Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914965877686272 |
|---|---|
| author | Charge, Shashank Dixit, Atul |
| author_facet | Charge, Shashank Dixit, Atul |
| contents | We derive Vorono\"{\dotlessi} summation formulas for the Liouville function $λ(n)$, the Möbius function $μ(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $λ(n)$ requires explicit evaluation of certain infinite series for which the use of the Vinogradov-Korobov zero-free region of the Riemann zeta function is indispensable. Several results of independent interest are obtained as special cases of these formulas. For example, a special case of the one for $μ(n)$ is a famous result of Ramanujan, Hardy, and Littlewood. Cohen type and Ramanujan-Guinand type identities are established for $λ(n)$ and $σ_a(n)σ_b(n)$, where $σ_s(n)$ is the generalized divisor function. As expected, infinite series over the non-trivial zeros of $ζ(s)$ now form an essential part of all of these formulas. A series involving $σ_a(n)σ_b(n)$ and product of modified Bessel functions occurring in one of our identities has appeared in a recent work of Dorigoni and Treilis in string theory. Lastly, we obtain results on oscillations of Riesz sums associated to $λ(n), μ(n)$ and of the error term of Riesz sum of $d^2(n)$ under the assumption of the Riemann Hypothesis, simplicity of the zeros of $ζ(s)$, the Linear Independence conjecture, and a weaker form of the Gonek-Hejhal conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04506 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities Charge, Shashank Dixit, Atul Number Theory Classical Analysis and ODEs Primary 11M06, 11M26, Secondary 33E20 We derive Vorono\"{\dotlessi} summation formulas for the Liouville function $λ(n)$, the Möbius function $μ(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $λ(n)$ requires explicit evaluation of certain infinite series for which the use of the Vinogradov-Korobov zero-free region of the Riemann zeta function is indispensable. Several results of independent interest are obtained as special cases of these formulas. For example, a special case of the one for $μ(n)$ is a famous result of Ramanujan, Hardy, and Littlewood. Cohen type and Ramanujan-Guinand type identities are established for $λ(n)$ and $σ_a(n)σ_b(n)$, where $σ_s(n)$ is the generalized divisor function. As expected, infinite series over the non-trivial zeros of $ζ(s)$ now form an essential part of all of these formulas. A series involving $σ_a(n)σ_b(n)$ and product of modified Bessel functions occurring in one of our identities has appeared in a recent work of Dorigoni and Treilis in string theory. Lastly, we obtain results on oscillations of Riesz sums associated to $λ(n), μ(n)$ and of the error term of Riesz sum of $d^2(n)$ under the assumption of the Riemann Hypothesis, simplicity of the zeros of $ζ(s)$, the Linear Independence conjecture, and a weaker form of the Gonek-Hejhal conjecture. |
| title | Voronoi summation formulas, oscillations of Riesz sums, and Ramanujan-Guinand and Cohen type identities |
| topic | Number Theory Classical Analysis and ODEs Primary 11M06, 11M26, Secondary 33E20 |
| url | https://arxiv.org/abs/2410.04506 |