Logarithmic derivative of Riemann zeta in terms of Mangoldt function

Theorem

The following formula holds: $$\dfrac{\zeta'(z)}{\zeta(z)}=-\displaystyle\sum_{k=1}^{\infty} \dfrac{\Lambda(k)}{k^z},$$ where $\zeta$ denotes the Riemann zeta and $\Lambda$ denotes the Mangoldt function.

References

• 1930: {{ #if: |{{{2}}}|Edward Charles Titchmarsh}}{{#if: |{{#if: |, [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]{{#if: |, [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]{{#if: |, [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]{{#if: |, [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]] and [[Mathematician:{{{author6}}}|{{ #if: |{{{2}}}|{{{author6}}}}}]]| and [[Mathematician:{{{author5}}}|{{ #if: |{{{2}}}|{{{author5}}}}}]]}}| and [[Mathematician:{{{author4}}}|{{ #if: |{{{2}}}|{{{author4}}}}}]]}}| and [[Mathematician:{{{author3}}}|{{ #if: |{{{2}}}|{{{author3}}}}}]]}}| and [[Mathematician:{{{author2}}}|{{ #if: |{{{2}}}|{{{author2}}}}}]]}}|}}: [[Book:Edward Charles Titchmarsh/The Zeta-Function of Riemann{{#if: |/Volume {{{volume}}}|}}{{#if: |/{{{edpage}}}}}|The Zeta-Function of Riemann{{#if: |: Volume {{{volume}}}|}}{{#if: |: {{{eddisplay}}}|{{#if: | ({{{ed}}} ed.)}}}}]]{{#if: | (translated by [[Mathematician:{{{translated}}}|{{ #if: |{{{2}}}|{{{translated}}}}}]])}}{{#if: |, {{{publisher}}}|}}{{#if: |, ISBN {{{isbn}}}|}}{{#if: Logarithmic derivative of Riemann zeta in terms of series over primes | ... (previous)|}}{{#if: Riemann zeta as integral of monomial divided by an exponential | ... (next)|}}{{#if: |: Entry: {{#if: |[[{{{entryref}}}|{{{entry}}}]]|{{{entry}}}}}|}}: § Introduction $(2{'}{'})$