CANT 2023.

Twenty-first Annual Workshop on
Combinatorial and Additive Number Theory
CUNY Graduate CenterMay 23 - 26, 2023

Abstracts(1) Ruben Ascoli, Princeton University
Email: rascoli@princeton.edu
Andrew W. Keisling, University of Michigan
Email: keislina@umich.edu
Title: Sum and difference sets in generalized dihedral groups
Abstract: Given a group G, we say that a set A ⊆ G has more sums thandifferences (MSTD) if |A + A| > |A − A|, has more differences than sums
(MDTS) if |A + A| < |A − A|, or is sum-difference balanced if |A + A| =|A − A|. Recently, Miller and Vissuet studied the frequency of these typesof subsets in arbitrary finite groups G and proved that almost all subsets
A ⊆ G are sum-difference balanced as |G| → ∞. For the dihedral group
D2n , they conjectured that of the remaining sets, most are MSTD.
We extend the conjecture to generalized dihedral groups D = Z2 n G,
where G is an abelian group of size n and the nonidentity element of Z2 acts
by inversion. We make further progress on the conjecture by considering
subsets with a fixed number of rotations and reflections. By bounding the
expected number of overlapping sums, we show that the collection SD,m ofsubsets of the generalized dihedral group
D of size m has√more MSTD sets√
than MDTS sets when 6 ≤ m ≤ cj n for cj = 1.3229/ 111 + 5j, where
j is the number of elements in G with order at most 2. We also analyze
the expectation for |A + A| and |A − A| for A ⊆ D2n , providing an explicit
formula for |A − A| when n is prime.
Joint work with Justin Cheigh, Guilherme Zeus Dantas e Moura, Ryan
Jeong, Astrid Lilly, Steven J. Miller, Prakod Ngamlamai, Matthew Phang.
(2) Emma Bailey, CUNY Graduate Center
Email: ebailey@gc.cuny.edu
Title: Atypical behaviour of the Riemann zeta function
Abstract: Selberg’s celebrated central limit theorem shows that log ζ(1/2 +it) at a typical point t at height T behaves like a complex, centered Gaussian random variable with variance log log T . In this talk I will present sometheoretical and numerical results regarding behavior of ζ(1/2 + it) outsideof the range where the central limit theorem is known to hold.

1(3) Jonathan Chapman, University of Bristol, UK
Email: jonathan.chapman@bristol.ac.uk
Title: Ramsey properties of polynomial equations
Abstract: A system of Diophantine equations is called partition regular if
it admits monochromatic solutions with respect to any finite colouring of
the positive integers. In the 1930s, Rado obtained necessary and sufficientconditions for arbitrary systems of linear equations to be partition regular.
However, it is only with recent breakthroughs in analytic number theory
and additive combinatorics that we have been able to obtain an analogous characterisation of partition regularity for large families of polynomial
equations. In this talk, I will report on these recent developments.
Joint work with Sam Chow (University of Warwick).
(4) Robert Donley, Queensborough Community College (CUNY)
Email: rdonley@qcc.cuny.edu
Title: Path enumeration for antimagic squares
Abstract: Students typically encounter path counting on rectangular posets
as an application of binomial coefficients. If one recasts this model in termsof order-raising operators, the Jordan canonical form gives an alternativedescription of Clebsch-Gordan coefficients without representation theory.
At the same time, the underlying parameter space consists of semimagic
squares of size three, and path counting methods there also yield such
coefficients in terms of certain hypergeometric series. Semimagic squares oflarger size present formidable difficulties, but antimagic squares, as definedby R. Stanley, retain basic features from the case of size three. We present
a model of antimagic squares for path counting and identify two families ofzeros for the corresponding hypergeometric series.
This talk reports on work from the Summer 2022 RAMMP REU at City
College (CUNY). Joint work with Arnav Krishna (Pierrepont School) andZixuan Ye (University of Buffalo).
(5) Jin-Hui Fang, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, PR China
Email: fangjinhui1114@163.com
Title: On the density of bounded bases
Abstract: For a nonempty set A of integers and an integer n, let rA (n) be
the number of representations of n in the form n = a + a0 , where a ≤ a0
and a, a0 ∈ A, and dA (n) be the number of representations of n in the formn = a − a0 , where a, a0 ∈ A. The binary support of a positive integer n isdefined as the subset S(n) of nonnegative integers
P consisting of the exponents in the binary expansion of n, i.e., n = i∈S(n) 2i , S(−n) = −S(n)and S(0) = ∅. For real number x, let A(−x, x) be the number of elementsa ∈ A with −x ≤ a ≤ x. The famous Erdős-Turán Conjecture statesthat if A is a set of positive integers such that rA (n) ≥ 1 for all sufficiently large n, then lim supn→∞ rA (n) = ∞. In 2004, Nešetřil and Serra
initially introduced the notation of “bounded” property and confirmed theErdős-Turán conjecture for a class of bounded bases. They also proved that,
there exists a set A of integers satisfying rA (n) = 1 for all integers n andS|S(x) S(y)| ≤ 4|S(x + y)| for x, y ∈ A. On the other hand, Nathanson
proved that there exists a set A of integers such that rA (n) = 1 for all integers n and 2 log x/ log 5 + c1 ≤ A(−x, x) ≤ 2 log x/ log 3 + c2 for all x ≥ 1,
where c1 , c2 are absolute constants. Following these results, we prove that,
there exists a set A of integers such that: SrA (n) = 1 for all integers n anddA (n) = 1 for all positive integers n, |S(x) S(y)| ≤ 4|S(x+y)| for x, y ∈ A
and A(−x, x) > (4/ log 5) log log x + c for all x ≥ 1, where c is an absolute
constant. Furthermore, we also construct a family of arbitrarily spare suchsets A.
(6) Leonid Fel, Technion – Israel Institute of Technology, Israel
Email: lfel@cv.technion.ac.il
Title: Symmetric (not complete intersection) numerical semigroups andsyzygy identities
Abstract: We consider symmetric (not complete intersection) numericalsemigroups hd1 , . . . , dm i of arbitrary embedding dimension m, minimallygenerated by a set of m positive integers, such that gcd(d1 , . . . , dm ) = 1.
We derive identities for degrees of syzygies of such semigroups and find thelower bound for their Frobenius numbers that generalizes recent results form = 4, 5, 6.
(7) Mikhail Gabdullin, Steklov Mathematical Institute, Moscow, Russia
Email: gabdullin.mikhail@yandex.ru
Title: Prime avoiding numbers is a basis of order two
Abstract: For a positive integer n, let F (n) be the distance from n to
the nearest prime number. In 2015 Ford, Heath-Brown, and Konyagin
introduced the notion of prime avoidance: a number n is called a prime
avoiding number with constant c, if
F (n) > clog n log log n log log log log n.
(log log log n)2They proved that for any positive integer k there exist c = c(k) > 0 andinfinitely many k-th powers which are prime avoiding with constant c. Later
Maier and Rassias extended this result to k-th prime powers and a stronger
prime avoidance.
We use the method from the recent breakthrough work of Ford, Konyagin, Maynard, Pomerance, and Tao on consecutive composite values ofpolynomials to prove the following theorem:
Every sufficiently large positive integer N can be represented as N =n1 + n2 , where
F (ni ) > (log N )(log log N )1/325565 ,i = 1, 2.

