Multifractal Tubes: Multifractal Zeta-Functions, Multifractal Steiner Formulas and Explicit Formulas

2013 ◽  
pp. 291-326 ◽  
Author(s):  
Lars Olsen
Keyword(s):  
Zeta Functions ◽  
Explicit Formulas

On the explicit formulas for zeta functions

2020 ◽  
Vol 43 (17) ◽  
pp. 10249-10261
Author(s):  
Sofiane Bouarroudj ◽  
Mounir Hajli
Keyword(s):  
Zeta Functions ◽  
Explicit Formulas

scholarly journals On zeta functions associated with polynomials

2000 ◽  
Vol 61 (3) ◽  
pp. 455-458 ◽  
Author(s):  
Andrzej Dąbrowski
Keyword(s):  
Complex Plane ◽  
Zeta Functions ◽  
Negative Integer ◽  
Explicit Formulas

We give direct proofs of meromorphic continuality on the whole complex plane of certain zeta functions ZP, Q (s) and Z (P/Q, s) associated with a pair of polynomials P, Q. We calculate ZP, Q (−q) (q a non-negative integer) and give explicit formulas for the residues of Z (P/Q, s) at poles.

scholarly journals Values of multiple zeta functions with polynomial denominators at non-positive integers

2021 ◽  
pp. 2150038
Author(s):  
Driss Essouabri ◽  
Kohji Matsumoto
Keyword(s):  
Positive Integer ◽  
Explicit Expression ◽  
Zeta Functions ◽  
Summation Formula ◽  
Bernoulli Numbers ◽  
Explicit Formulas ◽  
Integer Points ◽  
Special Values ◽  
Multiple Zeta ◽  
Positive Integers

We study rather general multiple zeta functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta functions at non-positive integer points. We first treat the case when the polynomials are power sums, and observe that some “trivial zeros” exist. We also prove that special values are sometimes transcendental. Then we proceed to the general case, and show an explicit expression of special values at non-positive integer points which involves certain period integrals. We give examples of transcendental values of those special values or period integrals. We also mention certain relations among Bernoulli numbers which can be deduced from our explicit formulas. Our proof of explicit formulas are based on the Euler–Maclaurin summation formula, Mahler’s theorem, and a Raabe-type lemma due to Friedman and Pereira.

Sign in / Sign up

Export Citation Format

Share Document