Power sums with applications to multizeta and zeta zero distribution for \(\mathbb F_q[t]\). (English) Zbl 1228.11139
Summary: We study the sum of integral powers of monic polynomials of a given degree over a finite field. The combinatorics of cancellations are quite complicated. We prove several results on the degrees of these sums giving direct or recursive formulas, congruence conditions and degree bounds for them. We point out a ‘duality’ between values for positive and negative powers. We show that despite the combinatorial complexity of the actual values, there is an interesting kind of a recursive formula (at least when the finite field is the prime field) which quickly leads to many interesting structural facts, such as Riemann hypothesis for Carlitz-Goss zeta function, monotonicity in degree, non-vanishing and special identity classification for function field multizeta, as easy consequences.
MSC:
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
References:
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