This is joint work with Artyom Radomskii (HSE University).
(8) Krystian Gajdzica, Jagiellonian University, Krakw, Poland
Email: krystian.gajdzica@im.uj.edu.pl
Title: Beyond the log-concavity of the restricted partition function
Abstract: For a fixed parameter k ∈ N+ and a sequence A = (ai )∞i=1
of positive integers, the restricted partition function pA (n, k) enumerates

those partitions of n whose parts belong to the multiset {a1 , a2 , . . . , ak }.
The main issues of our interests will be both the r-log-concavity problem
and the higher order Turán inequalities for pA (n, k). However, we will also
discuss some basic generalizations of the log-concavity which correspond to
the following inequalities:
p2A (n, k) > pA (n − m, k)pA (n + m, k),
p2A (an, k) > pA (an − bn, k)pA (an + bn, k),(n2 − 1)α p2A (n, k) > n2α pA (n − 1, k)pA (n + 1, k),where a, b, m ∈ N+ and α ∈ R+ are fixed such that a > b. For each of the
problems, the efficient solution will be presented.
(9) Zhenchao Ge, ShanghaiTech University, China
Email: gezc@shanghaitech.edu.cn, gezhenchao@gmail.com
Title: A Weyl-type inequality for irreducible elements in function fields,with applications
Abstract: We establish a Weyl-type estimate for exponential sums overirreducible elements in function fields. As an application, we generalize
an equidistribution theorem of Rhin. Our estimate works for polynomials
with degree higher than the characteristic of the field, a barrier to the
traditional Weyl differencing method. In this talk, we briefly introduce
Lê-Liu-Wooley’s original argument for ordinary Weyl sums (taken over all
elements), and how we generalize it to estimate bilinear exponential sums
with general coefficients.
This is joint work with Thái Hoàng Lê (Mississippi) and Yu-Ru Liu (Waterloo).
(10) Robert Groth, University of South Carolina
Email: rgroth@email.sc.edu
Title: Constructing generalized Sierpı́nski numbers
Abstract: A Sierpı́nski number is a positive odd integer k such that k ·2n +1
is composite for all n ∈ Z+ . Fix an integer A with 2 ≤ A. We show there
exists a positive odd integer k such that k·an +1 is composite for all integersa ∈ [2, A] and all n ∈ Z+ .
This is joint work with Michael Filaseta and Thomas Luckner.
(11) C. Sinan Gunturk, Courant Institute, NYU
Email: gunturk@cims.nyu.edu
Title: Density of the Bernstein lattices
Abstract: Let L(Pn ) be the lattice of polynomials of degree at most n withinteger coefficients. We define the Bernstein lattice L(Bn ) to be the set of
all polynomials with integer coefficients with respect to the Bernstein basis n kpn,k (x) :=x (1 − x)n−k , k = 0, . . . , n.
kQn
L(Bn ) is a coarse sublattice of L(Pn ), of index k=0 nk ∼ exp(cn2 ).
Let CZ ([0, 1]) consist of all real-valued continuous functions on [0, 1] thattake on integer values at 0 and 1. It is well-known, due to Pál, Kakeya, andS
Chlodovsky, that n L(Pn ) is dense in CZ ([0, 1]) with respect
to the uniform
S
metric. We extend this result and show that in fact n L(Bn ) is dense in
CZ ([0, 1]) as well. Our result comes with an approximation algorithm andan effective bound on the error of approximation.
Time permitting, the talk will also include some applications of thisresult to neural networks.
Joint work with Weilin Li (City College, CUNY).
(12) Norbert Hegyvari, Rényi Institute, Budapest, Hungary
Email: hegyvari@renyi.hu
Title: Covering shrinking polynomials by progressions
PT
Abstract: Erdős introduced the quantity S = T i=1 Xi , where X1 , . . . , XT
are arithmetic progressions, and cover the square numbers up to N . He conjectured that S is close to N , i.e. the square numbers cannot be covered”economically” by arithmetic progressions. Sárközy confirmed this conjecture and proved that S ≥ cN/ log2 N . In the talk, we extend this toshrinking polynomials and so-called {Xi } quasi-progressions.
(13) Harald Helfgott, Universität Göttingen, Germany and CNRS, France
Email: harald.helfgott@gmail.com
Title: Explicit bounds on sums of the Möbius functions P
Abstract: Let M (x) be the Mertens function M (x) = n≤x µ(n). Mostof us are used to thinking of the problem of estimating M (x) as being
essentially equivalent to the problem of estimating the number of primes
up to x (i.e., the Prime Number Theorem). If we want explicit bounds,
however, the problem of bounding M (x) becomes by far the harder of thetwo problems. The main reason is that, while the residue of −ζ 0 (s)/ζ(s)at a zero of ζ(s) is just the order of the zero, we do not have a good way
to control the residues of 1/ζ(s). (There are bounds for 1/ζ(s) inside thezero-free region, but their constants are very large.)
Up to now, the best bounds on M (x) have been either (a) based onelementary methods that amount to delicate refinements of ChebyshevMertens, or (b) based on iterative processes that combine (a) with explicit
versions of PNT. (The best results of these two kinds to date are due toDaval (unpublished) andP Ramaré, respectively.)
P The same is true of sumsof the form m(x) =µ(n)/n,m̌(x)=n≤xn≤x µ(n) log(x/n)/n, etc.,
which appear often in analytic number theory.
We give considerably stronger bounds on M (x), m(x) and m̌(x) bymeans of an analytic approach based on mean-value theorems.
Joint work with Andrés Chirre.
(14) Russell Jay Hendel, Towson University
Email: rhendel@towson.edu.
Title: Recursions, closed forms, and characteristic polynomials of the circuit array
Abstract: One modern graph metric represents an electrical circuit with agraph whose edges are replaced with resistors and the so-called resistancedistance between the nodes is determined by calculating the electrical resistance in the circuit. Electrical circuit theory provides functions that allow“reduction” of one circuit to another circuit where the resistance distance
between certain vertices are preserved. Recently there has been study of agraph, representable in the Cartesian plan as an n-grid, n rows of uprightequilateral triangles, all of whose edges are labeled one. It is possible toreduce the n-grid to an (n − 1)-grid with resistance preserving operations.
The collections of successive reductions has many interesting properties.
In this talk we continue to study a “slice” of this collection of grids represented by the Circuit Array, an infinite array of rational functions. We
show that certain closed forms, recursions, and characteristic polynomials(annihilators) emerge. One surprising result is that the annihilators of thenumerators and denominators of the underlying rational functions exclusively have roots which are integral powers of 9.
Joint work with Emily J. Evans.
(15) Robert Hough, Stony Brook University
Email: robert.hough@stonybrook.edu
Title: Subconvexity of Sato-Shintani zeta functions
Abstract: Selberg’s Eisenstein series in n complex variables, with a symmetric group of functional equations, is just one example of a huge family(containing the standard L-functions, conjecturally containing the Selberg
class) of zeta functions constructed by Sato and Shintani. I will describe an
approach, joint with Eun Hye Lee, for proving subconvexity in this family.
This work ’won the game’, which I can try to explain, also.
(16) Alex Iosevich, University of Rochester
Email: iosevich@gmail.com
Title: Point configurations in vector spaces over finite fields
Abstract: We are going to consider several variants of the Erdős-Falconer
distance problem in vector spaces over finite fields and emphasize the connection between discrete and continuous methods.
(17) Brad Isaacson, NYC College of Technology (CUNY)
Email: brad.isaacson71@citytech.cuny.edu
Title: On some polynomial reciprocity formulas
Abstract: Carlitz proved a powerful reciprocity theorem for generalized
Dedekind-Rademacher sums. Among its many consequences was an interesting polynomial reciprocity theorem which holds under a certain restriction of its parameters. Carlitz remarked that it was unclear how thisrestriction could be removed. In last year’s CANT talk, we removed thisrestriction and obtained a generalization of Carlitz’s polynomial reciprocity
theorem. In this talk, we generalize it further. As a corollary, we get thepolynomial reciprocity theorem of Beck and Kohl.

