _{d\mid n}d}$

then

${\displaystyle \sigma (n)<e^{\gamma }n\log \log n}$

for all n > 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant.

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:

${\displaystyle \sigma (n)<H_{n}+\log(H_{n})e^{H_{n}}}$

for every natural number n > 1, where ${\displaystyle H_{n}}$ is the nth harmonic number.^[11]

The Riemann hypothesis is also true if and only if the inequality

${\displaystyle {\frac {n}{\varphi (n)}}<e^{\gamma }\log \log n+{\frac {e^{\gamma }(4+\gamma -\log 4\pi )}{\sqrt {\log n}}}}$

is true for all n ≥ 120569#, where φ(n) is Euler's totient function and 120569# is the product of the first 120569 primes.^[12]

Another example was found by Jérôme Franel, and extended by Landau (see Franel & Landau (1924)). The Riemann hypothesis is equivalent to several statements showing that the terms of the Farey sequence are fairly regular. One such equivalence is as follows: if F_n is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all ε > 0

${\displaystyle \sum _{i=1}^{m}|F_{n}(i)-{\tfrac {i}{m}}|=O\left(n^{{\frac {1}{2}}+\epsilon }\right)}$

is equivalent to the Riemann hypothesis. Here

${\displaystyle m=\sum _{i=1}^{n}\varphi (i)}$

is the number of terms in the Farey sequence of order n.

For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of the symmetric group S_n of degree n, then Massias, Nicolas & Robin (1988) showed that the Riemann hypothesis is equivalent to the bound

${\displaystyle \log g(n)<{\sqrt {\operatorname {Li} ^{-1}(n)}}}$

for all sufficiently large n.

Lindelöf hypothesis and growth of the zeta function

The Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate of growth of the zeta function on the critical line, which says that, for any ε > 0,

${\displaystyle \zeta \left({\frac {1}{2}}+it\right)=O(t^{\varepsilon }),}$

as t → ${\displaystyle \infty }$.

The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions of the critical strip. For example, it implies that

${\displaystyle e^{\gamma }\leq \limsup _{t\rightarrow +\infty }{\frac {|\zeta (1+it)|}{\log \log t}}\leq 2e^{\gamma }}$

${\displaystyle {\frac {6}{\pi ^{2}}}e^{\gamma }\leq \limsup _{t\rightarrow +\infty }{\frac {1/|\zeta (1+it)|}{\log \log t}}\leq {\frac {12}{\pi ^{2}}}e^{\gamma }}$

so the growth rate of ζ(1 + it) and its inverse would be known up to a factor of 2.^[13]

Large prime gap conjecture

The prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is O(√p log p). This is a case in which even the best bound that can be proved using the Riemann hypothesis is far weaker than what seems true: Cramér's conjecture implies that every gap is O((log p)²), which, while larger than the average gap, is far smaller than the bound implied by the Riemann hypothesis. Numerical evidence supports Cramér's conjecture.^[14]

Analytic criteria equivalent to the Riemann hypothesis

Many statements equivalent to the Riemann hypothesis have been found, though so far none of them have led to much progress in proving (or disproving) it. Some typical examples are as follows. (Others involve the divisor function σ(n).)

The