Landau–Siegel zeros and zeros of the derivative of the Riemann zeta function

• Published in: Advances in mathematics (New York. 1965) Vol. 230; no. 4-6; pp. 2048 - 2064
• Main Authors: Farmer, David W., Ki, Haseo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2012
• Subjects: Gaps of zeros of the Riemann zeta function; Landau–Siegel zeros; The Riemann hypothesis; The Riemann zeta function; Zeros of the derivative of the Riemann zeta function; Landau–Siegel zeros; Gaps of zeros of the Riemann zeta function; The Riemann hypothesis; The Riemann zeta function; Zeros of the derivative of the Riemann zeta function
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2012.04.020

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.