(18) Ryan Jeong, University of Pennsylvania
Email: rsjeong@sas.upenn.edu
Title: Progress On the relative sizes of complements of generalized sumsets
Abstract: Given a subset A of {1, 2, . . . , N }, its sum set and difference set
are given by
A + A = {a1 + a2 : a1 , a2 ∈ A},

A − A = {a1 − a2 : a1 , a2 ∈ A}.

A problem of recent interest has been to understand the relative sizes of thesum and difference sets. We might expect that usually |A − A| > |A + A|
as addition commutes, but Martin and O’Bryant proved that a positive
proportion of the 2N subsets A of {1, 2, . . . , N } are sum-dominated, while
Hegarty and Miller establish a threshold phenomenon if we independentlyp−−−→include elements of {1, 2, . . . , N } in A with probability p(N ): |A−A||A+A| −
N →∞2 if p(N ) = o(N −1/2 ), this ratio decreases almost surely to 1 as c increasesp|(A+A)c |to ∞ for p(N ) = cN −1/2 , and |(A−A)−−−→ 2 if N −1/2 = o(p(N )).
c| −
N →∞

More recently, Hogan and Miller studied the relative sizes of generalized
sum and difference sets of the form As,d = A + A + · · · + A − A − A − · · · − A,with s sums and d differences, s+d = h and p(N ) = cN −δ for δ ≥ (h−1)/h.
They generalized the results of Hegarty and Miller by establishing thatp|As1 ,d1 |h−12!
−−−→ ss21 !d
|As ,d | −!d1 ! if δ > h and this ratio decreases almost surely to 1
2

2

N →∞

h−1

as c increases to ∞ for p(N ) = cN − h . The behavior in the “slow decay
regime δ < (h − 1)/h was left open. We make progress in this direction by
showing that, with probability 1 − o(1),|Ach,0 |≤ d + 1 + o(1),|Ach−d,d |which, in particular, establishes that the slow decay results of Hegarty andMiller do not generalize like the other two regimes for all choices of s andd.
Joint work with Steven J. Miller.
(19) Mizan Khan, Eastern Connecticut State University
Email: khanm@easternct.edu
Title: Interior hulls and continued fractions
Abstract: The interior hull of a lattice polygon is the convex closure of
the lattice points in the interior of the polygon. We will give a concrete
description of the interior hull of any lattice parallelogram.
We begin by focusing on clean parallelograms. A clean parallelogramin R2 is a lattice parallelogram whose boundary contains no lattice points
other than its vertices. Using unimodular maps we can identify a clean parallelogram with a parallelogram whose vertices are (0, 0), (1, 0), (a, n) and(a + 1, n) with 0 < a < n and gcd(a, n) = 1. Following Stark’s geometric approach to continued fractions we show that the convergents of thecontinued fraction of n/a (viewed as lattice points) appear in a one-to-two
correspondence with the vertices of the interior hull of this parallelogram.
Once we have given a concrete description of the interior hull of any cleanparallelogram, we can easily adapt our work to give a concrete description

of the interior hull of an arbitrary lattice parallelogram.
This is joint work with Gabriel Khan, Riaz Khan and Peng Zhao.
(20) Gergely Kiss, Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Email: kigergo57@gmail.com
Title: Recent progress on Fuglede’s conjecture on Z3p
Abstract: In my talk I present the recent developments in Fuglede’s conjecture on Z3p . This problem is particularly interesting, since it is known
that the conjecture holds on Zp and on Z2p , and does not hold on Zdp ford ≥ 4. Although the problem on Z3p is still open in general, I show a newconcept, which leads to some combinatorial and finite geometric problems
by using the concept of weak tilings and discrete Fourier analysis. My talkis based on joint work with D. Matolcsi, M. Matolcsi and G. Somlai and I
also discuss the results of R. D. Malikiosis.
(21) Sándor Kiss, Budapest University of Technology and Economics, Hungary
Email: ksandor@math.bme.hu
Title: Problems and results on additive representation functions associatedto linear forms
Abstract: Let k ≥ 2 and λ1 , . . . , λk be fixed positive integers. For a set A
of nonnegative integers, the additive representation function associated tolinear forms is
RA,λ (n) = |{(a1 , . . . , ak ) ∈ Ak : λ1 a1 + . . . + λk ak = n}|.
In my talk I would like to summarize our recent results about representation functions associated to linear forms. We will extend an earlier resultof Nathanson to representation functions associated for linear forms. Furthermore, we will describe all the k-tuples λ = (λ1 , . . . , λk ) and the sets ofnonnegative integers A with RA,λ (n) = 1 for every nonnegative integer n.
We also have several open problems for further research.
This is joint work with Csaba Sándor.
(22) Sergei Konyagin
Email: konyagin@mi-ras.ru
Title: Large gaps between sums of two squarefull numbers
Abstract: A positive integer n is called squarefull or powerful if in its
αr1
factorization n = pα
1 . . . pr we have αi ≥ 2 for all i. We consider that 0 is
also a squarefull number. Thus, a number is squarefull if and only if it canbe represented as n = a2 b3 for some a, b ∈ Z+ .
Let W be the set of all sums of two squarefull numbers. Blomer (2005)
proved that
W (x) := |W ∩ [1, x]| =x(log x)α+o(1)(x → ∞),

