Dhairya R Shah Teacher of Mathematics · Applied Mathematician

Mathematical research · Dhairya Shah

Floor–Ceiling Research

Four connected directions: applications, zeta roots, prime counting and rationality.

These preprints build on a shared counting idea: group inputs by the integer value produced by a floor or ceiling map, then use the resulting multiplicities to reorganise sums, products and arithmetic tests. Each paper develops one direction of this programme and can be read separately.

Applications

Preprint; not peer reviewed

Floor and ceiling level counts: sums, products and applications

Website v1 31 pages

Develops inverse floor and ceiling level counts into a common framework for finite and infinite sums and products. The main applications connect spherical spectral cutoffs, determinants and Gaussian integrals, with explicit gamma and Barnes evaluations, compatible normalizations and quantitative tail bounds. Further constructions cover weighted partitions, rational scales and power-bounded double sums.

Spectral determinants · Gaussian integrals · Weighted partitions

Read PDF 646 kB Download PDF

Zeta roots

Preprint; not peer reviewed

Floor–ceiling root zeta functions: zero geometry and Mellin criteria

Website v1 24 pages

Studies the zero geometry of integer-root floor and ceiling zeta functions. It proves a unique simple real zero near each sufficiently distant negative reference point and derives paired asymptotics. Möbius-weighted interpolation and a regularized geometric kernel connect the transforms to reciprocal-zeta poles and Mellin decay criteria. The RH-equivalent decay estimates remain unproved.

Zeta zeros · Möbius interpolation · Mellin transforms

Read PDF 728 kB Download PDF

Prime counting

Preprint; not peer reviewed

Floor–ceiling transforms of prime counting measures

Website v1 27 pages

Develops rounded level counts for prime and von Mangoldt measures. A positive cubic-difference kernel has an explicit Mellin transform, noncancellation at nontrivial Riemann zeros, a separated prime-square bias and accelerated evaluation bounds. The paper also quantifies rational-scale corrections and a prime-only approximation obstruction. Its RH-related conclusions depend on global arithmetic estimates that remain unproved.

Prime measures · Positive kernels · Certified evaluation

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Rationality

Preprint; not peer reviewed

Floor and ceiling in rationality problems and certified denominator exclusions

Website v1 21 pages

Combines exact integer-hit tests with positive series whose ordered partial sums detect rationality through equality of their totals, alongside an obstruction under complete termwise ordering. Certified intervals and adjacent-fraction certificates give exact minimum candidate denominators, with computations for π + e, ζ(7) and the Euler–Mascheroni constant γ at a million-place decimal scale. These finite denominator exclusions do not determine the rationality of the tested constants.

Rationality tests · Continued fractions · Interval certificates

Read PDF 532 kB Download PDF

Versions and status

Website v1 identifies the initial release of each PDF on this site. Dates above record the website release, rather than a manuscript revision or an arXiv version. All four papers and the supplementary note are preprints and have not been peer reviewed. Links to arXiv records and journal publications will be added when available.

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