Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A FUNCTION-FIELD ANALOGUE OF THE GOLDBACH COUNTING FUNCTION AND THE ASSOCIATED DIRICHLET SERIES

  • Shigeki Egami (1-2-704 Naito-machi) ;
  • Kohji Matsumoto (Graduate School of Mathematics Nagoya University and Center for General education Aichi Institute of Technology)
  • Received : 2023.02.07
  • Accepted : 2023.10.05
  • Published : 2024.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We consider a function-field analogue of Dirichlet series associated with the Goldbach counting function, and prove that it can, or cannot, be continued meromorphically to the whole plane. When it cannot, we further prove the existence of the natural boundary of it.

Keywords

Acknowledgement

The second named author was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research No.22K03267.

References

  1. A. Baker, Transcendental Number Theory, Cambridge Univ. Press, London, 1975. 
  2. G. Bhowmik and J.-C. Schlage-Puchta, Meromorphic continuation of the Goldbach generating function, Funct. Approx. Comment. Math. 45 (2011), part 1, 43-53. https://doi.org/10.7169/facm/1317045230 
  3. S. Egami and K. Matsumoto, Convolutions of the von Mangoldt function and related Dirichlet series, in "Number Theory: Sailing on the Sea of Number Theory", Proc. 4th China-Japan Seminar (Weihai, China), S. Kanemitsu and J.-Y. Liu (eds.), Ser. Number Theory Appl. 2, World Sci. Publ. Co., 2007, pp. 1-23. https://doi.org/10.1142/9789812770134_0001 
  4. A. Fujii, An additive problem of prime numbers, Acta Arith. 58 (1991), 173-179. https://doi.org/10.4064/aa-58-2-173-179 
  5. A. Fujii, An additive problem of prime numbers. II, Proc. Japan Acad. 67A (1991), 248-252. https://doi.org/10.3792/pjaa.67.248 
  6. A. Fujii, An additive problem of prime numbers. III, Proc. Japan Acad. 67A (1991), 278-283. https://doi.org/10.3792/pjaa.67.278 
  7. A. O. Gel'fond, On the approximation of transcendental numbers by algebraic numbers, Dokl. Akad. Nauk 2 (1935), 177-182. 
  8. G. H. Hardy and J. E. Littlewood, Some problems of "partitio numerorum (V ) : A further contribution to the study of Goldbach's problem, Proc. London Math. Soc. (2) 22 (1924), 46-56. https://doi.org/10.1112/plms/s2-22.1.46 
  9. M. Rosen, Number Theory in Function Fields, Graduate Texts in Mathematics, 210, Springer, New York, 2002. https://doi.org/10.1007/978-1-4757-6046-0 
  10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, reprint of the fourth (1927) edition, Cambridge Mathematical Library, Cambridge Univ. Press, Cambridge, 1996. https://doi.org/10.1017/CBO9780511608759