where α = 1 − 2−1/3 = 0.20 . . . .
As suggested by Shparlinski, we study large gaps between elements of
W . Namely, for x > 1 define M (x) as the length of the largest subintervalof [1, x] without elements of W . Blomer’s result implies that M (x) ≥(log x)α+o(1) as x → ∞ since the largest gap is at least as large as theaverage gap. We improve this estimate.
Theorem: For x ≥ 3 we have M (x) ≥ c(log x)/(log log x)2 where c > 0
is an absolute constant.
Joint work with Alexander Kalmynin.
(23) Tomasz Kowalczyk, Jagiellonian University in Kraków, Poland
Email: tomek.kowalczyk@uj.edu.pl
Title: On Waring numbers of henselian rings II
Abstract: Let n > 1 be a positive integer and R be a commutative ring withunity. We define the nth Waring number wn (R) as the smallest positive
integer g such that every sum of nth powers of elements of R can be written
as a sum of at most g nth powers of elements of R.
Let R be a henselian local ring with residue field k of nth level sn (k).
We give some upper and lower bounds for the nth Waring number wn (R) in
terms of wn (k) and sn (k). In large number of cases we are able to computewn (R). Similar results for the nth Waring number of the total ring of
fractions of R are obtained. We then provide applications. In particular
we compute wn (Zp ) and wn (Qp ) for n ∈ {3, 4, 5} and any prime p.
Joint work with Piotr Miska.
(24) Noah Kravitz, Princeton University
Email: nkravitz@princeton.edu
Title: Lonely Runner spectra, revisited
Abstract: Dirichlet’s Theorem says that for any real number t, there issome v in {1, 2, ..., n} such that tv is within 1/(n + 1) of an integer. The
Lonely Runner Conjecture of Wills and Cusick asserts that the constant
1/(n + 1) in this theorem cannot be improved by replacing {1, 2, ..., n} witha different set of n nonzero real numbers. The conjecture, although now
more than 50 years old, remains wide open for n larger than 6. In this talkI will describe the “Lonely Runner spectra” that arise when one considers
the “inverse problem” for the Lonely Runner Conjecture, and I will explain
the (a priori surprising) “hierarchical” relations among these spectra.
Joint work with Vikram Giri.
(25) Andrzej Kukla, Jagiellonian University in Kraków, Poland
Email: andrzej.kukla@student.uj.edu.pl
Title: Practical sets and A-practical numbers
Abstract: Set A ⊂ ZP
+ is said to be practical if and only if all positive
integers smaller than a∈A a are representable as some sum of distinct elements from A. On the basis of this concept we define A-practical numbers,
which generalize the notion of practical numbers. In the talk I will discussbasic properties of practical sets and their connection to practical numbers,
but also basic results concerning A-practical numbers with emphasis on
expansion and removal theorems.
Joint work with Maciej Ulas and Piotr Miska, Jagiellonian University.

(26) Angel Kumchev, Towson University
Email: akumchev@towson.edu
Title: New versions of old results on gaps between squarefree integers
Abstract: The study of the shortest interval (x, x + h] that contains squarefree integers for all sufficiently large x goes back to the 1940s. A period of
active development in the 1980s and early 1990s culminated in a celebrated
result of Filaseta and Trifonov that if c is a sufficiently large constant, one
can take h ≤ cx1/5 log x. However, since then progress has stalled, andthis is still the best known upper bound on the maximum gap between
squarefree integers.
In this talk, I will report on some related work completed by an REU
group I supervised jointly with Nathan McNew in the summer of 2022.
First, through a thorough examination and careful optimization of the techniques of Filaseta and Trifonov we were able to prove a fully explicit versionof their gap result: The interval (x, x + 11x1/5 log x] contains a squarefree
integer for all x ≥ 2. We also proved versions with smaller explicit constants that hold when x ≥ x0 , where x0 is given explicitly. The researchteam proved also such results on gaps between cubefree and general k-free
integers by developing ideas of Trifonov from the mid 1990s.
Further, we are able to establish analogs of several bounds on the gaps
between squarefree integers for squarefree polynomials over a finite field Fq .
In this context, Keating and Rudnick have shown that when q → ∞ one
can prove an essentially best possible result. However, the case when q isfixed and the size of the polynomials is determined by their degrees does
not appear to have been studied. We show that in that case, the problem
resembles the classical problem much more closely. In particular, we provea variant of the Filaseta-Trifonov theorem for polynomials over finite fields.
(27) Jeffrey C. Lagarias, University of Michigan
Email: lagarias@umich.edu
Title: The floor quotient partial order
The floor quotient order is given by a relation on the positive integers whichsays d is less than n if d = bn/kc for some integer k. This relation defines apartial order. This talk reports on properties of the partial order restricted
to order intervals [d, n], with emphasis on the initial intervals [1, n]. These
include results on the size of intervals, the internal structure of intervals,
and the two-variable Mobius function of the order on intervals. The initialinterval has extra structure arising from an involution d → bn/dc acting on
its elements. The restriction to the “top half” of these elements has Móbius
function related to the usual divisibility Möbius function. Experimentallyµ(1, n) may sometimes take large values. We bound its size and show it
has infinitely many sign changes.
Joint work with D. Harry Richman (U. Washington).

(28) Noah Lebowitz-Lockard, University of Texas, Tyler, TX
Email: nlebowitzlockard@uttyler.edu
Title: Long runs of numbers with the same number of divisors
Abstract: In 1952, Erdős and Mirsky asked for an upper bound on thelongest sequence of consecutive numbers ≤ x with the same number of divisors. No one made any progress until this past January, when Spătaru
found the first non-trivial upper bound. We discuss a slight improvement on
Spătaru’s bound and discuss the analogous problem for related functions,such as the sum-of-proper-divisors function and the Carmichael function.
Joint work with Joseph Vandehey.
(29) Paolo Leonetti, Universitá degli Studi dell’Insubria, Varese, Italy
Email: leonettipaolo@gmail.com
Title: How many sets of integers have a small perturbation of the type
A + B?
Abstract: We show, from a topological and from a measure viewpoint,that most sets of nonnegative integers have the property that all their“small” perturbations cannot be written as A + B, for some A, B ⊆ N with
|A|, |B| ≥ 2.
References: P.L., Almost all sets of nonnegative integers and their small
perturbations are not sumsets, Proc. Amer. Math. Soc., to appear.
(30) Jared Duker Lichtman, University of Oxford, UK
Email: jared.d.lichtman@gmail.com
Title: Primes in large arithmetic progressions and smooth shifted primes
Abstract: We prove the infinitude of shifted primes p − 1 without prime
factors above p0.2844 . This refines p0.2961 from Baker and Harman in 1998.
Consequently, we obtain an improved lower bound on the distribution of
Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we
estimate primes of size x with quadrilinear forms of moduli up to x0.5312 .
This result extends moduli beyond x0.5233 due to Zhang and Polymath,and x0.5238 recently obtained by Maynard. These results all build on wellknown 1986 work of Bombieri, Friedlander, and Iwaniec.
(31) Florian Luca, University of the Witwatersrand, South Africa
Email: florian.luca@wits.ac.za
Title: Carmichael numbers of the form 2n k + 1
Abstract: In the first part of the talk we will survey some results concerning
Carmichael numbers of the form 2n k + 1. Then we will present the proofof the fact that there is no Carmichael number of the above form with kprime.
Joint work with A. Alahmadi.

