Sum Of Digits Research Articles

A 3D photon-to-digital converter readout for low-power and large-area applications

A new trend in large area noble liquid experiments is to measure the scintillation light with photodetectors and their electronics inside the active volume. Compared to the typical approach of using silicon photomultipliers (SiPM) with an analog readout chain leading to an analog-to-digital converter, this paper presents a new 3D photon-to-digital converter (PDC) readout that takes advantage of the binary nature of the single-photon avalanche diodes (SPAD). The readout contains 4096 pixels over 25 mm2, each including a 3D bonding pad and a quenching circuit. The readout features three different outputs: a fast flag to get the timestamp of each event from an external time-to-digital converter, a digital sum to retrieve the number of pixels triggered during an event, and an analog monitor to generate an analog SiPM-like output. The analog monitor is also used to validate the two former digital outputs. The readout also includes 61 2D CMOS SPADs for validation purpose prior to the final 3D integration with SPADs custom made according to our design by Teledyne DALSA (Bromont, Canada). As a first system integration toward large-area detector applications, a mini-tile of 2 × 2 readouts has been developed to test all the functionalities. The measured single-photon timing resolution ranges from 72 to 93 ps FWHM across the mini-tile SPAD channels population (i.e. 4 × 61 channels). The flag timing resolution is below 95 ps RMS, which includes the contribution of the optimized flag H-tree but also an additional trigger tree that replaces the 3D SPAD array at this stage of development. Once bonded with the 3D SPADs, the trigger tree won't be required to measure the flag timing resolution. With the removed contribution of the trigger tree, the estimated flag timing resolution should be below 45 ps RMS. The extent of the benefits of the digital sum output depend on the application, and this paper focuses on two cases. First, a low-power coincidence scheme such as required by the nEXO liquid xenon experiment, leading to a power consumption as low as 140 μW per PDC. With a finer sampling of the scintillation light such as required for pulse shape discrimination in liquid argon, the power consumption remains below 100 μW per PDC. Overall, this readout is designed as a replacement for a typical analog SiPM chain, without compromise on the performances.

Kempner-like Harmonic Series

Inspired by a question asked on the list mathfun, we revisit Kempner-like series, i.e., harmonic sums ∑ ′ 1 / n where the integers n in the summation have “restricted” digits. First we give a short proof that lim k → ∞ ( ∑ s 2 ( n ) = k 1 / n ) = 2 log 2 , where s 2 ( n ) is the sum of the binary digits of the integer n. Then we give a generalization that addresses the case where s 2 ( n ) is replaced with s b ( n ) , the sum of b-ary digits in base b: we prove that lim k → ∞ ∑ s b ( n ) = k 1 / n = ( 2 log b ) / ( b − 1 ) . Finally we indicate that other generalizations could be studied: the sum of digits in base 2 could be replaced with, e.g., the function a 11 ( n ) of—possibly overlapping—11 in the base-2 expansion of n, for which one can obtain lim k → ∞ ∑ a 11 ( n ) = k 1 / n = 4 log 2 .

The error term of the sum of digital sum functions in arbitrary bases

Let $k$ be a non-negative integer and $q > 1$ be a positive integer. Let $s_q(k)$ be the sum of digits of $k$ written in base $q.$ In 1940, Bush proved that $A_q(x)=\sum_{k \leq x} s_q (k)$ is asymptotic to $\frac{q-1}{2}x \log_q x.$ In 1968, Trollope proved an explicit formula for the error term of $A_2(n-1),$ labeled by $-E_2(n),$ where $n$ is a positive integer. In 1975, Delange extended Trollope's result to an arbitrary base $q$ by another method and labeled the error term $nF_q(\log_q n).$ When $q=2,$ the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope's method to extend $-E_2(n)$ to $-E_q(n)$ with a general base $q,$ and we also proved directly that $nF_q(\log_q n)$ and $-E_q(n)$ are equal for any $q.$

Summing the sum of digits

We revisit and generalize inequalities for the summatory function of the sum of digits in a given integer base. We prove that several known results can be deduced from a theorem in a 2023 paper by Mohanty, Greenbury, Sarkany, Narayanan, Dingle, Ahnert, and Louis, whose primary scope is the maximum mutational robustness in genotype-phenotype maps.

