African Music: Amadou & Mariam, Thomas Mapfumo

• Amadou Bagayoko (born 1954) and Mariam Doumbia (born 1958) are a musical duo from Mali.
• After meeting at Mali's Institute for the Young Blind, Amadou and Mariam became a musical duo.
• Their music as a duo is known as Afro-blues, which blends traditional Mali sounds with international multicultural sounds.
• They use instruments like Syrian violins and tablas.
• Thomas Mapfumo (born 1945) from Zimbabwe created chimurenga music.
• Chimurenga music is based on traditional Shona mbira music but is played on electric instruments.
• His music contains lyrics about Zimbabwe's social and political struggle.
• N'Faly Kouyaté, active since the 1990s, comes from Guinea.
• Born into a respected family of Mandinka griots, Kouyaté is a master performer of the kora (a stringed musical instrument).
• Kouyaté works with the band Afro Celt Sound System and his own group Dunyakan, bringing new vibrancy to griot music.
• Etran Finatawa formed in 2004, is from Niger.
• The five band members come from the two neighboring ethnic groups, the Tuareg and Wodaabe peoples.
• They combined the two groups' music, yielding a hypnotic musical style known as the nomad blues
• Wizkid, born in Lagos, Nigeria in 1990, introduced Afrobeats to a new audience .
• He collaborated with Canadian rapper Drake on "One Dance".
• Afrobeats is nonpolitical pop music with lyrical phrases in a mix of English and West African languages.
• Wiyaala, born in Ghana in 1986, does Afropop, blending African and Western pop influences.
• Noella Wiyaala sings in four languages, as well as several Ghanaian dialects.
• Her music contains a strong message of female empowerment.

Riemann Zeta Function

• Riemann Zeta Function is defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} +...$
• s is a complex number where the real part is greater than 1, or $\Re(s) > 1$

Function Properties

• It has a pole at $s = 1$
• It has trivial zeros $s = -2, -4, -6,...$
• Non-trivial zeros exist on the critical strip $0 < \Re(s) < 1$
• The Riemann Hypothesis states that all non-trivial zeros have a real part equal to $\frac{1}{2}$, or $\Re(s) = \frac{1}{2}$

Special Values

• $\zeta(0) = -\frac{1}{2}$
• $\zeta(2) = \frac{\pi^2}{6}$
• $\zeta(4) = \frac{\pi^4}{90}$

Integral Representation

• Defined as $\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx$
• $\Gamma(s)$ represents the gamma function

Functional Equation

• $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$
• Relates the value of the zeta function at $s$ to its value at $1-s$

Applications

• Number theory
• Physics
• Cryptography

Riemann Hypothesis

• This hypothesis is an unsolved problem in mathematics.
• The hypothesis states that all non-trivial zeros of the Riemann zeta function have a real part equal to $\frac{1}{2}$.
• If true, it will provide important information for understanding the distribution of prime numbers.