Mathematics › Algebra and Number Theory

Analytic Number Theory Research

Works Count: 35425

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This cluster of papers focuses on advanced research in prime number theory, L-functions, and their connections to topics such as the Riemann Zeta function, random matrix theory, arithmetic progressions, Vinogradov's mean value theorem, modular forms, and the distribution of primes. It also delves into the exploration of Euler's constant and its implications in number theory.

Keywords

Primes; L-Functions; Riemann Zeta Function; Random Matrix Theory; Arithmetic Progressions; Vinogradov's Mean Value Theorem; Modular Forms; Number Theory; Distribution of Primes; Euler's Constant

ON THE REPRESENTATION OF A LARGER EVEN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF AT MOST TWO PRIMES

1973-02-20

Chen Jing-Run

In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is … In this paper we shall prove that every sufficiently large even integer is a sum of a prime and a product of at most 2 primes. The method used is simple without any complicated numerical calculations.

Approximate formulas for some functions of prime numbers

1962-03-01

J. Barkley Rosser , Lowell Schoenfeld

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Elementary Methods in Number Theory

2000-01-01

Melvyn B. Nathanson

The best constants in the Khintchine inequality

1981-01-01

Uffe Haagerup

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The Distribution of Prime Numbers

1933-11-01

A. E. Ingham

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Modular Functions and Dirichlet Series in Number Theory

1990-01-01

Tom M. Apostol

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Limit Theorems for the Riemann Zeta-Function

1996-01-01

Antanas Laurinčikas

On the Functional Equations Satisfied by Eisenstein Series

1976-01-01

R. P. Langlands

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Random Matrix Theory and L-Functions at s = 1/2

2000-10-01

Jonathan P. Keating , N. C. Snaith

On a Problem of Sidon in Additive Number Theory, and on some Related Problems

1941-10-01

P. Erdős , P. Túrán

�ber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet scher Reihen durch Funktionalgleichungen

1949-12-01

Hans Maaß

Riemannian Coverings and Isospectral Manifolds

1985-01-01

Toshikazu Sunada

Exponential Sums and Lattice Points III

2003-10-23

M. N. Huxley

The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a … The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri–Iwaniec–Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri–Iwaniec–Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate O(RK (log R)Λ) in terms of the maximum radius of curvature R, we reduce K from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes K/2 =131/416. 2000 Mathematics Subject Classification 11P21, 11L07.

A Hecke Algebra of (Z/rZ)Sn and Construction of Its Irreducible Representations

1994-07-01

S. Ariki , Kazuhiko Koike

Hecke operators on ?0(m)

1970-06-01

A. O. L. Atkin , Joseph Lehner

Cryptographic Hash Functions from Expander Graphs

2007-09-14

Denis Charles , Kristin Lauter , Eyal Z. Goren

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Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis

1994-10-11

Hugh L. Montgomery

Uniform distribution van der Corput sets Exponential sums I: The methods of Weyl and van der Corput Exponential sums II: Vinogradov's method An introduction to Turan's method Irregularities of distribution … Uniform distribution van der Corput sets Exponential sums I: The methods of Weyl and van der Corput Exponential sums II: Vinogradov's method An introduction to Turan's method Irregularities of distribution Mean and large values of Dirichlet polynomials Distribution of reduced residue classes in short intervals Zeros of $L$-functions Small polynomials with integral coefficients Some unsolved problems Index.

Topics in Analytic Number Theory

1973-01-01

Hans Rademacher

On Certain L-Functions

1981-04-01

Freydoon Shahidi

Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes

1916-01-01

G. H. Hardy , J. E. Littlewood

solution of I~ ' The idea which dominates the critical stage of the argument is also LANvxu's, but is to be found in another of his papers ('(~ber die Anzahl … solution of I~ ' The idea which dominates the critical stage of the argument is also LANvxu's, but is to be found in another of his papers ('(~ber die Anzahl der Gitterpunkte in gewissen Bereichen',

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Linear equations in primes

2010-04-25

Ben Green , Terence Tao

Consider a system ‰ of nonconstant affine-linear forms 1 ; : : : ; t W ‫ޚ‬ d !‫,ޚ‬ no two of which are linearly dependent.Let N be a large … Consider a system ‰ of nonconstant affine-linear forms 1 ; : : : ; t W ‫ޚ‬ d !‫,ޚ‬ no two of which are linearly dependent.Let N be a large integer, and let K Â OE N; N d be convex.A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N ! 1, for the number of integer points n 2 ‫ޚ‬ d \ K for which the integers 1 .n/;: : : ; t .n/are simultaneously prime.This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture.It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime.In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms 1 ; : : : ; t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions.Our result assumes two families of conjectures, which we term the inverse Gowersnorm conjecture (GI.s/) and the Möbius and nilsequences conjecture (MN.s/),where s 2 f1; 2; : : : g is the complexity of the system and measures the extent to which the forms i depend on each other.The case s D 0 is somewhat degenerate, and follows from the prime number theorem in APs.Roughly speaking, the inverse Gowers-norm conjecture GI.s/ asserts the Gowers U sC1 -norm of a function f W OEN !OE 1; 1 is large if and only if f correlates with an s-step nilsequence, while the Möbius and nilsequences conjecture MN.s/ asserts that the Möbius function is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity.These conjectures have long been known to be true for s D 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s D 2 in two papers of the authors.Thus our results in the case of complexity s 6 2 are unconditional.In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 6 N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

