A Limit Formula for ζ(2k+1)

✍ Scribed by Yizhong Lan

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 140 KB
• Volume: 78
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1999.2408

✦ Synopsis

In this paper, we gave a limit formula for `(2k+1). This formula is related to a tamely ramified cyclic field of degree 2k+1.

1999 Academic Press

Let k be a positive integer. We know that the value of the Riemann zeta function `(s) at s=2k is

where B 2k is the Bernoulli number. According to this formula, (2k) is a transcendental number. As yet no simple formula analogous to the above formula is known for (2k+1). In this paper, we give a limit formula for (2k+1). This formula is related to some arithmetic invariants of a tamely ramified cyclic field of degree 2k+1. It is new information about (2k+1).

Let K be a normal extension field of degree n over the rational number field Q. We denote by O K the integer ring of K. Throughout this paper we use the following notation: P, the set of all prime numbers; d(K), the discriminant of K; I(K), the set of all non-zero ideals of O K ; Na, the absolute norm of an ideal a # I(K); Tr :, the trace of : # K over Q.