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ArtinL–functions, Artin primitive roots Conjecture and applications CIMPA-ICTP Research School, Nesin Mathematics Village 2017

An introduction to the Riemann zeta function Exercises

Michel Waldschmidt

1. Check the formulae

µ?1= , µ?j=', |µ|? =1, µ? = ,

|µ|?1= 2!, jk?1= k, 1?1=⌧, µ?log =⇤.

2. Check the formulae for the Dirichlet seriesD(f, s) =X

n 1

f(n)n s.

D(jk, s) =⇣(s k), D(, s) =⇣(2s), D(µ, s) = 1

⇣(s), D(⌧, s) =⇣(s)2, D( k, s) =⇣(s k)⇣(s), D(', s) = ⇣(s 1)

⇣(s) , D(|µ|, s) = ⇣(s)

⇣(2s), D( , s) = ⇣(2s)

⇣(s) , D(2!, s) = ⇣(s)2

⇣(2s)· 3. Check that the multiplication by log is a derivation in the ring A of arithmetic functions:

log·(f ?g) = (log·f)?g+f?(log·g).

Check

D(log, s) = ⇣0(s), D(⇤, s) = ⇣0(s)

⇣(s)· 3. Check the formula

⇣0(s)

⇣(s) = 1

2log⇡ 1 2

0(s/2) (s/2)

1 s

1

s 1+ (2s 1) X

⇢2Z+

1

(s ⇢)(s (1 ⇢))· Deduce

⇣0(1/2)

⇣(1/2) = 1

2log(8⇡) + 2 +⇡

4·

Hint: one can use (without proof) the formula

0(1/4)

(1/4) = 3 log 2 ⇡ 2·

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