A central limit theorem for the variation of the sum of digits

A central limit theorem for the variation of the sum of digits

Along the path of the great Kaprekar: A-function, repunits and their properties

Following the famous Indian mathematician D. Kaprekar, in the paper[5], the author presented a new method for obtaining integers ????(????) = ⅑ (???? − ????(????)) where S(n) is the sum of digits of the number n In decimal notation. This A-function turned out to be related to a remarkable class of numbers the class of integers Rn repunit. In the paper, new properties are found Rn. The properties A-function. It is proved that the set of a-self numbers is infinite and each a-self number has exactly 10 generators. The hypothesis about the distribution of a-self numbers is justified and formulated. The hypothesis about the distribution of the number of chains of «neighboring» a-self numbers is justified and formulated and the complete consistency of the two hypotheses is proved. Formulas for the chains of «neighboring» a-self numbers are found. The multiple relations between the Kaprekar function K(n) and the function introduced by us A(n). We studied and found all solutions of the functional equation A(qn) = qA(n).

Weyl Sums over Integers with Digital Restrictions

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs, we use ideas that go back to a paper by Banks, Conflitti, and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem established by Bourgain, Demeter, and Guth (2016) and Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.

A new semiring and its cryptographic applications

This paper introduced a novel semiring structure involving nonnegative integers, where operations depended on the comparison of the magnitudes of decimal digit sums. Consequently, a corresponding matrix semiring can be established on this commutative semiring. We showed that the 3-satisfiability problem can be polynomial-time reduced to solving systems of quadratic polynomial equations over this semiring. We proposed a key exchange protocol based on this matrix semiring, with its security relying on the two-sided digital circulant matrix action problem over this semiring. This scheme provides a novel cryptographic primitive for post-quantum cryptography.

On the Sum of Digits of Expansions of a Pair of Consecutive Numbers over a Linear Recurrent Sequence

On the Sum of Digits of Expansions of a Pair of Consecutive Numbers over a Linear Recurrent Sequence

Collisions of digit sums in bases 2 and 3

Collisions of digit sums in bases 2 and 3

Physical assets life cycle analysis in a food industry company - case study

Monitoring the life cycle of physical assets (PA) implies addressing issues, such as PA’s energy efficiency and its replacement and definition of the most proper moment to renew. The goals of this article are: to present a characterization of energy sources and analyze the PA life cycle in a food sector company. First, it will be characterized the costs and the expenses of the organization’s energy sources, then, a study about the replacement of PA is presented; Traditional methods were used, such as economic life; The models that underlie it are discussed throughout the article, using actual data, for validation. Three methods for depreciation of PA are used: Linear Depreciation; Sum of Digits and Exponential. Other methods were used to determine the Economic Cycle for replacing PA: Uniform Annual Expenditure (MRAU); Minimizing the Average Total Cost (MCMT); and the MCMT-Reduced to Present Value (MCMT-RVP). The equipment of the study was bakery ovens (gas and electrical). Results and conclusions from the application of the methods used in the evaluation of these PAs are presented.

On the binary digits of n and n2

Let s(n) denote the sum of digits in the binary expansion of the integer n. Hare, Laishram and Stoll (2011) studied the number of odd integers such that s(n)=s(n2)=k, for a given integer k≥1. The remaining cases that could not be treated by these authors were k∈{9,10,11,14,15}. In this paper we show that there is only a finite number of solutions for k∈{9,10,11} and comment on the difficulties to settle the two remaining cases k∈{14,15}. A related problem is to study the solutions of s(n2)=4 for odd integers. Bennett, Bugeaud and Mignotte (2012) proved that there are only finitely many solutions and conjectured that n=13,15,47,111 are the only solutions. In this paper, we give an algorithm to find all solutions with fixed sum of digits value, supporting this conjecture, as well as show related results for s(n2)=5.