(32) Ariane Masuda, New York City College of Technology (CUNY)
Title: Involutions over finite fields
Email: amasuda@citytech.cuny.edu
Abstract: In the context of memory-limited environments, involutions aredesirable because they allow for efficient use of limited memory resources.
Specifically, in many applications, both the permutation and its inverse
must be stored in memory, which can be a challenge in resource-constrained
environments. By using an involution as the interleaver, the same structure
and technology used for encoding can be used for decoding as well. Fixed
points are points that remain unchanged under the permutation. In cryptographic applications, such as the design of S-boxes, it is desirable to have
permutations with a small number of fixed points to increase the securityof the system. Therefore, understanding the number of fixed points andhow to construct involutions with few fixed points is an important area of
research. In the talk I will present some new families of involutions overfinite fields, including explicit constructions based on a prescribed numberof fixed points.
Part of this work is joint with Juliane Capaverde and Virgı́nia Rodrigues.
Another part is joint with Lillian González-Albino and Ivelisse Rubio.
(33) Karyn McLellan, Mount Saint Vincent University
Email: karyn.mclellan@msvu.ca
Title: Counting nth differences of the floor of an irrational sequence
Abstract: Consider the following open problem: Let r be an irrational
number with fractional part between 13 and 21 . Let Cn be the number ofdistinct nth differences of the sequence (bkrc), where k ∈ Z+ . Prove or
disprove that Cn = (2, 3, 3, 5, 4, 7, 5, 9, 6, 11, 7, 13, 8, 15, . . .), which is simply
a riffle of (2, 3, 4, 5, 6, . . .) and (3, 5, 7, 9, 11, . . .). In this talk I will discuss
the problem and provide some interesting results.
Joint work with Danielle Cox and Shayne Breen, MSVU.
(34) Nathan McNew, Towson University
Email: nmcnew@towson.edu
Title: The distribution of intermediate prime factors
1
Abstract: Let P ( 2 ) (n) denote the middle prime factor of n (taking into1
account multiplicity). It has previously been shown that log log P ( 2 ) (n)has normal order 21 log log x, and its values follow a Gaussian distribution
around this value. We extend this work by obtaining an asymptotic for1
mula for the count of n ≤ x for which P ( 2 ) (n) = p, for primes p in a wide
range up to x. We also generalize the “middle prime factor” to any percentile α ∈ (0, 1) (and find that the results are subtly different when α isirrational). This result has a variety of applications, for example we canuse it to obtain an asymptotic expression for the geometric mean of the
middle prime factors, and discover that the golden ratio makes a surprising appearance. We also find that P (α) (n) obeys Benford’s leading digit
law, and that P (α) (n) is equidistributed in coprime residue classes, in an
essentially optimal range of uniformity in the modulus.

Joint work with Paul Pollack and Akash Singha Roy at University ofGeorgia.
(35) Piotr Miska, Jagiellonian University in Kraków, Poland
Email: piotr.miska@uj.edu.pl
Title: On Waring numbers of henselian rings I
Abstract: Let n > 1 be a positive integer and R be a commutative ring withunity. We define the nth Waring number wn (R) as the smallest positive
integer g such that every sum of nth powers of elements of R can be written
as a sum of at most g nth powers of elements of R.
Let R be a henselian local ring with residue field k of nth level sn (k).
We give some upper and lower bounds for the nth Waring number wn (R) in
terms of wn (k) and sn (k). In large number of cases we are able to computewn (R). Similar results for the nth Waring number of the total ring of
fractions of R are obtained. We then provide applications. In particular wecompute the Waring numbers of rings of p-adic integers and fields of p-adic
numbers for n ∈ {3, 4, 5} and any prime p.
Joint work with Tomasz Kowalczyk.
(36) Md Ibrahim Molla, Ramakrishna Mission Vivekananda Educational andResearch Institute (RKMVERI), India
Email: mdibimya@gmail.com
Title: Some zero-sum constants and their associated inverse problems
Abstract: Let G be a finite abelian group (written additively) of exponentm and A ⊂ [1, m − 1] a non-empty set. Then the Davenport constant of
G with weight A, denoted by DA (G), is defined to be the smallest positive
integer k such that every sequence S = (x1 , . . . , xk ) over G of length k
has a non-empty A-weighted zero-sum subsequence, that is, there exist a
non-empty subsequence (xj1 , . . . , xjt ) of S and a1 , . . . , at ∈ A such thatPt
i=1 ai xji = 0, where 0 is the identity element of G. Similarly, for any suchweight set A and for a finite abelian group G of order n, the constant EA (G)is defined to be the least t ∈ N such that for any sequence (x1 , . . . , xt ) of telements with xi ∈ G, there exists an A-weighted zero-sum subsequence oflength n.
The case A = {1} corresponds to the classical Davenport constant D(G)and E(G).
For a particular DA (G) (respectively, EA (G)), the associated inverse
problem refers to the investigation of all sequences over G of length DA (G)−1 (respectively, EA (G)-1) not having any non-empty A-weighted zero-sum
subsequence of any length (respectively, of length n).
The study of the inverse problems corresponding to different zero-sum
constants with various weight sets is a fascinating topic and has been ofgrowing interest. In this talk, our plan is to discuss some results in thisdirection.

(37) Mel Nathanson, Lehman College (CUNY)
Email: melvyn.nathanson@lehman.cuny.edu
Title: Patterns in the iteration of an arithmetic function
Abstract: Let Ω be a set of positive integers and let S : Ω → Ω be an arithmetic function. Let V = (vi )ni=1 be a finite sequence of positive integers.
An integer m ∈ Ω has increasing-decreasing pattern V with respect to S if,
for all odd integers i ∈ {1, . . . , n},
S v1 +···+vi−1 (m) < S v1 +···+vi−1 +1 (m) < · · · < S v1 +···+vi−1 +vi (m)and, for all even integers i ∈ {2, . . . , n},
S v1 +···+vi−1 (m) > S v1 +···+vi−1 +1 (m) > · · · > S v1 +···+vi−1 +vi (m).
The arithmetic function S is wildly increasing-decreasing if, for every finite
sequence V of positive integers, there exists an integer m ∈ Ω such that m
has increasing-decreasing pattern V with respect to S. It is proved that
the Collatz function is wildly increasing-decreasing.
(38) Miquel Ortega, Universitat Politècnica de Catalunya, Barcelona, Spain
Email: miquel.ortega.sanchez-colomer@upc.edu
Title: Product-free sets in the free group
Abstract: We prove that product-free sets of the free group over a finite
alphabet have maximum density 1/2 with respect to the natural measurethat assigns total weight one to each set of irreducible words of a given
length. This confirms a conjecture of Leader, Letzter, Narayanan and Walters. In more general terms, we actually prove that strongly k-product-freesets have maximum density 1/k in terms of the said measure. The bounds
are tight.
Joint work with Juanjo Rué and Oriol Serra.
(39) Cormac O’Sullivan, Bronx Community College (CUNY)
Email: cormac.osullivan@bcc.cuny.edu
Title: A hidden link between two of Ramanujan’s approximations
Abstract: In consecutive notebook entries, Ramanujan gave asymptotic approximations to the exponential function and the exponential integral. The
asymptotic expansion coefficients seem to agree up to an alternating sign,
as we conjectured in an earlier paper. We establish this hidden link with a
combinatorial proof that involves Stirling numbers, second-order Eulerian
numbers, modifications of both of these, and Stirling’s approximation to the
gamma function. An analytic second proof has also been provided by a referee. It is not yet clear if this result is an oddity or part of a broader picture.