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Zeroes of zeta functions and symmetry

1999-01-01

Nicholas Katz , Peter Sarnak

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, … Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence is quite convincing. Firstly, there are the “function field” analogues, that is zeta functions of curves over finite fields and their generalizations. For these a spectral interpretation for their zeroes exists in terms of eigenvalues of Frobenius on cohomology. Secondly, the developments, both theoretical and numerical, on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection. Moreover, the low-lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups. In this paper we review these developments. In order to present the material fluently, we do not proceed in chronological order of discovery. Also, in concentrating entirely on the subject matter of the title, we are ignoring the standard body of important work that has been done on the zeta function and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>-functions.

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An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers

1998-08-31

E M Matveev

In this paper we study linear forms with rational integer coefficients (, ), where the are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an … In this paper we study linear forms with rational integer coefficients (, ), where the are algebraic numbers satisfying the so-called strong independence condition. In standard notation, we prove an explicit estimate of the form

An approximation of partial sums of independent RV's, and the sample DF. II

1976-01-01

J. Koml�s , Péter Major , G. Tusn�dy

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Ein Satz �ber Untermengen einer endlichen Menge

1928-12-01

Emanuel Sperner

The Difference Between Consecutive Primes, II

2001-11-01

Roger C. Baker , G. Harman , J. Pintz

The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the … The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995) on a mean value containing the fourth power of the zeta function. 2000 Mathematical Subject Classification: 11N05.

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On the normal concentration of divisors, 2

2009-07-06

Helmut Maier , Gérald Tenenbaum

Abstract We improve the current upper and lower bounds for the normal order of the Erdős–Hooley Δ–function obtaining, for almost all integers n , the inequalities where the exponent γ … Abstract We improve the current upper and lower bounds for the normal order of the Erdős–Hooley Δ–function obtaining, for almost all integers n , the inequalities where the exponent γ := (log 2)/log((1−1/log 27)/(1 − 1/log 3)) ≈ 0.33827 is conjectured to be optimal.

Integral moments of L-functions

2005-06-22

J. Brian Conrey , D. W. Farmer , Jonathan P. Keating +2 more

We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading … We give a new heuristic for all of the main terms in the integral moments of various families of primitive L-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical L-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures. 2000 Mathematics Subject Classification 11M26, 15A52.

