SafekipediaIn mathematics, the multiple zeta functions are an exciting extension of the famous Riemann zeta function. They help mathematicians understand patterns in numbers by adding up special kinds of fractions. These functions are defined using sums where each number in the sequence must be smaller than the one before it, making the math both elegant and challenging.
When the numbers used in these functions are whole numbers, the results are called multiple zeta values or Euler sums. These special numbers have deep connections to other areas of math, like analytic continuation and meromorphic functions. Researchers have been studying them since the late 1990s, uncovering surprising relationships and properties.
The "depth" of a multiple zeta value tells us how many numbers are in the sequence, while the "weight" is the total of all those numbers. Mathematicians often use a special shorthand to write these functions more neatly, like grouping repeating numbers together with braces and exponents. These functions open doors to new discoveries in number theory and beyond.
Definition
Multiple zeta functions are special types of mathematical sums that build on the idea of the Riemann zeta function. They involve adding up fractions where the denominators are powers of numbers, arranged in a specific decreasing order.
These functions are linked to another concept called multiple polylogarithms, which themselves are generalizations of polylogarithm functions. When certain values are used, these become what mathematicians call "colored multiple zeta values" or, in simpler 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 expressed using special kinds of integrals. These integrals help us understand the relationships between different parts of the function.
One important idea is that when you multiply certain kinds of integrals together, the result can be written as a sum of new integrals. This helps mathematicians find patterns and simplify complicated calculations involving multiple zeta functions.
| 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 becomes a special case. We can write it as a sum involving three integers, where each term is a fraction. This helps us understand how the function behaves with three values instead of many. The sums can also be broken down into smaller parts, making it easier to study their patterns.
Euler reflection formula
The multiple zeta functions follow special rules similar to the Euler reflection formula. For two numbers a and b that are both greater than 1, the formula shows 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, another formula connects the multiple zeta functions with combinations of the regular zeta functions. These rules help mathematicians understand how these special functions behave and relate to each other.
Symmetric sums in terms of the zeta function
The multiple zeta function is a generalization of the Riemann zeta function. It is defined by a sum where the terms involve products of powers of integers, similar to the Riemann zeta function but with additional conditions on the ordering of the integers.
These functions are important in number theory and have properties that connect them to symmetric sums and partitions. The relationships between these functions and their symmetric counterparts involve intricate combinatorial structures and partitions, showing deep connections between number theory and algebra.
The sum and duality conjectures
The sum conjecture in multiple zeta functions states that for certain sequences of numbers, the sum of their multiple zeta values equals the regular Riemann zeta value of their total. This idea was first suggested by C. Moen and later explored by others. For example, when breaking down the number 7 into two parts, the sum of specific multiple zeta values equals the zeta value of 7 itself.
There is also a duality conjecture, which suggests that certain sequences of numbers have matching zeta values. This means that if two sequences are considered "dual" to each other, their multiple zeta functions will have the same value. These ideas help mathematicians understand the relationships between different zeta functions.
Main article: Partitions
Euler sum with all possible alternations of sign
The Euler sum with alternations of sign is a special kind of mathematical series that studies sums with alternating signs. These sums are connected to a concept called the non-alternating Euler sum.
One important idea is the use of generalized harmonic numbers. These numbers are like regular harmonic numbers, but they include an extra parameter. For example, a generalized harmonic number can be written as a sum like (+1 + \frac{1}{2^b} + \frac{1}{3^b} + \cdots), where (b) changes the way the terms are added.
These alternating sums can be linked to the multiple zeta function. This function is a more complex version of the regular Riemann zeta function, which itself deals with sums of reciprocals of numbers raised to a power. The multiple zeta function adds another layer by considering sums with several indices, like (\sum_{n_1 > n_2 > \cdots > n_k > 0} \prod_{i=1}^{k} \frac{1}{n_i^{s_i}}).
The study of these alternated sums helps mathematicians understand deeper properties of numbers and their relationships.
Other results
The multiple zeta function has some interesting patterns when we add up certain values. For example, if we add up the multiple zeta function for different numbers, we sometimes get a simpler result that relates to the regular zeta function.
Scientists have found many of these patterns, which help us understand how multiple zeta functions behave and connect to each other. These results show that even though the 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) and 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 and sums in numbers.
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