- Analytic Number Theory
- Chebyshev functions
- Complex Analysis
- Dirichlet product
- Dirichlet series
- Divisor function
- Euler's totient function
- Little Picard theorem
- Möbius function
- Multiplicative functions
- Prime number theorem
- Ramanujan's sum
- Riemann zeta function
- Summation by parts
- Uncategorized
- Von Mangoldt function
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
Category Archives: Riemann zeta function
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
Bounds for the Riemann zeta function in the critical strip
In this article we obtain upper bounds for $\zeta(s)$ in the strip $\delta \leq \sigma \leq 2$, where $\delta > 0$. We will first show that for a fixed $0 < \varepsilon < \delta \leq 1$ we have \[\zeta(s) \ll … Continue reading
Posted in Analytic Number Theory, Riemann zeta function
Tagged bounds, Riemann zeta function
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