(40) Péter Pál Pach, Budapest University of Technology and Economics, Hungary
Email: p.p.pach@gmail.com
Title: The Alon-Jaeger-Tarsi conjecture via group ring identities
Abstract: The Alon-Jaeger-Tarsi conjecture states that for any finite field Fof size at least 4 and any nonsingular matrix A over F there exists a vector
x such that neither x nor Ax has a 0 component. In this talk we discuss
the proof of this result for sufficiently large primes and further applicationsof our method about coset covers and additive bases.
Joint work with János Nagy and István Tomon.
(41) Firdavs Rakhmonov, University of Rochester
Email: frakhmon@ur.rochester.edu
Title: Distribution of similar configurations in subsets of Fdq
Abstract: Let Fq be a finite field of order q and E be a set in Fdq . The
distance set of E is defined by ∆(E) := {kx − yk : x, y ∈ E}, wherekαk = α12 + · · · + αd2 . Iosevich, Koh and Parshall (2018) proved that if d ≥ 2
is even and |E| ≥ 9q d/2 , thenFq =o∆(E) n a=: a ∈ ∆(E), b ∈ ∆(E) \ {0} .
∆(E)bIn other words, for each r ∈ F∗q there exist (x, y) ∈ E 2 and (x0 , y 0 ) ∈ E 2
such that kx − yk =6 0 and kx0 − y 0 k = rkx − yk.
Geometrically, this means that if the size of E is large, then for any givenr ∈ F∗q we can find a pair of edges in the complete graph K|E| with vertex
set E such that one of them is dilated by r ∈ F∗q with respect to the other.
A natural question arises whether it is possible to generalize this result toarbitrary subgraphs of K|E| with vertex set E and this is the main goal of
the talk.
In this talk, Ill explain the proof of the problem for k-paths (k ≥ 2),
simplexes and 4-cycles. The proof is based on a mixture of tools from different areas such as enumerative combinatorics, group actions and Turntype theorems.
(42) Alex Rice, Millsaps College
Email: riceaj@millsaps.edu
Title: Generalized arithmetic progressions and diophantine approximation
by polynomials
Abstract: We discuss two related notions of “approximate subgroups” inside finite sets of integers: Bohr sets, which capture simultaneous Diophantine approximation, and symmetric generalized arithmetic progressions (GAPs). For example, fix natural numbers N and d and consider
the following pair of questions:(a) For fixed α1 , . . . , αd ∈ R, how close to an integer can we simultaneously
make n2 α1 , . . . , n2 αd for some 1 ≤ n ≤ N ?
(b) How large can a set of the form {x1 `1 +· · ·+xd `d : |`i | ≤ Li } ⊆ [−N, N ]be before it is guaranteed to contain a perfect square?

Our discussions range from classical facts like the Kronecker approximation
theorem and Linnik’s theorem, to a recent breakthrough result of Maynard
and its potential future applications. In between we survey results including previous joint work with Neil Lyall and Ernie Croot.
(43) Asher Roberts, CUNY Graduate Center
Email: aroberts@gradcenter.cuny.edu
Title: The rate of convergence for Selberg’s multivariate CLT
Abstract: We discuss the approach of Radziwill and Soundararajan to proving Selberg’s central limit theorem for the Riemann zeta function on thecritical line. We will see that this approach can be extended to a multivariate context, providing another proof of a theorem of Bourgade. By tracking
the error resulting from each step, with the help of a carefully chosen metric, we obtain a rate of convergence for both the original theorem and themultivariate extension.
(44) David A. Ross, University of Hawai’i
Email: ross@math.hawaii.edu
Title: Some proofs about sequences in the spirit of Paul du Bois-Reymond
Abstract: Across several papers in the nineteenth century, du Bois-Reymonddeveloped a ‘calculus’ of scales of infinity, that is, relative sizes in thelimit of functions increasing to infinity. Other mathematicians—notably
Hadamard, Borel, and Hardy—later developed this machinery further. I’ll
apply these ideas to three examples: a Fekete-style theorem; a result about
almost additive sequences due to Pólya and Szegő; and an asymptotic fixed
point theorem.
(45) Misha Rudnev, University of Bristol, UK
Email: m.rudnev@bristol.ac.uk
Title: On sums and products of integers with few prime factors
Abstract: I will sketch the proof of a new sum-product result showing thatfor a set A of N integers, each of which has o(log N) prime factors, either
the product set A*A is big or there is a large subset with small additiveenergy. The proof relies on martingale maxmal inequalities and Schmidt’s
subspace theorem. The bound is optimal, up to sub-polynomial factors ofN, matching the example by Balog and Wooley.
This is joint work with Hanson, Shkredov and Zhelezov.
(46) Anurag Sahay, University of Rochester
Email: anuragsahay@rochester.edu
Title: The VC-dimension of multiplicative structures in finite fields
Abstract: Let Γ ⊆ F×
q be a multiplicative subgroup of a finite field. We
consider the VC dimension of the set system H = {Γ+x : x ∈ Fq } consistingof its additive translates. An example of particular interest is the set ofquadratic residues, Γ = Sq = {x2 : x ∈ F×q }.
We present some conjectures regarding this VC-dimension; the basis
for these conjectures is the expectation that additive and multiplicative
structure in finite fields must be loosely correlated. We also provide somepartial progress towards our conjectures. The main technical tool is theWeil bound for multiplicative character sums.
Joint work with Brian McDonald and Emmett Wyman.
(47) Aliaksei Semchankau, Carnegie Mellon University
Email: asemchan@andrew.cmu.edu
Title: On the sequence n! mod p
Abstract: Let p be a prime, and let A be a set of residues of the sequence
1!, 2!, 3!, . . . , (p − 1)! modulo p. We prove√√|A| > ( 2 + o(1)) p.
Now consider an interval I ⊆ {0, 1, . . . , p − 1} of length N > p7/8+ε anddenote by AI the set of residues modulo p it produces. Then we prove√|AI | > (1 + o(1)) p.
Tools used are results from Algebraic Geometry as black boxes and simple
combinatorial observations.
This is joint work with Alexandr Grebennikov, Arsenii Sagdeev, Aliaksei
Vasilevskii, and available on arXiv.
(48) Steven Senger, Missouri State University
Email: stevensenger@gmail.com
Title: Recent developments in point configuration problems in vector spaces
over various fields
Abstract: We discuss recent results involving the ubiquity of distinct isomorphic copies of point configurations in vector spaces over fields, both
finite and infinite. This work is inspired by the results in the fractal setting
on trees due to Yumeng Ou and Krystal Taylor.
(49) Sayak Sengupta, SUNY-Binghamton
Email: ssengup1@binghamton.edu
Title: Locally nilpotent polynomials over Z
Abstract: Let K be a number field and OK be the ring of integers of K.
For a polynomial u(x) in OK [x] and r ∈ OK , we can construct a dynamicalsequence u(r), u(2) (r), . . .. Let P (u(n) (r)) := {p ∈ MSpec(OK ) | u(n) (r) ∈p, for some n ∈ N}. For which polynomials u(x) and r ∈ OK do we expect
to have P (u(n) (r)) = MSpec(OK )? If we hit 0 somewhere in the above
sequence, then we obviously have the equality. If we do not hit zero for any
iteration then the question becomes very interesting. In this talk, we willdefine such polynomials for a general number field K and then we will lookat some results in the particular case of K = Q.

