arXiv subjects

Siegfred Baluyot

Publications and source records attributed to Siegfred Baluyot.

8 recordsLinked to original sources

arXiv2025

Twisted moments of characteristic polynomials of random matrices in the unitary group

Recently, Keating and the second author of this paper devised a heuristic for predicting asymptotic formulas for moments of the Riemann zeta-function $\zeta(s)$. Their approach indicates how lower twisted moments of $\zeta(s)$ may be used to evaluate higher moments. In this paper, we present a rigorous random matrix theory analogue of their heuristic. To do this, we develop a notion of "twisted moment" of characteristic polynomials of matrices in the unitary group $U(N)$, and we prove several identities involving Schur polynomials. Our results may be viewed as a proof of concept of the heuristic for $\zeta(s)$.

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arXiv2025

Multiple Dirichlet series predictions for moments of $L$-functions: unitary, symplectic and orthogonal examples

We devise heuristics using multiple Dirichlet series to predict asymptotic formulas for shifted moments of (1) the family of Dirichlet $L$-functions of all even primitive characters of conductor $\leq Q$, with $Q$ a parameter tending to infinity, (2) the family of quadratic Dirichlet $L$-functions, (3) the family of quadratic twists of an $L$-function associated to a fixed Hecke eigencuspform for the full modular group, and (4) the family of quadratic twists of an $L$-function of a fixed arbitrary elliptic curve over $\mathbb{Q}$ that has a non-square conductor. For each of these families, the resulting predictions agree with the predictions of the recipe developed by Conrey, Farmer, Keating, Rubinstein, and Snaith, except for (4), where the recipe requires a slight modification due to a correlation between the Dirichlet coefficients and the root number of the corresponding $L$-functions. We find a one-to-one correspondence between the residues from the multiple Dirichlet series analysis and the terms from the recipe prediction.

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arXiv2023

Low-lying zeros of a large orthogonal family of automorphic $L$-functions

We study a new orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q \asymp Q$. To illustrate our methods, we prove a one level density result for this family with the support of the Fourier transform of the test function being extended to be inside $(-4, 4)$. The main techniques developed in this paper will be useful in developing further results for this family, including estimates for high moments, information on the vertical distribution of zeros, as well as critical line theorems.

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arXiv2023

Sign changes of the error term in the Piltz divisor problem

We study the function $Δ_k(x):=\sum_{n\leq x} d_k(n) - \mbox{Res}_{s=1} ( ζ^k(s) x^s/s )$, where $k\geq 3$ is an integer, $d_k(n)$ is the $k$-fold divisor function, and $ζ(s)$ is the Riemann zeta-function. For a large parameter $X$, we show that if the Lindelöf hypothesis is true, then there exist at least $X^{\frac{1}{k(k-1)}-\varepsilon}$ disjoint subintervals of $[X,2X]$, each of length $X^{1-\frac{1}{k}-\varepsilon}$, such that $|Δ_k(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$ for all $x$ in the subinterval. If the Riemann hypothesis is true, then we can improve the length of the subintervals to $\gg X^{1-\frac{1}{k}} (\log X)^{-k^2-2}$. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case $k=2$, and Cao, Tanigawa, and Zhai, who studied the case $k=3$. The first main ingredient of our proofs is a bound for the second moment of $Δ_k(x+h)-Δ_k(x)$. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of $Δ_k(x)$, which we obtain by combining a method of Tsang with a technique of Lester.

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arXiv2022

Moments of zeta and correlations of divisor-sums: stratification and Vandermonde integrals

We refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta-function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

math.NT↗

arXiv2022

Twisted $2k$th moments of primitive Dirichlet $L$-functions: beyond the diagonal

We study the family of Dirichlet $L$-functions of all even primitive characters of conductor at most $Q$, where $Q$ is a parameter tending to $\infty$. For an arbitrary positive integer $k$, we approximate the twisted $2k$th moment of this family by using Dirichlet polynomial approximations of $L^k(s,χ)$ of length $X$, with $Q<X<Q^2$. Assuming the Generalized Lindelöf Hypothesis, we prove an asymptotic formula for these approximations of the twisted moments. Our result agrees with the prediction of Conrey, Farmer, Keating, Rubinstein, and Snaith for this family of $L$-functions, and provides the first rigorous evidence beyond the diagonal terms for their conjectured asymptotic formula for the general $2k$th moment of this family.

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arXiv2018

Dirichlet $L$-functions of quadratic characters of prime conductor at the central point

We prove that more than nine percent of the central values $L(\frac{1}{2},χ_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $χ_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},χ_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.

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arXiv2018

Small zeros of Dirichlet $L$-functions of quadratic characters of prime modulus

In this paper, we investigate the distribution of the imaginary parts of zeros near the real axis of Dirichlet $L$-functions associated to the quadratic characters $χ_{p}(\cdot)=(\cdot |p)$ with $p$ a prime number. Assuming the Generalized Riemann Hypothesis (GRH), we compute the one-level density for the zeros of this family of $L$-functions under the condition that the Fourier transform of the test function is supported on a closed subinterval of $(-1,1)$. We also write down the ratios conjecture for this family of $L$-functions a la Conrey, Farmer and Zirnbauer and derive a conjecture for the one-level density which is consistent with the main theorem of this paper and with the Katz-Sarnak prediction and includes lower order terms. Following the methods of Özlük and Snyder, we prove that GRH implies $L(\frac{1}{2},χ_p)\neq 0$ for at least $75\%$ of the primes.

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