A Generalization to Bellman and Shapiro's Method on the Sum of Digital Sum Functions

Consider the digital sum function in base 2. In 1948, R. Bellman and H.N. Shapiro proved a formula for the sum of digital sum functions in base 2 up to a certain number with an error term. However, there was a mistake in their proof. We are able to correct the mistake and walk in their footsteps while retaining their idea and method to prove the theorem. We also generalized their method to prove the same theorem for general base.

Computational aspects of sturdy and flimsy numbers

Following Stolarsky, we say that a natural number n is flimsy in base b if some positive multiple of n has smaller digit sum in base b than n does; otherwise it is sturdy. When n is proven flimsy by multiplier k, we say n is k-flimsy. We study computational aspects of sturdy and flimsy numbers.We provide some criteria for determining whether a number is sturdy. We study the computational problem of checking whether a given number is sturdy, giving several algorithms for the problem, focusing particularly on the case b=2.We find two additional, previously unknown sturdy primes. We develop a method for determining which numbers with a fixed number of 0's in binary are flimsy. Finally, we develop a method that allows us to estimate the number of k-flimsy numbers with n bits, and we provide explicit results for k=3 and k=5. Our results demonstrate the utility (and fun) of creating algorithms for number theory problems, based on methods of automata theory.

Dirichlet series associated to sum-of-digits functions

We study the Dirichlet series [Formula: see text], where [Formula: see text] is the sum of the base-b digits of the integer n, and [Formula: see text], where [Formula: see text] is the summatory function of [Formula: see text]. We show that [Formula: see text] and [Formula: see text] have analytic continuations to the plane [Formula: see text] as meromorphic functions of order at least 2, determine the locations of all poles, and give explicit formulas for the residues at the poles. We give a continuous interpolation of the sum-of-digits functions [Formula: see text] and [Formula: see text] to non-integer bases using a formula of Delange, and show that the associated Dirichlet series have a meromorphic continuation at least one unit left of their abscissa of absolute convergence.

Soma iterada de algarismos de um número racional

The digital roots S* (x), of a n positive integer is the digit 0 ≤ b ≤ 9 obtained through an iterative digit sum process, where each iteration is obtained from the previous result so that only the b digit remains. For example, the iterated sum of 999999 is 9 because 9 + 9 + 9 + 9 + 9 + 9 = 54 and 5 + 4 = 9. The sum of the digits of a positive integer, and even the digital roots, is a recurring subject in mathematical competitions and has been addressed in several papers, for example in Ghannam (2012), Ismirli (2014) or Lin (2016). Here we extend the application Sast to a positive rational number x with finite decimal representation. We highlight the following result: given a rational number x, with finite decimal representation, and the sum of its digits is 9, so when divided x by powers of 2, the number resulting also has the sum of its digits 9. Fact that also occurs when the x number is divided by powers of 5. Similar results were found when the x digit sum is 3 or 6.

The Monomial Lattice in Modular Symmetric Power Representations

Let p be a prime. We study the structure of and the inclusion relations among the terms in the monomial lattice in the modular symmetric power representations of $\text {GL}_{2}(\mathbb {F}_{p})$ . We also determine the structure of certain related quotients of the symmetric power representations which arise when studying the reductions of local Galois representations of slope at most p. In particular, we show that these quotients are periodic and depend only on the congruence class modulo p(p − 1). Many of our results are stated in terms of the sizes of various sums of digits in base p-expansions and in terms of the vanishing or non-vanishing of certain binomial coefficients modulo p.

The sum of digits functions of the Zeckendorf and the base phi expansions

We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.

On the correlation of the sum of digits along prime numbers

Let $s_{q}$ denote the sum of digits function in base $q$. The aim of this work is to estimate the exponential sums involving the sum of digits of shifted prime numbers, of the form $\sum _{p\leq x}e(\alpha _{0}s_{q}(p)+\cdots +\alpha _{k}s_{q}(p+k))$, wh

On Gelfond-type problem for generalized Zeckendorf representations

Gelfond proved that for coprime 𝑏 − 1 and 𝑑 sums of digits of 𝑏-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference 𝑑. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences. We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case 𝑑 = 2, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for 𝑑 = 3 the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers. In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference 2 with the logarithmic remainder term.