SafekipediaIn mathematics, the multiple zeta functions are a fun way to extend the famous Riemann zeta function. They help mathematicians see patterns in numbers by adding up special fractions. These functions are made using sums where each number must be smaller than the one before it.
When the numbers are whole numbers, the results are called multiple zeta values or Euler sums. These numbers connect to many other parts of math, like analytic continuation and meromorphic functions. People have been studying them since the late 1990s and found many interesting facts.
The "depth" of a multiple zeta value shows how many numbers are in the sequence. The "weight" is the total of all those numbers. Mathematicians use special ways to write these functions more simply, like grouping repeating numbers together. These functions can lead to new discoveries in number theory and more.
Definition
Multiple zeta functions are special kinds of mathematical sums. They are based on the Riemann zeta function. They add up fractions where the bottom numbers are powers of numbers, listed in a special decreasing order.
These functions are connected to another idea called multiple polylogarithms. Multiple polylogarithms are broader versions of polylogarithm functions. When we use certain values, they become what mathematicians name "colored multiple zeta values" or, in easier cases, "multiple zeta values."
Integral structure and identities
Mathematicians have found that multiple zeta functions, which are like more complex versions of the Riemann zeta function, can be shown using special kinds of integrals. These integrals help us see how different parts of the function relate to each other.
One key idea is that when you multiply some types of integrals together, the answer can be written as a sum of new integrals. This helps mathematicians find patterns and make complicated calculations easier.
| s | t | approximate value | explicit formulae | OEIS |
|---|---|---|---|---|
| 2 | 2 | 0.811742425283353643637002772406 | 3 4 ζ ( 4 ) {\displaystyle {\tfrac {3}{4}}\zeta (4)} | A197110 |
| 3 | 2 | 0.228810397603353759768746148942 | 3 ζ ( 2 ) ζ ( 3 ) − 11 2 ζ ( 5 ) {\displaystyle 3\zeta (2)\zeta (3)-{\tfrac {11}{2}}\zeta (5)} | A258983 |
| 4 | 2 | 0.088483382454368714294327839086 | ( ζ ( 3 ) ) 2 − 4 3 ζ ( 6 ) {\displaystyle \left(\zeta (3)\right)^{2}-{\tfrac {4}{3}}\zeta (6)} | A258984 |
| 5 | 2 | 0.038575124342753255505925464373 | 5 ζ ( 2 ) ζ ( 5 ) + 2 ζ ( 3 ) ζ ( 4 ) − 11 ζ ( 7 ) {\displaystyle 5\zeta (2)\zeta (5)+2\zeta (3)\zeta (4)-11\zeta (7)} | A258985 |
| 6 | 2 | 0.017819740416835988362659530248 | A258947 | |
| 2 | 3 | 0.711566197550572432096973806086 | 9 2 ζ ( 5 ) − 2 ζ ( 2 ) ζ ( 3 ) {\displaystyle {\tfrac {9}{2}}\zeta (5)-2\zeta (2)\zeta (3)} | A258986 |
| 3 | 3 | 0.213798868224592547099583574508 | 1 2 ( ( ζ ( 3 ) ) 2 − ζ ( 6 ) ) {\displaystyle {\tfrac {1}{2}}\left(\left(\zeta (3)\right)^{2}-\zeta (6)\right)} | A258987 |
| 4 | 3 | 0.085159822534833651406806018872 | 17 ζ ( 7 ) − 10 ζ ( 2 ) ζ ( 5 ) {\displaystyle 17\zeta (7)-10\zeta (2)\zeta (5)} | A258988 |
| 5 | 3 | 0.037707672984847544011304782294 | 5 ζ ( 3 ) ζ ( 5 ) − 147 24 ζ ( 8 ) − 5 2 ζ ( 6 , 2 ) {\displaystyle 5\zeta (3)\zeta (5)-{\tfrac {147}{24}}\zeta (8)-{\tfrac {5}{2}}\zeta (6,2)} | A258982 |
| 2 | 4 | 0.674523914033968140491560608257 | 25 12 ζ ( 6 ) − ( ζ ( 3 ) ) 2 {\displaystyle {\tfrac {25}{12}}\zeta (6)-\left(\zeta (3)\right)^{2}} | A258989 |
| 3 | 4 | 0.207505014615732095907807605495 | 10 ζ ( 2 ) ζ ( 5 ) + ζ ( 3 ) ζ ( 4 ) − 18 ζ ( 7 ) {\displaystyle 10\zeta (2)\zeta (5)+\zeta (3)\zeta (4)-18\zeta (7)} | A258990 |
| 4 | 4 | 0.083673113016495361614890436542 | 1 2 ( ( ζ ( 4 ) ) 2 − ζ ( 8 ) ) {\displaystyle {\tfrac {1}{2}}\left(\left(\zeta (4)\right)^{2}-\zeta (8)\right)} | A258991 |
Three parameters case
When we look at the multiple zeta function with just three parameters, it is a special case. We can write it as a sum with three whole numbers. Each part of the sum is a small piece. This helps us see how the function works with three values. The sums can also be split into smaller pieces. This makes it easier to find patterns.
