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 mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by, after whom it is named.
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 is a function whose argument may be any complex number other than 1, and whose values are also complex. It has zeros at the negative even integers; that is, when is one of These are called its trivial zeros. The zeta function is also zero for other values of, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that:
Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers where is a real number and is the imaginary unit.

Riemann zeta function

The Riemann zeta function is defined for complex with real part greater than 1 by the absolutely convergent infinite series
Leonhard Euler considered this series in the 1730s for real values of, in conjunction with his solution to the Basel problem. He also proved that it equals the Euler product
where the infinite product extends over all prime numbers.
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. 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
within the region of convergence for both series. But the eta function series on the right converges not just when the real part of is greater than one, but more generally whenever has positive real part. Thus, the zeta function can be redefined as, extending it from to the larger domain, except for the points where is zero. These are the points, where can be any nonzero integer; the zeta function can be extended to these values too by taking limits, giving a finite value for all values of with positive real part except the simple pole at.
In the strip this extension of the zeta function satisfies the functional equation
One may then define for all remaining nonzero complex numbers by applying this equation outside the strip, and letting equal the right side of the equation whenever has non-positive real part.
If is a negative even integer, then, because the factor vanishes; these are the zeta function's trivial zeros.
The value ζ = −1/2 is not determined by the functional equation, but is the limiting value of as 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 has real part between 0 and 1.

Origin

Riemann's original motivation for studying the zeta function and its zeros was their occurrence in his explicit formula for the number of primes less than or equal to a given number, 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
which counts the primes and prime powers up to, counting a prime power as. The number of primes can be recovered from this function by using the Möbius inversion formula:
where is the Möbius function. Riemann's formula is then
where the sum is over the nontrivial zeros of the zeta function and where is a slightly modified version of that replaces its value at its points of discontinuity by the average of its upper and lower limits:
The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function occurring in the first term is the logarithmic integral function given by the Cauchy principal value of the divergent integral
The terms involving the zeros of the zeta function need some care in their definition as has branch points at 0 and 1, and are defined by analytic continuation in the complex variable in the region ; i.e., they should be considered as. The other terms also correspond to zeros: the dominant term comes from the pole at, considered as a zero of multiplicity, 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 or.
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, and he knew that all of its non-trivial zeros must lie in the range. He checked that a few of the zeros lay on the critical line with real part and suggested that they all do; this is the Riemann hypothesis.

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

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 is the upper bound of the real parts of the zeros, then
, where is the prime-counting function and is the logarithmic integral function.
It is already known that.
Helge von Koch 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, says that the Riemann hypothesis implies
for all. also showed that the Riemann hypothesis implies
for all, where is Chebyshev's second function.
Adrian Dudek proved that the Riemann hypothesis implies that for, there is a prime satisfying
The constant may be reduced to provided that 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
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
then the claim that
for every positive ε is equivalent to the Riemann hypothesis. The determinant of the order n Redheffer matrix is equal to M, 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, Edward Charles Titchmarsh, Helmut Maier and Hugh Montgomery, and Kannan Soundararajan. Soundararajan's result is that, conditional on the Riemann hypothesis,
The Riemann hypothesis puts a rather tight bound on the growth of M, since disproved the slightly stronger Mertens conjecture
Another closely related result is due to, 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 for all .
The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic functions aside from μ. A typical example is Robin's theorem, which states that if σ is the sigma function, given by
then
for all 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:
for every natural number, where is the nth harmonic number.
The Riemann hypothesis is also true if and only if the inequality
is true for all, where φ is Euler's totient function and 120569# is the product of the first 120569 primes.
Another example was found by Jérôme Franel, and extended by Landau. 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 Fn is the Farey sequence of order n, beginning with 1/n and up to 1/1, then the claim that for all
is equivalent to the Riemann hypothesis. Here
is the number of terms in the Farey sequence of order n.
For an example from group theory, if g is Landau's function given by the maximal order of elements of the symmetric group Sn of degree n, then showed that the Riemann hypothesis is equivalent to the bound
for all sufficiently large n.