@article{22191, abstract = {Montgomery and Soundararajan showed that the distribution of ψ(x+H)−ψ(x), for 0≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log(N/H), when N δ ≤ H ≤ N 1−δ . Their work depends on showing that sums Rk (h) of k-term singular series are µk (−h log h + Ah) k/2 + Ok (h k/2−1/(7k)+ε ), where A is a constant and µk are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when k is odd, Rk (h) ≍ h (k−1)/2 (log h) (k+1)/2 . We prove an upper bound with the correct power of h when k = 3, and prove analogous upper bounds in the function field setting when k = 3 and k = 5. We provide further evidence for this conjecture in the form of numerical computations.}, author = {Kuperberg, Vivian Zieve}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {4}, pages = {617--666}, publisher = {Mathematical Sciences Publishers}, title = {{Odd moments in the distribution of primes}}, doi = {10.2140/ant.2025.19.617}, volume = {19}, year = {2025}, } @article{22195, abstract = {Sums of the singular series constants that appear in the Hardy–Littlewood k-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo r or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo r of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats. }, author = {Kuperberg, Vivian Zieve}, issn = {1793-7310}, journal = {International Journal of Number Theory}, number = {01}, pages = {53--74}, publisher = {World Scientific Publishing}, title = {{Sums of singular series along arithmetic progressions and with smooth weights}}, doi = {10.1142/s1793042125500046}, volume = {21}, year = {2025}, } @article{22200, abstract = {In Kuperberg and Lal´ın [Forum Math. 34 (2022), pp. 711–747], the authors studied the mean-square of certain sums of the divisor function dk(f) over the function field Fq[T] in the limit as q →∞ and related these sums to integrals over the ensemble of symplectic matrices, along similar lines as previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z. 288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue a more detailed study of both the symplectic and orthogonal matrix integrals, relating them to symmetric function theory. The function field results lead to conjectures concerning analogous questions over number fields.}, author = {Kuperberg, Vivian Zieve and Lalín, Matilde}, issn = {2330-0000}, journal = {Transactions of the American Mathematical Society, Series B}, number = {10}, pages = {323--370}, publisher = {American Mathematical Society}, title = {{Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime}}, doi = {10.1090/btran/186}, volume = {12}, year = {2025}, } @article{22199, abstract = {Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function dk(n) over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.}, author = {Kuperberg, Vivian Zieve and Lalín, Matilde}, issn = {0025-5793}, journal = {Mathematika}, number = {3}, publisher = {Wiley}, title = {{Arithmetic constants for symplectic variances of the divisor function}}, doi = {10.1112/mtk.70029}, volume = {71}, year = {2025}, } @article{22201, abstract = {We consider an analog of a conjecture of Montgomery and Soundararajan on the moments of primes in short intervals in number fields; this analog was discussed and heuristically derived in a paper of the second author, Rodgers, and Roditty-Gershon. Adapting work of the first author and Fiorilli in the integer case, we establish lower bounds on a weighted version of these moments which agree with the conjectured values.}, author = {de la Bretèche, Régis and Kuperberg, Vivian Zieve}, issn = {1565-8511}, journal = {Israel Journal of Mathematics}, number = {1}, pages = {437--461}, publisher = {Springer Nature}, title = {{Lower bounds on weighted moments of primes in short intervals in number fields}}, doi = {10.1007/s11856-024-2711-0}, volume = {267}, year = {2025}, } @article{22203, abstract = {We prove near-optimal upper bounds for the oddmoments of the distribution of coprime residues inshort intervals, confirming a conjecture of Montgomeryand Vaughan. As an application, we prove near-optimalupper bounds for the average of the refined singularseries in the Hardy–Littlewood conjectures concerningthe number of prime 𝑘-tuples for 𝑘 odd. The mainnew ingredient is a near-optimal upper bound for thenumber of solutions to ∑1 ⩽𝑖 ⩽𝑘𝑎 𝑖𝑞𝑖∈ ℤ when 𝑘 is odd,with gcd(𝑎𝑖 , 𝑞𝑖 ) = 1 and restrictions on the size of thenumerators and denominators, which is of indepen-dent interest.