Euler reflection formula
The multiple zeta functions have special rules, much like the Euler reflection formula. For two numbers a and b that are both greater than 1, there is a relationship between the multiple zeta functions and the regular zeta functions.
For three numbers a, b, and c that are all greater than 1, there is another formula that links the multiple zeta functions with combinations of the regular zeta functions. These rules help mathematicians learn more about how these functions work and connect to each other.
Symmetric sums in terms of the zeta function
The multiple zeta function is a way to expand the Riemann zeta function. It adds more rules about how numbers are arranged in its sums.
These functions are useful in number theory. They connect to patterns in numbers and how numbers can be grouped, showing links between number theory and algebra.
The sum and duality conjectures
The sum conjecture in multiple zeta functions says that for some groups of numbers, the total of their multiple zeta values is the same as the regular Riemann zeta value of their total. This idea was first suggested by C. Moen and studied by others. For example, when you break the number 7 into two parts, the total of some multiple zeta values equals the zeta value of 7.
There is also a duality conjecture. It says that some groups of numbers have matching zeta values. If two groups are "dual" to each other, their multiple zeta functions will have the same value. These ideas help mathematicians learn more about how different zeta functions are related.
Main article: Partitions
Euler sum with all possible alternations of sign
The Euler sum with alternations of sign is a special kind of math series. It looks at sums where the signs change up and down.
This idea is linked to generalized harmonic numbers. These numbers are like regular harmonic numbers, but they have an extra part. For example, a generalized harmonic number can be shown as a sum like (+1 + \frac{1}{2^b} + \frac{1}{3^b} + \cdots), where (b) changes how the numbers are added.
These changing sums are also connected to the multiple zeta function. This function is a more detailed version of the regular Riemann zeta function. The regular function works with sums of the reciprocals of numbers raised to a power. The multiple zeta function adds more by looking at sums with several parts, like (\sum_{n_1 > n_2 > \cdots > n_k > 0} \prod_{i=1}^{k} \frac{1}{n_i^{s_i}}).
Studying these changing sums helps mathematicians learn more about numbers and how they relate to each other.
Other results
The multiple zeta function has some interesting patterns. When we add up certain values, we sometimes get a simpler result that relates to the regular zeta function.
Scientists have found many of these patterns. They help us understand how multiple zeta functions behave and connect to each other. These results show that even though multiple zeta functions look complicated, they follow some neat rules.
Mordell–Tornheim zeta values
The Mordell–Tornheim zeta function is a special kind of zeta function. It was introduced by Matsumoto (2003). It is based on earlier work by Mordell (1958) and Tornheim (1950). This function is also a special case of the Shintani zeta function. It helps mathematicians study special patterns in numbers.
This article is a child-friendly adaptation of the Wikipedia article on Multiple zeta function, available under CC BY-SA 4.0.