(50) Adam Sheffer, Baruch College (CUNY)
Email: adamsh@gmail.com
Title: Distinct distances in the complex plane
Abstract. We prove that if P is a set of n points in C2 , then either the
points in P determine Ω(n1−ε ) complex distances, or P is contained in aline with slope ±i. If the latter occurs, then each pair of points in P havecomplex distance 0.
Joint work with Joshua Zahl.
(51) Liyang Shen, New York University
Email: Liyang.Shen@nyu.edu
Title: Linear recurrences of order at most two in nontrivial small divisorsand large divisors
Abstract: For each positive integer N , define√√0SN= {1 < d < N : d|N } and L0N = { N < d < N : d|N }.
Recently, Chentouf characterized
all positive integers N such that the set√of small divisors {d ≤ N : d|N } satisfies a linear recurrence of order at
most two. We nontrivially extend the result by excluding the trivial divisor
1 from consideration, which dramatically increases the analysis complexity.
0
satisfies
Our first result characterizes all positive integers N such that SN
a linear recurrence of order at most two. Moreover, our second result characterizes all positive N such that L0N satisfies a linear recurrence of orderat most two, thus extending considerably a recent result that characterizes
N with L0N being in an arithmetic progression.
Joint work with Hùng Viê.t Chu, Kevin Huu Le, and Steven J. Miller,
Yuan Qiu.
(52) I.D. Shkredov, Steklov Mathematical Institute, Moscow, Russia
Email: ilya.shkredov@gmail.com
Title: On some multiplicative properties of large difference sets
Abstract: We study multiplicative properties of difference sets A − A forlarge sets A ⊆ Z/qZ in the case of composite q. A quantitative version ofa result of A. Fish about the structure of the product sets (A − A)(A − A)is obtained. We show that the multiplicative covering number of any difference set is always small. Also, we find several applications of our results
to various sums with multiplicative characters.
(53) Satyanand Singh, New York College of Technology (CUNY)
Email: ssingh@citytech.cuny.edu
Title: On the Pythagorean reciprocal identity a−2 + b−2 = d−2 and thegeneration of all solutions for fixed d > 0
Abstract: The Pythagorean reciprocal identity originates by way of a special perpendicular construct in a right triangle. Roger Vogel in 1999 showed
how to generate solutions to a−2 + b−2 = d−2 , with gcd(a, b, d) = 1, via
the primitive triples of the right triangle. We remove the requirement thatgcd(a, b, d) = 1 and show that this diophantine equation has solutions if andonly if d ≡ 0 mod 12. For fixed d, we generate all solutions to the identity
by two different methods. Our techniques employ appropriate bounds onthe variables and one of them is connected to the OEIS sequence A020884.
As an example, for d = 120, (a, b) ∈ {(130, 312), (136, 225), (150, 200)}
which as we will see results in three distinct triangles. You can switch theorder of a and b to get three additional algebraic solutions. This example
refers to a problem I created for the MAA Metro NY problem of the monthfor March 2023.
This is joint work with Alexander Rozenblyum.
(54) Bartosz Sobelewski, Jagiellonian University, Kraków, Poland
Email: bartosz.sobelewski@uj.edu.pl
Title: Values of the colored binary partition function represented as a sumof three squares
Abstract: For a fixed positive integer m, let bm (n) denote the number ofpartitions of n ∈ N with parts being powers of 2, where each part can takeone of m colors. In particular, b1 is the classical binary partition function,
already studied by Euler, while the case m ≥ 2 was introduced rather
recently. We are interested in the existence of a representation
bm (n) = x2 + y 2 + z 2 ,where x, y, z ∈ Z. In the case m = 2k −1 we give a “nice” characterization of
the set Sm of n such that this equation has a solution, which involves termsof the Thue–Morse sequence. This allows us to calculate the natural density
of Sm and provide an accurate approximation of its counting function. We
also discuss the harder case m 6= 2k − 1 and present some numerical results
concerning related equations.
Joint work with Maciej Ulas.
(55) Gábor Somlai, ELTE/CUNY Graduate Center
Enmail: zsomlei@gmail.com
Title: Cutting the positive real numbers into disjoint sets closed under
addition and multiplication
Abstract: Róbert Freud asked whether the positive real numbers can be
cut into two disjoint sets both closed under addition and multiplication.
Even if it might sound surprising, there is such a decomposition. This ismostly due to the fact that Q(α) can be cut into two pieces both closed
under addition and multiplication, if α is transcendental. However, any
finite extension of Q is indecomposable.
The tools used are just a combination of elementary ideas from geometry,
analysis and c
Joint work with Gergely Kiss and Tamás Terpai.
(56) Christoph Spiegel, Zuse Institute Berlin, Germany
Email: spiegel@zib.de
Title: Towards flag algebras in additive combinatorics
Abstract: We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of Fn3 as well as obtaining the first non-trivial
lower bound for the minimum number of monochromatic 4-APs in 2-coloringsof Fn5 . The former parallels results by Cumings et al (2013) in extremal

graph theory and the latter improves upon results of Saad and Wolf (2017)
The lower bounds are notably obtained by extending the flag algebra calculus of Razborov (2007) to additive structures in vector spaces over finite
fields.
This is joint work with Juanjo Ru and available on arXiv:
https://arxiv.org/abs/2304.00400(57) Christian Táfula, Université de Montréal, Canada
Email: christian.tafula.santos@umontreal.ca
Title: On the structure of t-representable sumsets
Abstract: Let A ⊆ Z≥0 be a finite set with minimal element 0, maximum
element m, and ` elements in between. Write (hA)(t) for the set of integers
that can be written in at least t ways as a sum of h elements of A. In 1970,
Nathanson showed that hA = (hA)(1) enjoys a notion of “structure” forlarge h, allowing us to determine hA in a relatively simple fashion. Thiswas revisited by Granville, Shakan and Walker in 2020, who showed that
hA is structured for every h ≥ m − `, and that finite sets in Zd also have asimilar notion of structure.
Also in 2020 (50 years later!), Nathanson showed that (hA)(t) enjoys anotion of structure as well for large h. In this talk, we will study the problemof giving upper bounds to how large h should be, showing that (hA)(t) is
structured if h & 1e m`t1/` . This estimate is asymptotically sharp, as we
can construct a family of sets A = A(m, `, t) ⊆ Z≥0 for which (hA)(t) is not
structured for h ≤ (1 − o(1)) 1e m`t1/` .
If time allows, we will also discuss how (hA)(t) for finite sets A ⊆ Zd
enjoys a similar notion of structure for large h.
(58) Ruiwen (Raven) Tang, Stuyvesant High School
Email: ruiwentang@gmail.com
Title: The constant of point-line incidence constructions
Abstract: We study a lower bound for the constant of the Szemerédi–Trotter theorem. In particular, we show that a recent infinite family ofpoint-line configurations satisfies I(P, L) ≥ (c + o(1))|P|2/3 |L|2/3 , withc ≈ 1.27. Our technique is based on studying a variety of properties ofEuler’s totient function. We also improve the current best constant for
Elekes’s construction from 1 to about 1.27.
Joint work with Martin Balko and Adam Sheffer.
(59) Marc Technau, Graz University of Technology, Austria
Email: mtechnau@math.tugraz.at
Title: On the Farey fraction spin chain
Abstract: In 1999, Kleban and Özlük introduced a ‘Farey fraction spin
chain’ and made a conjecture regarding its asymptotic number of stateswith given energy, the latter being given (up to some normalisation) by thenumber Φ(N ) of 2 × 2 matrices arising as products of 11 01 and 10 11 whose
trace equals N . Although their conjecture was disproved by Peter (2001),

