Dirichlet's divider problem

The Dirichlet divisor problem is a mathematical problem from the field of analytic number theory . It makes a statement about the asymptotic behavior of sums over number functions . To this day, the problem is considered open.

Early 20th century, the problem of was Adolf Piltz significantly to Piltzschen divider problem generalized.

formulation

Designates the function that counts the number of divisors of , then applies ${\displaystyle d(n)}$ ${\displaystyle n}$

${\displaystyle \sum _{n\leq x}d(n)=x\log(x)+(2\gamma -1)x+\Delta (x).}$

The assessment was shown by Peter Gustav Lejeune Dirichlet . Dirichlet's divider problem now asks about the exact nature of the error . The set of all real numbers with the property is considered . The problem is: how big is ? ${\displaystyle \Delta (x)=O({\sqrt {x}})}$ ${\displaystyle \Delta (x)}$ ${\displaystyle D}$ ${\displaystyle \beta }$ ${\displaystyle \Delta (x)=O(x^{\beta })}$ ${\displaystyle \inf D}$

generalization

This problem can be generalized. For this one defines

${\displaystyle d_{k}(n):=\sum _{u_{1},\dotsc ,u_{k} \atop u_{1}\cdots u_{k}=n}1.}$

During and all pairs with counts down (in other words, the divisor of ), counts all tuples with from. It is known that then ${\displaystyle d_{2}(n)=d(n)}$ ${\displaystyle (u_{1},u_{2})}$ ${\displaystyle u_{1}u_{2}=n}$ ${\displaystyle n}$ ${\displaystyle d_{k}(n)}$ ${\displaystyle (u_{1},\dots ,u_{k})}$ ${\displaystyle u_{1}\cdots u_{k}=n}$

${\displaystyle \sum _{n\leq x}d_{k}(n)=xP_{k}(\log x)+\Delta _{k}(x)}$

with a polynomial function of degrees holds. The Piltz divider problem now asks about the nature of the error . ${\displaystyle P_{k}}$ ${\displaystyle k-1}$ ${\displaystyle \Delta _{k}(x)}$

Possible solutions

An important step towards a (general) solution would be the proof of Lindelöf's conjecture , which makes a statement about the growth of the Riemann zeta function in the so-called critical strip.

Via integration of a closed curve and Perron's formulas follows

${\displaystyle \sum _{n\leq x}d_{k}(n)=xP_{k}(\log x)+O\left({\frac {x^{1+\varepsilon }}{T}}+x^{\sigma }\int \limits _{0}^{T}{\frac {|\zeta (\sigma +\mathrm {i} t)|^{k}}{|\sigma +\mathrm {i} t|}}\mathrm {d} t\right).}$

Here is a polynomial function of degree , and . The term is given by the residual of the function at position 1. The background to this connection is that the Dirichlet series of the function is generated by the. By choosing one of the following : ${\displaystyle P_{k}}$ ${\displaystyle k-1}$ ${\displaystyle x\geq 2T^{k/2}\geq 1}$ ${\displaystyle 0<\sigma <1}$ ${\displaystyle xP_{k}(\log x)}$ ${\displaystyle \zeta (s)^{k}x^{s}/s}$ ${\displaystyle \zeta (s)^{k}}$ ${\displaystyle d_{k}(n)}$ ${\displaystyle T=x^{2/(k+4)}}$ ${\displaystyle \textstyle \zeta ({\tfrac {1}{2}}+\mathrm {i} t)=O(t^{1/4+\varepsilon })}$

${\displaystyle \sum _{n\leq x}d_{k}(n)=xP_{k}(\log x)+O(x^{1-{\frac {2}{k+4}}+\varepsilon }).}$

However, this approach via curve integration is probably still far from a final solution, since more detailed knowledge of the Riemann zeta function must be available for further improvements.

Progress

Over the years, better and better estimates have been found. Better values ​​were given by GF Woronoi (1903, ), J. van der Corput (1922, ) and MN Huxley ( ). On the other hand, GH Hardy and E. Landau showed that must apply. ${\displaystyle x^{\tfrac {1}{3}}\log x}$ ${\displaystyle \beta ={\tfrac {33}{100}}}$ ${\displaystyle \beta ={\tfrac {131}{416}}}$ ${\displaystyle \beta \geq {\tfrac {1}{4}}}$

Individual evidence

1. ↑ J. Brüdern: Introduction to analytic number theory. Springer Verlag, Berlin / Heidelberg 1995, p. 133.
2. ↑ J. Brüdern: Introduction to analytic number theory. Springer Verlag, Berlin / Heidelberg 1995, p. 132.
3. ↑ J. Brüdern: Introduction to analytic number theory. Springer Verlag, Berlin / Heidelberg 1995, p. 133.
4. ↑ G. Voronoï : Sur un problème du calcul des fonctions asymptotiques. In: J. Reine Angew. Math. 126 (1903) pp. 241-282.
5. ↑ JG van der Corput: tightening of the estimation in the divider problem. In: Math. Ann. 87 (1922) 39-65. Corrections 89 (1923) p. 160.
6. ^ MN Huxley: Exponential Sums and Lattice Points III . In: Proc. London Math. Soc. tape 87 , no. 3 , 2003, p. 591-609 .
7. ^ GH Hardy: On Dirichlet's divisor problem. In: Lond. MS Proc. (2) 15 (1915) 1-25.
   See GH Hardy, EM Wright: An Introduction to the Theory of Numbers. 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 , p. 272.