- Analytic Number Theory
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abscissa of convergence Blaschke factors Borel-Carathéodory lemma bounds Cauchy's theorem characteristic function of divisors Chebyshev function Dirichlet series divisor function entire function Euler's totient function holomorphic function Jensen's inequality Kronecker's lemma Menon's identity Möbius function Picard theorem prime number theorem Ramanujan's sum Riemann zeta function summation by parts units zero-free region
Tag Archives: zero-free region
Absence of zeros of $\zeta(s)$ on the line $\sigma = 1$ under prime number theorem
In this article we show that prime number theorem implies nonvanishing of $\zeta(s)$ on the line $\sigma = 1$ and the argument we present here follows closely the approach taken in Ingham’s book The Distribution of Prime Numbers. The key … Continue reading