Remarks on number theory over additive arithmetical semigroups
Abstract
UDC 511
We deal with additive arithmetical semigroups and present old and new proofs for the distribution of zeros of the corresponding $\zeta$-functions. We use these results to prove prime number theorems and a Selberg formula for such semigroups.
References
Barát, A.; Indlekofer, K.-H. On mean-value theorems for multiplicative functions in additive arithmetical semigroups. Ann. Univ. Sci. Budapest. Sect. Comput. 33 (2010), 49--72. MR2891033
Barát, Anna; Indlekofer, Karl-Heinz. Mean-value theorems for uniformly summable multiplicative functions on additive arithmetical semigroups. Ann. Univ. Sci. Budapest. Sect. Comput. 40 (2013), 105--122. MR3129159
Beurling, Arne. Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I. (French) Acta Math. 68 (1937), no. 1, 255--291. doi: 10.1007/BF02546666
Carlitz, L. The distribution of irreducible polynomials in several indeterminates. Illinois J. Math. 7 1963 371--375. MR0153665 DOI: https://doi.org/10.1215/ijm/1255644947
Fogels, E. On the abstract theory of primes. I. Acta Arith. 10 (1964/65), 137--182. doi: 10.4064/aa-10-2-137-182
Flajolet, Philippe; Sedgewick, Robert. Analytic combinatorics. Cambridge University Press, Cambridge, 2009. xiv+810 pp. ISBN: 978-0-521-89806-5 doi: 10.1017/CBO9780511801655
Indlekofer, K.-H. The abstract prime number theorem for function fields. Acta Math. Hungar. 62 (1993), no. 1-2, 137--148. doi: 10.1007/BF01874225
Indlekofer, K.-H. Some remarks on additive arithmetical semigroups. ; translated from Liet. Mat. Rink. 42 (2002), no. 2, 185--204 Lithuanian Math. J. 42 (2002), no. 2, 146--162 doi: 10.1023/A:1016110209755
Indlekofer, Karl-Heinz. Some remarks on additive arithmetical semigroups. II. Šiauliai Math. Semin. 4(12) (2009), 83--104. MR2530200
Indlekofer, K.-H. Tauberian theorems with applications to arithmetical semigroups and probabilistic combinatorics. Ann. Univ. Sci. Budapest. Sect. Comput. 34 (2011), 135--177. MR2891061
Indlekofer, Karl-Heinz. Remarks on Tauberian theorems for exp-log functions. Šiauliai Math. Semin. 8(16) (2013), 83--93. MR3145619
Indlekofer, K.-H.; Kátai, I.; Klesov, O. I. On random arithmetical functions. I. Lith. Math. J. 50 (2010), no. 3, 271--283. doi: 10.1007/s10986-010-9085-1
Indlekofer, K.-H.; Kátai, I.; Klesov, O. I. On random arithmetical functions II. Ann. Univ. Sci. Budapest. Sect. Comput. 38 (2012), 295--308. MR3046522
Indlekofer, Karl-Heinz; Manstavičius, Eugenijus; Warlimont, Richard. On a certain class of infinite products with an application to arithmetical semigroups. Arch. Math. (Basel) 56 (1991), no. 5, 446--453. doi: 10.1007/BF01200088
Indlekofer, Karl-Heinz; Warlimont, Richard. Remarks on the infinite product representations of holomorphic functions. Publ. Math. Debrecen 41 (1992), no. 3-4, 263--276. MR1189108
Indlekofer, K.-H.; Wehmeier, S. Mean behaviour of multiplicative functions on additive arithmetical semigroups. Comput. Math. Appl. 52 (2006), no. 3-4, 577--592. doi: 10.1016/j.camwa.2005.04.019
Knopfmacher, John. Analytic arithmetic of algebraic function fields. Lecture Notes in Pure and Applied Mathematics, 50. Marcel Dekker, Inc., New York, 1979. {rm iii}+130 pp. ISBN: 0-8247-6907-4 0545904
Knopfmacher, John; Zhang, Wen-Bin. Number theory arising from finite fields. Analytic and probabilistic theory. Monographs and Textbooks in Pure and Applied Mathematics, 241. Marcel Dekker, Inc., New York, 2001. vi+404 pp. ISBN: 0-8247-0577-7 doi: 10.1201/9780203908150
Manstavičius, Eugenijus. An analytic method in probabilistic combinatorics. Osaka J. Math. 46 (2009), no. 1, 273--290. MR2531150
Wright, E. M. A relationship between two sequences. III. J. London Math. Soc. 43 (1968), 720--724. doi: 10.1112/jlms/s1-43.1.720
Zhang, Wen-Bin. The prime element theorem in additive arithmetic semigroups. I. Illinois J. Math. 40 (1996), no. 2, 245--280. MR1398093 DOI: https://doi.org/10.1215/ijm/1255986103
Zhang, Wen-Bin. Probabilistic number theory in additive arithmetic semigroups. I. Analytic number theory, Vol. 2 (Allerton Park, IL, 1995), 839--885, Progr. Math., 139, Birkhäuser Boston, Boston, MA, 1996. MR1409397
This work is licensed under a Creative Commons Attribution 4.0 International License.
