Explicit formulae for L-functions: Difference between revisions

Revision as of 12:16, 27 June 2021

Revision as of 13:56, 24 October 2022

Line 9:

:<math>\pi_0(x) = \sum_n\frac{1}{n}\,\mu(n)\,f(x^{1/n}) = f(x) - \frac{1}{2}\,f(x^{1/2}) - \frac{1}{3}\,f(x^{1/3}) - \frac{1}{5}\,f(x^{1/5}) + \frac{1}{6}\,f(x^{1/6}) - \cdots,</math>

where {{math|''&mu;''(''n'')}} is the [[Möbius function]]. Riemann's formula is then

:<math>f(x) = \operatorname{li}(x) - \sum_\rho \operatorname{li}(x^\rho) - \log(2) + \int_x^\infty \frac{\operatorname{d}tdt}{~t\,(t^2-1)~\log(t)~}</math>

involving a sum over the non-trivial zeros {{mvar|ρ}} of the Riemann zeta function. The sum is not [[Absolute convergence|absolutely convergent]], but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function {{math|li}} occurring in the first term is the (unoffset) [[logarithmic integral function]] given by the [[Cauchy principal value]] of the divergent integral

:<math>\operatorname{li}(x) = \int_0^x \frac{\operatorname{d}tdt}{\,\log(t)\,}\,.</math>

The terms {{math|li(''x''^''ρ'')}} involving the zeros of the zeta function need some care in their definition as {{math|li}} has [[branch point]]s at 0 and 1, and are defined by [[analytic continuation]] in the complex variable {{mvar|ρ}} in the region {{math|''x''&nbsp; >&nbsp; 1}} and {{math|Re(''ρ'')&nbsp; >&nbsp; 0}}. The other terms also correspond to zeros: The dominant term {{math|li(''x'')}} comes from the pole at {{math|''s''&nbsp; {{=}}&nbsp; 1}}, considered as a zero of multiplicity &minus;1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see {{harvnb|Zagier|1977}}.)

The first rigorous proof of the aforementioned formula was given by von Mangoldt in 1895: it started with a proof of the following formula for the [[Chebyshev's function]] {{mvar|ψ}}&nbsp;<ref>Weisstein, Eric W. [http://mathworld.wolfram.com/ExplicitFormula.html Explicit Formula] on MathWorld.</ref>

:<math>\psi_0(x) = \dfrac{1}{2\pi i} \int_{\sigma-i \infty}^{\sigma+i \infty}\left(-\dfrac{\zeta'(s)}{\zeta(s)}\right)\dfrac{x^s}{s}\operatorname{d}s, ds = x - \sum_\rho\frac{~x^\rho\,}{\rho} - \log(2\pi) -\dfrac{1}{2}\log(1-x^{-2})</math>

where the LHS is an inverse Mellin transform with

:<math>\quad\sigma > 1\,, \quad \psi(x) = \sum_{p^k \le x} \log p\,,

Line 23:

This series is also conditionally convergent and the sum over zeroes should again be taken in increasing order of imaginary part:<ref name=Ing77>Ingham (1990) p.77</ref>

:<math>\sum_\rho\frac{x^\rho}{\rho} = \lim_{T \rightarrow \infty} S(x,T) \quad</math> where <math>\quad S(x,T) = \sum_{\rho:|\Im \rho| \le T} \frac{x^\rho}{\rho}\,.</math>.

The error involved in truncating the sum to {{math|''S''(''x'',''T'')}} is always smaller than {{math|ln(''x'')}} in absolute value, and when divided by the [[natural logarithm]] of {{mvar|x}}, has absolute value smaller than {{math|{{frac|''x''|''T''}}}} divided by the distance from {{mvar|x}} to the nearest prime power.<ref>[https://math.stackexchange.com/q/497949 Confused about the explicit formula for ψ0(x)]</ref>