arXiv subjects

Djordje Milićević

Publications and source records attributed to Djordje Milićević.

At least 19 recordsLinked to original sources

arXiv2026

The non-spherical sup-norm problem for $\mathrm{GL}(n)$ and generalized spherical functions

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $π$ of $\mathrm{GL}(n,\mathbb{Z})\backslash\mathrm{GL}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $π$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.

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arXiv2026

Sums of products of Kloosterman sums to prime power moduli

We prove new bounds on complete sums of products of k additively shifted Kloosterman sums to odd high prime power moduli q=p^n, which feature substantially stronger power savings (about q^(-1/ceil(k/2)) in generic configurations) and a novel quantification of the alignment among the shifts. We prove our bounds by developing a method to estimate complete exponential sums with a broad class of phases well-controlled by a specified number of initial terms in a property reflecting p-adic differentiability.

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arXiv2025

Bilinear forms with Kloosterman sums and moments of twisted L-functions

We establish power-saving estimates for general bilinear forms with Kloosterman sums modulo arbitrary q, including when both variables are shorter than the Polya-Vinogradov range. As an application, we obtain power-saving asymptotics for the second moment of (holomorphic or Maass) modular L-functions twisted with Dirichlet characters to an arbitrary large admissible modulus q. The bounds obtained are independent of the Ramanujan-Petersson conjecture and remove all factorability conditions on q in the work of Blomer, Fouvry, Kowalski, Michel, Milicevic, and Sawin.

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arXiv2025

The shape of quadratic Gauss paths

We consider the distribution of quadratic Gauss paths, polygonal paths joining partial sums of quadratic Gauss sums to square-free fundamental discriminant moduli in a dyadic range [Q,2Q]. We prove that this striking ensemble converges in law, as Q->\infty, to a random Fourier series we explicitly describe, and we prove a convergence in probability result and a classification result for the limiting shapes that explain the visually remarkable properties of these Gauss paths.

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arXiv2023

Counting cusp forms by analytic conductor

Let $F$ be a number field and $n\geqslant 1$ an integer. The universal family is the set $\mathfrak{F}$ of all unitary cuspidal automorphic representations on ${\rm GL}_n$ over $F$, ordered by their analytic conductor. We prove an asymptotic for the size of the truncated universal family $\mathfrak{F}(Q)$ as $Q\rightarrow\infty$, under a spherical assumption at the archimedean places when $n\geqslant 3$. We interpret the leading term constant geometrically and conjecturally determine the underlying Sato--Tate measure. Our methods naturally provide uniform Weyl laws with logarithmic savings in the level and strong quantitative bounds on the non-tempered discrete spectrum for ${\rm GL}_n$.

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arXiv2022

Beyond the spherical sup-norm problem

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of G=SL_2(C) with K=SU_2(C). Our results cover the case of vector-valued Maass forms as well as all the individual scalar-valued Maass forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong localization estimates for generalized spherical functions of high K-type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally.

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arXiv2022

Ambient prime geodesic theorems on hyperbolic 3-manifolds

We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.

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arXiv2021

Conductor zeta function for the GL(2) universal family

We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The method proceeds by establishing the requisite analytic properties of the underlying conductor zeta function.

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arXiv2020

The density hypothesis for horizontal families of lattices

We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations are uniform in the volume and spectral aspects.

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arXiv2020

Twelfth moment of Dirichlet L-functions to prime power moduli

We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic method of stationary phase for sums of products and complement Nunes' bound for smooth square-free moduli.

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arXiv2020

Distribution of Kloosterman paths to high prime power moduli

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p^n of a fixed odd prime p, a pure depth-aspect analogue of theorems of Kowalski-Sawin and Ricotta-Royer-Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as n->\infty to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every p converges in law as p->\infty, and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the p-adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments.

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arXiv2020

Nonvanishing of Dirichlet L-functions, II

We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $χ$ of large prime modulus, the central value $L(\frac{1}{2},χ)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.

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arXiv2019

The second moment theory of families of L-functions

For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q, and prove that it satisfies such formulas. We derive arithmetic consequences: - a positive proportion of central values L(f x chi, 1/2) are non-zero, and indeed bounded from below; - there exist many characters chi for which the central L-value is very large; - the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.

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arXiv2018

The sup-norm problem for GL(2) over number fields

We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature a Weyl-type exponent in the level aspect, they reproduce or improve upon all known special cases, and over totally real fields they are as strong as the best known hybrid result over the rationals.

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arXiv2016

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.

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arXiv2016

On moments of twisted $L$-functions

We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ and $g$ is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted $L$-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M.~Young's asymptotic evaluation of the fourth moment of Dirichlet $L$-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.

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arXiv2015

Non-vanishing of Dirichlet L-functions in Galois orbits

A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by intermediate Galois groups as soon as they are larger than roughly the square-root of the prime-power conductor.

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arXiv2015

Bounds for eigenforms on arithmetic hyperbolic 3-manifolds

On a family of arithmetic hyperbolic 3-manifolds of squarefree level, we prove an upper bound for the sup-norm of Hecke-Maass cusp forms, with a power saving over the local geometric bound simultaneously in the Laplacian eigenvalue and the volume. By a novel combination of diophantine and geometric arguments in a noncommutative setting, we obtain bounds as strong as the best corresponding results on arithmetic surfaces.

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