}, author = {Bloom, Thomas F. and Kuperberg, Vivian Zieve}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{Odd moments and adding fractions}}, doi = {10.1112/plms.70068}, volume = {131}, year = {2025}, } @article{22204, abstract = {We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes a1 and a2 mod q, and any r1,r2 ≥ 1, a positive density of sums of two squares begin a chain of r1 consecutive sums of two squares, all of which are a1 mod q, followed immediately by a chain of r2 consecutive sums of two squares, all of which are a2 mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any r ≥ 1, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q}, author = {Kimmel, Noam and Kuperberg, Vivian Zieve}, issn = {1475-3030}, journal = {Journal of the Institute of Mathematics of Jussieu}, number = {5}, pages = {1995--2046}, publisher = {Cambridge University Press}, title = {{Positive density for consecutive runs of sums of two squares}}, doi = {10.1017/s1474748025000131}, volume = {24}, year = {2025}, } @article{22190, abstract = {We consider the set of 𝑚×𝑛 matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions.}, author = {Afifurrahman, Muhammad and Kuperberg, Vivian Zieve and Ostafe, Alina and Shparlinski, Igor E.}, issn = {1435-5337}, journal = {Forum Mathematicum}, number = {4}, publisher = {De Gruyter}, title = {{Statistics of ranks, determinants and characteristic polynomials of rational matrices}}, doi = {10.1515/forum-2024-0114}, volume = {37}, year = {2024}, } @article{22197, abstract = {We study the distribution of consecutive sums of two squares in arithmetic progressions. If {En}n∈N is the sequence of sums of two squares in increasing order, we show that for any modulus q and any congruence classes a1, a2, a3 mod q which are admissible in the sense that there are solutions to x2 + y2 ≡ ai mod q, there exist infinitely many n with En+i−1 ≡ ai mod q, for i =1, 2, 3. We also show that for any r1, r2 ≥ 1, there exist infinitely many n with En+i−1 ≡ a1 mod q for 1 ≤ i ≤ r1 and En+i−1 ≡ a2 mod q for r1 +1 ≤ i ≤ r1 + r2}, author = {Kimmel, Noam and Kuperberg, Vivian Zieve}, issn = {0022-314X}, journal = {Journal of Number Theory}, pages = {135--147}, publisher = {Elsevier}, title = {{Consecutive runs of sums of two squares}}, doi = {10.1016/j.jnt.2024.05.003}, volume = {264}, year = {2024}, } @article{22192, abstract = {In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime k-tuples imply that the distribution of primes in log-size intervals is Poissonian. He did so by computing average values of the singular series constants over different sets of a fixed size k contained in an interval [1,h] as h → ∞, and then using this average to compute moments of the distribution of primes. In this paper, we study averages where k is relatively large with respect to h. We then apply these averages to the tail of the distribution. For example, we show, assuming appropriate Hardy–Littlewood conjectures and in certain ranges of the parameters, the number of intervals [n,n + λlogx] with n ≤ x containing at least k primes is ≪ x exp(−k/(λe)).}, author = {Kuperberg, Vivian Zieve}, issn = {1464-3847}, journal = {The Quarterly Journal of Mathematics}, number = {4}, pages = {1457--1479}, publisher = {Oxford University Press}, title = {{Sums of singular series with large sets and the tail of the distribution of primes}}, doi = {10.1093/qmath/haad030}, volume = {74}, year = {2023}, } @article{22193, abstract = {Gross and Smith have put forward generalizations of Hardy–Littlewood twin prime conjectures for algebraic number fields. We estimate the behaviour of sums of a singular series that arises in these conjectures, up to lower-order terms. More exactly, where S(η) is the singular series, we find asymptotic formulas for smoothed sums of S(η)−1. Based upon Gross and Smith’s conjectures, we use our result to suggest that for large enough ‘short intervals’ in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramér model prediction by a universal factor, independent of K. The conjecture over number fields generalizes a classical conjecture of Goldston and Montgomery over the integers. Numerical data are provided supporting the conjecture.}, author = {Kuperberg, Vivian Zieve and Rodgers, Brad and Roditty-Gershon, Edva}, issn = {1572-9303}, journal = {The Ramanujan Journal}, keywords = {Primes, Short intervals, Ramanujan sums, Singular series, Number fields}, number = {2}, pages = {291--317}, publisher = {Springer Nature}, title = {{Sums of singular series and primes in short intervals in algebraic number fields}}, doi = {10.1007/s11139-022-00561-9}, volume = {58}, year = {2022}, } @article{22194, abstract = {In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽𝑞⁢[𝑡] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function 𝑑𝑘⁢(𝑓) over short intervals and over arithmetic progressions for the function field 𝔽𝑞⁢[𝑇] to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function.}, author = {Kuperberg, Vivian Zieve and Lalín, Matilde}, issn = {1435-5337}, journal = {Forum Mathematicum}, number = {3}, pages = {711--747}, publisher = {De Gruyter}, title = {{Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions}}, doi = {10.1515/forum-2021-0171}, volume = {34}, year = {2022}, } @article{22196, abstract = {We explore two questions about pseudo-polynomials, which are functions f : N → Z such that k divides f(n + k) − f(n) for all n, k. First, for certain arbitrarily sparse sets R, we construct pseudo-polynomials f with p|f(n) for some n only if p ∈ R. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan. We also consider α-primary pseudo-polynomials, where the pseudo-polynomial condition is only required for k lying in a set of primes of density α. We show that if an α-primary pseudo-polynomial is O(e(β−)n), where β = √7 3 − 1 6 ≈ 0.715, then it is a polynomial.}, author = {Kuperberg, Vivian Zieve}, issn = {0022-314X}, journal = {Journal of Number Theory}, keywords = {Pseudo-polynomials, Chinese remainder theorem, Ruzsa’s conjecture}, pages = {531--541}, publisher = {Elsevier}, title = {{On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials}}, doi = {10.1016/j.jnt.2022.04.006}, volume = {241}, year = {2022}, } @article{22205, abstract = {A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag–Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways. We construct variations on Higman's group replacing the Baumslag–Solitar group by other groups G. We give an elementary condition on G enjoyed for example by Z≀Z and the integral Heisenberg group, under which the resulting group is sofic. We then use soficity to deduce that there exist permutations of Z/nZ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.}, author = {Kassabov, Martin and Kuperberg, Vivian Zieve and Riley, Timothy R.}, issn = {2415-6302}, journal = {Journal of Combinatorial Algebra}, number = {1}, pages = {41--70}, publisher = {European Mathematical Society}, title = {{Soficity and variations on Higman’s group}}, doi = {10.4171/jca/26}, volume = {3}, year = {2019}, } @article{22206, abstract = {Cools, Draisma, Payne, and Robeva proved that generic metric graphs that are "paths of loops" are Brill-Noether general. We show that Brill-Noether generality does not hold for "trees of loops": the only trees of loops that are Brill-Noether general are paths of loops. We study various notions of generality and examine which of these graphs satisfy them.}, author = {Kailasa, Sameer and Kuperberg, Vivian Zieve and Wawrykow, Nicholas}, issn = {1077-8926}, journal = {The Electronic Journal of Combinatorics}, keywords = {Chip-firing, Metric graphs, Brill-Noether generality}, number = {1}, publisher = {The Electronic Journal of Combinatorics}, title = {{Chip-firing on trees of loops}}, doi = {10.37236/7244}, volume = {25}, year = {2018}, } @article{22198, abstract = {Packings of equal disks in the plane are known to have density at most π/ √ 12, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a square torus that we conjecture to be the most dense for certain numbers of packing disks, using continued fractions to approximate 1/ √ 3 and 2 − √ 3. We also define a constant to measure the efficiency of a packing motived by a related constant due to Markov for continued fractions. One idea is to use the unique factorization property of Gaussian integers to prove that there is an upper bound for the Markov constant for grid-like packings. By way of contrast, we show that an upper bound by Gruber [In many cases optimal configurations are almost regular hexagonal, vol. 65, pp. 121–145, 1999;Geom Dedicata 84(1–3):271–320, 2001] for the error for the limiting density of a packing of equal disks in a planar square, which is on the order of 1/ √ N, is the best possible, whereas for our examples for the square torus, the error for the limiting density is on the order of 1/N, where N is the number of packing disks.}, author = {Connelly, Robert and Funkhouser, Matthew and Kuperberg, Vivian Zieve and Solomonides, Evan}, issn = {1432-0444}, journal = {Discrete & Computational Geometry}, number = {3}, pages = {614--642}, publisher = {Springer Nature}, title = {{Packings of equal disks in a square torus}}, doi = {10.1007/s00454-016-9843-x}, volume = {58}, year = {2017}, } @article{22202, abstract = {We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7-modular and 11-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p-modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p-modular version of the Hadamard conjecture for all but finitely many cases when p ≡ 3(mod 4), and prove a weaker result for p ≡ 1 (mod 4). Finally, we look at constraints on the existence of m-modular Hadamard matrices when the size of the matrix is small compared to m.}, author = {Kuperberg, Vivian Zieve}, issn = {1520-6610}, journal = {Journal of Combinatorial Designs}, keywords = {modular hadamard matrices, modular symmetric designs}, number = {9}, pages = {393--405}, publisher = {Wiley}, title = {{Hadamard matrices modulo p and small modular Hadamard matrices}}, doi = {10.1002/jcd.21522}, volume = {24}, year = {2016}, }