Mangoldt function

In mathematics , the Mangoldt function , named after the German mathematician Hans von Mangoldt , is a number-theoretic function that is usually referred to as. ${\ displaystyle \ Lambda}$

Definitions and basic properties

The Mangoldt function is defined as

} } \ mathrm {l {\ ddot {a}} sst,} {\ text {where}} p {\ text {prime and}} k \ in \ mathbb {N} ^ {+} \\ 0 & {\ text { otherwise}} \ end {cases}}}

${\ displaystyle \ exp (\ Lambda (n))}$ can be specified explicitly as

{\ displaystyle e ^ {\ Lambda (n)} = {\ frac {\ operatorname {kgV} (1,2,3, \ dotsc, n)} {\ operatorname {kgV} (1,2,3, \ dotsc , n-1)}}}

where denotes the least common multiple . ${\ displaystyle {\ rm {kgV}}}$

The first values of the sequence are ${\ displaystyle \ exp (\ Lambda (n))}$

1, 2, 3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, ... (sequence A014963 in OEIS )

The summed Mangoldt function,

{\ displaystyle \ psi (n) = \ sum _ {i = 1} ^ {n} \ Lambda (i),}

is also known as the Chebyshev function . It plays a role in proving the prime number theorem .

{\ displaystyle \ sum _ {d | n} \ mu \ left ({\ frac {n} {d}} \ right) \ cdot \ log d = \ Lambda (n)}

{\ displaystyle \ sum _ {d | n} \ Lambda (d) = \ log n \,}

{\ displaystyle \ sum _ {d | n} \ mu (d) \ log d = - \ Lambda (n) \,}

{\ displaystyle \ sum _ {d | n} \ mu \ left ({\ frac {n} {d}} \ right) \ Lambda (d) = - \ mu (n) \ log n}

where denotes the Möbius function . ${\ displaystyle \ mu (n)}$

The Mangoldt function plays an important role in the theory of the Dirichlet series .

It applies

{\ displaystyle \ log \ zeta (s) = \ sum _ {n = 2} ^ {\ infty} {\ frac {1} {n ^ {s}}} {\ frac {\ Lambda (n)} {\ log n}} \ qquad \ quad \ mathrm {f {\ ddot {u}} r \; Re} (s)> 1.}

The logarithmic derivation of this provides a connection between the Riemann function ${\ displaystyle \ zeta}$ and the Mangoldt function:

{\ displaystyle {\ frac {\ zeta ^ {\ prime} (s)} {\ zeta (s)}} = - \ sum _ {n = 1} ^ {\ infty} {\ frac {\ Lambda (n) } {n ^ {s}}} \ qquad \ quad \ mathrm {f {\ ddot {u}} r \; Re} (s)> 1.}

More generally, it is even true: is multiplicative and its Dirichlet series ${\ displaystyle f}$ ${\ displaystyle F}$

{\ displaystyle F (s) = \ sum _ {n = 1} ^ {\ infty} {\ frac {f (n)} {n ^ {s}}}}

converges for certain , then applies ${\ displaystyle s}$

{\ displaystyle {\ frac {F ^ {\ prime} (s)} {F (s)}} = \ sum _ {n = 1} ^ {\ infty} {\ frac {f (n) \ Lambda (n )} {n ^ {s}}}.}