Tohoku Mathematical Journal
2025
September
SECOND SERIES VOL. 77, NO. 3

Tohoku Math. J.
77 (2025), 375-419

Title MEROMORPHIC CONTINUATION AND NON-POLAR SINGULARITIES OF LOCAL ZETA FUNCTIONS IN SOME SMOOTH CASES

Author Toshihiro Nose

(Received November 2, 2022, revised November 6, 2023)
Abstract. It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, certain cases of specific (non-real analytic) smooth functions are precisely investigated. In particular, we give asymptotic limits of local zeta functions at some singularities along one direction. It follows from these behaviors that the respective local zeta functions have singularities different from poles. Then we show the optimality of the lower estimates of a certain quantity concerning with meromorphic continuation of local zeta functions in some smooth model cases.

Mathematics Subject Classification. Primary 30E15; Secondary 14B05, 14M25.
Key words and phrases. Local zeta functions, meromorphic continuation, non-polar singularities, flat functions.

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