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LOCAL AVERAGE OF THE HYPERBOLIC CIRCLE PROBLEM FOR FUCHSIAN GROUPS
Author(s) -
Biró András
Publication year - 2018
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579317000419
Subject(s) - mathematics , selberg trace formula , fuchsian group , generalization , trace (psycholinguistics) , term (time) , group (periodic table) , upper half plane , plane (geometry) , radius , unit circle , mathematical analysis , pure mathematics , geometry , riemann hypothesis , philosophy , linguistics , physics , chemistry , organic chemistry , computer security , quantum mechanics , computer science
Let Γ ⊆ PSL ( 2 , R ) be a finite‐volume Fuchsian group. The hyperbolic circle problem is the estimation of the number of elements of the Γ ‐orbit of z in a hyperbolic circle around w of radius R , where z and w are given points of the upper half plane and R is a large number. An estimate with error term e ( 2 / 3 ) Ris known, and this has not been improved for any group. Recently, Risager and Petridis proved that in the special case Γ = PSL ( 2 , Z ) taking z = w and averaging over z in a certain way the error term can be improved to e ( 7 / 12 + ε ) R . Here we show such an improvement for a general Γ ; our error term is e ( 5 / 8 + ε ) R(which is better than e ( 2 / 3 ) Rbut weaker than the estimate of Risager and Petridis in the case Γ = PSL ( 2 , Z ) ). Our main tool is our generalization of the Selberg trace formula proved earlier.