quite precise results are known on average by works of Kallies–Özlük–Peter–
Snyder (2001), Boca (2007) and Ustinov (2013). We shall attempt to convey
some of the ideas underlying the above works.
Precise asymptotics for Φ(N ) had hitherto only been known conditionally on the availability of zero-free strips for certain Dirichlet L-functions.
However, in this talk we shall see that the question regarding asymptotics
for Φ(N ) can be reduced to a problem studied (and solved!) much earlier
by Hooley (1958) in a special case and, quite recently, in full generality by
Bykovskiı̆ and Ustinov (2019). The reduction itself is short and completely
elementary.
(60) Ognian Trifonov, University of South Carolina
Email: trifonov@math.sc.edu
Title: Representing positive integers as a sum of a squarefree number anda small prime
Abstract: We prove that every positive integer n which is not equal to 1, 2,
3, 6, 11, 30, 155, or 247 can √be represented as a sum of a squarefree number
and a prime not exceeding n.
Joint work with Jack Dalton.
(61) Tim Trudgian, UNSW Canberra at the Australian Defence Force Academy
Email: t.trudgian@adfa.edu.au
Title: A hair’s breadth of a half
Abstract: The now-classical method of Montgomery and Odlyzko showsthat there are infinitely many pairs of zeroes of the zeta-function closer
than the average spacing. It has been known for almost 40 years that thismethod cannot show an infinitude of gaps less than half of the average
spacing. But could it show infinitely many gaps of exactly half the average
spacing? This is connected to the question of existence of Siegel zeroes. I
shall outline the history of this problem, and show that even getting very
close to one half is not possible by this method.
Joint work with Dan Goldston (San Jose State University) and Caroline
Turnage-Butterbaugh (Carleton College).
(62) Daniel Tsai, National Taiwan University,Taiwan (R.O.C.)
Email: tsaidaniel@ntu.edu.tw
Abstract: We define an additive function v : N → Z by setting v(p) = p
for primes p and v(pα ) = p + α for prime powers pα with α ≥ 2. Similarfunctions such as those from A008474, A001414, and A000026 in the OEIS
have been studied. Denote by rb (n) the integer formed by reversing thebase b digits of an integer n ≥ 1; we write r(n) for r10 (n). For example, if
n = 18, then r(18) = 81 but r2 (18) = r2 (100102 ) = 10012 = 9. Spiegelhofer
has obtained a family of pairs (f, b), where f : N → C and b ≥ 2, such thatf (n) = f (rb (n)) for all n ≥ 1.
Considering the equality v(n) = v(r(n)), we say an integer n ≥ 1 isa v-palindrome if 10 - n, n 6= r(n), and v(n) = v(r(n)); this generalizes
decimal palindrome, integers n where n = r(n). Tsai proved the following

sequences are v-palindromes:
18, 198, 1998, 19998, 199998, . . . ,18, 1818, 181818, 18181818, 1818181818, . . . .
The sequence of v-palindromes is A338039 in the OEIS.
It was conjectured that no prime v-palindromes exist. We prove thatprime v-palindromes are precisely the primes of the form 5 · 10m − 1 suchthat 5 · 10m − 3 is also prime, and thus standard heuristics suggests that
there are only finitely many prime v-palindromes.
Joint work with Muhammet Boran, Steven J. Miller, Jesse Purice, andGaram Choi.
(63) Maciej Ulas, Jagiellonian University, Kraków, Poland
Email: maciel.ulas@gmail.com
Title: Geometric progressions in the sets of values of rational functions
Abstract: Let a, Q ∈ Q be given and consider the set G(a, Q) = {aQi : i ∈N} of terms of geometric progression with 0th term equal to a and the quotient Q. Let f ∈ Q(x, y) and Vf be the set of finite values of f . We consider
the problem of existence of a, Q ∈ Q such that G(a, Q) ⊂ Vf . In the firstpart of the talk we describe several classes of rational function for which our(x,y)
problem has a positive solution. In particular, if f (x, y) = ff21 (x,y), wheref1 , f2 ∈ Z[x, y] are homogenous forms of degrees d1 , d2 and |d1 − d2 | = 1,
we prove that G(a, Q) ⊂ Vf if and only if there are u, v ∈ Q such thata = f (u, v). In the second, experimental, part of the talk we study thestated problem for the rational function f (x, y) = (y 2 − x3 )/x. We relate
the problem to the existence of rational points on certain elliptic curves andpresent interesting numerical observations which allow us to state several
questions and conjectures.
(64) Aled Walker, King’s College London, UK
Email: aled.walker@kcl.ac.uk
Title: On a Bohr set analogue of Chowla’s conjecture
Abstract: Let λ denote the Liouville function. In 1965, Chowla made aconjecture on the natural average of the sequence λ(n)λ(n + 1) (and moregenerally the average of λ(f (n)) for polynomial f ). From this the subject
has recently expanded, and there are now a suite of conjectures on the natural averages (and logarithmic averages) of sequences λ(a1 (n)) · · · λ(ak (n)),
where a1 (n), . . . , ak (n) are certain arithmetic sequences. In this talk we will
discuss recent joint work with Joni Teräväinen, in which we established thatthe logarithmic average of λ(bα1 nc)λ(bα2 nc) is 0 whenever α1 , α2 are positive reals with α1 /α2 irrational. We will also prove non-trivial cancellationon the logarithmic average of λ(bα1 nc) · · · λ(bαk nc) when k > 3. These
results answer the two-point case of a conjecture of Frantzikinakis, andprovide some progress on the higher order cases.

(65) Trevor Wooley, Purdue University
Email: twooley@purdue.edu
Title: Waring’s problem and Freı̆man’s theorem
Abstract: Freı̆man proved that when (ki ) is an increasing sequence of positive integers, then for each j, there exists s = s(j) having the property that
all large integers n are represented as a sum of positive integral ki -th powers(with i ∈ {j, j + 1, . . . , s}) if and only if the series 1/k1 + 1/k2 + . . . diverges.
We describe recent work joint with Jörg Brüdern making Freı̆man’s theorem effective. Some concrete examples will be described.
(66) Kiseok Yeon, Purdue University
Email: kyeon@purdue.edu
Title: The Hasse principle for homogeneous polynomials with random coefficients over thin sets
Abstract: In this talk, we introduce a framework via the circle method inorder to confirm the Hasse principle for random projective hypersurfaces
over Q. We first give a motivation for developing this framework by providing an overall history of the problems of confirming the Hasse principle
for projective hypersurfaces over Q. Next, we provide a sketch of the proofof our main result and show a part of the estimates used in the proof.
Furthermore, we introduce an auxiliary mean value theorem which plays acrucial role in our argument and may be of independent interest.