Perron's formula




In mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetical function, by means of an inverse Mellin transform.




Contents






  • 1 Statement


  • 2 Proof


  • 3 Examples


  • 4 Generalizations


  • 5 References





Statement


Let {a(n)}{displaystyle {a(n)}}{a(n)} be an arithmetic function, and let


g(s)=∑n=1∞a(n)ns{displaystyle g(s)=sum _{n=1}^{infty }{frac {a(n)}{n^{s}}}} g(s)=sum_{n=1}^{infty} frac{a(n)}{n^{s}}

be the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for ℜ(s)>σ{displaystyle Re (s)>sigma }Re(s)>sigma. Then Perron's formula is


A(x)=∑n≤x′a(n)=12πi∫c−i∞c+i∞g(z)xzzdz.{displaystyle A(x)={sum _{nleq x}}'a(n)={frac {1}{2pi i}}int _{c-iinfty }^{c+iinfty }g(z){frac {x^{z}}{z}}dz.}{displaystyle A(x)={sum _{nleq x}}'a(n)={frac {1}{2pi i}}int _{c-iinfty }^{c+iinfty }g(z){frac {x^{z}}{z}}dz.}

Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is an integer. The integral is not a convergent Lebesgue integral, it is understood as the Cauchy principal value. The formula requires c > 0, c > σ, and x > 0 real, but otherwise arbitrary.



Proof


An easy sketch of the proof comes from taking Abel's sum formula


g(s)=∑n=1∞a(n)ns=s∫1∞A(x)x−(s+1)dx.{displaystyle g(s)=sum _{n=1}^{infty }{frac {a(n)}{n^{s}}}=sint _{1}^{infty }A(x)x^{-(s+1)}dx.}{displaystyle g(s)=sum _{n=1}^{infty }{frac {a(n)}{n^{s}}}=sint _{1}^{infty }A(x)x^{-(s+1)}dx.}

This is nothing but a Laplace transform under the variable change x=et.{displaystyle x=e^{t}.}x=e^t. Inverting it one gets Perron's formula.



Examples


Because of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoretic sums. Thus, for example, one has the famous integral representation for the Riemann zeta function:


ζ(s)=s∫1∞⌊x⌋xs+1dx{displaystyle zeta (s)=sint _{1}^{infty }{frac {lfloor xrfloor }{x^{s+1}}},dx}zeta(s)=sint_1^infty frac{lfloor xrfloor}{x^{s+1}},dx

and a similar formula for Dirichlet L-functions:


L(s,χ)=s∫1∞A(x)xs+1dx{displaystyle L(s,chi )=sint _{1}^{infty }{frac {A(x)}{x^{s+1}}},dx}L(s,chi)=sint_1^infty frac{A(x)}{x^{s+1}},dx

where


A(x)=∑n≤xχ(n){displaystyle A(x)=sum _{nleq x}chi (n)}A(x)=sum_{nle x} chi(n)

and χ(n){displaystyle chi (n)}chi (n) is a Dirichlet character. Other examples appear in the articles on the Mertens function and the von Mangoldt function.



Generalizations


Perron's formula is just a special case of the Mellin discrete convolution


∑n=1∞a(n)f(n/x)=12πi∫c−i∞c+i∞F(s)G(s)xsds{displaystyle sum _{n=1}^{infty }a(n)f(n/x)={frac {1}{2pi i}}int _{c-iinfty }^{c+iinfty }F(s)G(s)x^{s}ds}{displaystyle sum _{n=1}^{infty }a(n)f(n/x)={frac {1}{2pi i}}int _{c-iinfty }^{c+iinfty }F(s)G(s)x^{s}ds}


where G(s)=∑n=1∞a(n)ns{displaystyle G(s)=sum _{n=1}^{infty }{frac {a(n)}{n^{s}}}}{displaystyle G(s)=sum _{n=1}^{infty }{frac {a(n)}{n^{s}}}} and F(s)=∫0∞f(x)xs−1dx{displaystyle F(s)=int _{0}^{infty }f(x)x^{s-1}dx}{displaystyle F(s)=int _{0}^{infty }f(x)x^{s-1}dx} the Mellin transform. The Perron formula is just the special case of the test function f(1/x)=θ(x−1){displaystyle f(1/x)=theta (x-1)}{displaystyle f(1/x)=theta (x-1)} Heaviside step function



References



  • Page 243 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}

  • Weisstein, Eric W. "Perron's formula". MathWorld.


  • Tenenbaum, Gérald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics. 46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN 0-521-41261-7. Zbl 0831.11001.




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