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Dinesh S. Thakur

Publications and source records attributed to Dinesh S. Thakur.

7 recordsLinked to original sources
arXiv2026

Four-digit Kaprekar dynamics in odd bases

Start with four digits, arrange them in both descending and ascending order, subtract, and repeat. This simple process is known as the Kaprekar routine, famous in base ten for sending every nonconstant four-digit string to $6174$. We show that in every odd base $B>3$, the four-digit Kaprekar map has an unexpectedly rigid structure. After at most three iterations, every nonconstant orbit enters an explicit triangular region $\mathcal{T}_B$, and on this region the map is conjugate to projective doubling: \[ \{[r],[s]\}\longmapsto \{[2r],[2s]\}. \] This gives a complete finite description of all nonconstant terminal cycles, including an explicit formula for their lengths and counts. In particular, the longest terminal cycle has length at most $(B-1)/2$, and equality can occur only when $B$ is prime. For primes $p>5$, equality occurs precisely when the least positive $m$ with $2^m\equiv\pm1\pmod p$ is $m=(p-1)/2$. The results proved here were first formulated by Schwartz and Thakur. As a test case for AI-assisted formal mathematics, AxiomProver produced Lean/mathlib formalizations of these results.

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arXiv2020

Zeta-like Multizeta Values for higher genus curves

We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or conjecturally equivalently algebraic). These are the first known relations between multizetas, which are not with prime field coefficients. We seem to have one universal family. We also find that interestingly the mechanism with which the relations work is quite different from the rational function field case, raising interesting questions about the expected motivic interpretation in higher genus. We provide some data in support of the guesses.

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arXiv2016

Surprising symmetries in distribution of prime polynomials

The primes or prime polynomials (over finite fields) are supposed to be distributed `irregularly' , despite nice asymptotic or average behavior. We provide some conjectures/guesses/hypotheses with `evidence' of surprising symmetries in prime distribution. At least when the characteristic is $2$, we provide conjectural rationality and characterization of vanishing for families of interesting infinite sums over irreducible polynomials over finite fields. The cancellations responsible do not happen degree by degree or even for degree bounds for primes or prime powers, so rather than finite fields being responsible, interaction between all finite field extensions seems to be playing a role and thus suggests some interesting symmetries in the distribution of prime polynomials. Primes are subtle, so whether there is actual vanishing of these sums indicating surprising symmetry (as guessed optimistically), or whether these sums just have surprisingly large valuations indicating only some small degree cancellation phenomena of low level approximation symmetry (as feared sometimes pessimistically!), remains to be seen. In either case, the phenomena begs for an explanation. (Version 2: A beautiful explanation by David Speyer and important updates are added at the end after the first version.)

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arXiv2014

Multiplicative relations between coefficients of logarithmic derivatives of $\mathbb{F}_q$-linear functions and applications

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli-Carlitz fractions and power sums as well as their multi-variable generalizations. Resulting understanding of their factorizations has arithmetic significance, as well as applications to function field zeta and multizeta values evaluations and relations between them. Using specialization/generalization arguments, we provide much more general identities on linear forms providing a switch between power sums for positive and negative powers.

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arXiv2013

Zeta-like Multizeta Values for $\mathbb{F}_q[t]$

We prove and conjecture several relations between multizeta values for $\mathbb{F}_q[t]$, focusing on zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or equivalently algebraic). In particular, we describe them conjecturally fully for $q=2$, or more generally for any $q$ for `even' weight (`eulerian' tuples). We provide some data in support of the guesses.

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arXiv1999

Diophantine approximation and deformation

We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower exponents. If we transport Vojta's conjecture on height inequality to finite characteristic by modifying it by adding suitable deformation theoretic condition, then we see that the numbers giving rise to general curves approach Roth's bound. We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree.

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