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Linear Recurring Sequences

1959-03-01

Neal Zierler

Previous article Next article Linear Recurring SequencesNeal ZierlerNeal Zierlerhttps://doi.org/10.1137/0107003PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. A. Albert, Fundamental Concepts of Higher Algebra, Univ. of Chicago Press, 1956 0073.00802 Google … Previous article Next article Linear Recurring SequencesNeal ZierlerNeal Zierlerhttps://doi.org/10.1137/0107003PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] A. A. Albert, Fundamental Concepts of Higher Algebra, Univ. of Chicago Press, 1956 0073.00802 Google Scholar[2] J. L. Brenner, Linear recurrence relations, Amer. Math. Monthly, 61 (1954), 171–173 MR0059934 0055.03508 CrossrefGoogle Scholar[3] W. H. Bussey, Galois field tables for $p^{n}\leqq{169}$, Bull. Amer. Math. Soc., 12 (1905), 22–38 CrossrefGoogle Scholar[4] W. H. Bussey, Galois field tables of order less than 1000, Bull. Amer. Math. Soc., 16 (1909), 188–206 CrossrefGoogle Scholar[5] R. D. Carmichael, A Simple Principle of Unification in the Elementary Theory of Numbers, Amer. Math. Monthly, 36 (1929), 132–143 MR1521682 CrossrefGoogle Scholar[6] Claude Chevalley, Introduction to the Theory of Algebraic Functions of One Variable, Mathematical Surveys, No. VI, American Mathematical Society, New York, N. Y., 1951xi+188 MR0042164 0045.32301 CrossrefGoogle Scholar[7] Randolph Church, Tables of irreducible polynomials for the first four prime moduli, Ann. of Math. (2), 36 (1935), 198–209 MR1503219 0011.00501 CrossrefGoogle Scholar[8] G. E. Forsythe, , H. H. Germond and , A. S. Householder, Monte Carlo Methods, Nat. Bur. Standards. Appl. Math. Ser., 12, 1951 Google Scholar[9] Marshall Hall, An isomorphism between linear recurring sequences and algebraic rings, Trans. Amer. Math. Soc., 44 (1938), 196–218 MR1501967 0019.19301 CrossrefGoogle Scholar[10] C. Jordan, The Calculus of finite differences, Chelsea, New York, 1950 0041.05401 Google Scholar[11] R. Lerner, Signals having uniform ambiguity functions, I.R.E. Convention Record, Information Theory Section, New York, 1958, March Google Scholar[12] Oystein Ore, Contributions to the theory of finite fields, Trans. Amer. Math. Soc., 36 (1934), 243–274 MR1501740 0009.10003 CrossrefGoogle Scholar[13] R. Price and , P. E. Green, Jr., An anti-multipath communication system, Proc. I.R.E., to appear Google Scholar[14] Raphael M. Robinson, Mersenne and Fermat numbers, Proc. Amer. Math. Soc., 5 (1954), 842–846 MR0064787 0058.27504 CrossrefISIGoogle Scholar[15] W. McC. Siebert, A radar detection philosophy, Trans. I.R.E. (Information Theory), IT-2 (1956), 204–221, Sept. 10.1109/TIT.1956.1056805 CrossrefISIGoogle Scholar[16] B. L. van der Waerden, Modern Algebra. Vol. I, Frederick Ungar Publishing Co., New York, N. Y., 1949xii+264, and 1953 MR0029363 0039.00902 Google Scholar[17] Morgan Ward, The arithmetical theory of linear recurring series, Trans. Amer. Math. Soc., 35 (1933), 600–628 MR1501705 0007.24901 CrossrefGoogle Scholar[18] Edwin Weiss and , Neal Zierler, Locally compact division rings, Pacific J. Math., 8 (1958), 369–371 MR0121432 0087.03101 CrossrefGoogle Scholar[19] Neal Zierler, A decomposition theorem for the integers modulo q, Amer. Math. Monthly, 65 (1958), 31–32 MR0098084 0103.27204 CrossrefGoogle Scholar[20] Neal Zierler, On the theorem of Gleason and Marsh, Proc. Amer. Math. Soc., 9 (1958), 236–237 MR0094332 0090.24202 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails The Fast m-Transform: A Fast Computation of Cross-Correlations with Binary m-Sequences13 July 2006 | SIAM Journal on Computing, Vol. 20, No. 4AbstractPDF (798 KB)Doubly-Periodic Sequences and Two-Dimensional RecurrencesSteven Homer and Jerry Goldman2 August 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 6, No. 3AbstractPDF (1147 KB)Factoring Polynomials over a Finite FieldMichael Willett12 July 2006 | SIAM Journal on Applied Mathematics, Vol. 35, No. 2AbstractPDF (527 KB)The Index of an M-SequenceMichael Willett28 July 2006 | SIAM Journal on Applied Mathematics, Vol. 25, No. 1AbstractPDF (328 KB)On the Distribution of the Coefficients of Some PolynomialsNazmi M. 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Solomon10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 2AbstractPDF (430 KB) Volume 7, Issue 1| 1959Journal of the Society for Industrial and Applied Mathematics History Submitted:19 December 1957Published online:10 July 2006 InformationCopyright © 1959 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0107003Article page range:pp. 31-48ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

Low lying zeros of families of L-functions

2000-12-01

Henryk Iwaniec , Wenzhi Luo , Peter Sarnak

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Bounded gaps between primes

2014-02-20

Zhang Yitang

It is proved that lim inf n→∞ (pn+1 -pn) < 7 × 10 7 , where pn is the n-th prime.Our method is a refinement of the recent work of … It is proved that lim inf n→∞ (pn+1 -pn) < 7 × 10 7 , where pn is the n-th prime.Our method is a refinement of the recent work of Goldston, Pintz and Yıldırım on the small gaps between consecutive primes.A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose.* l( mod q) e q (la).We adopt the following conventions throughout our presentation.The set H given by (1.3) is assumed to be admissible and fixed.We write ν p for ν p (H); similar abbreviations will be used in the sequel.Every quantity depending on H alone is regarded as a constant.For example, the absolutely convergent product

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The development of the number field sieve

1993-01-01

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Spectral Methods of Automorphic Forms

2002-11-19

Henryk Iwaniec

Introduction Harmonic analysis on the Euclidean plane Harmonic analysis on the hyperbolic plane Fuchsian groups Automorphic forms The spectral theorem. Discrete part The automorphic Green function Analytic continuation of the … Introduction Harmonic analysis on the Euclidean plane Harmonic analysis on the hyperbolic plane Fuchsian groups Automorphic forms The spectral theorem. Discrete part The automorphic Green function Analytic continuation of the Eisenstein series The spectral theorem. Continuous part Estimates for the Fourier coefficients of Maass forms Spectral theory of Kloosterman sums The trace formula The distribution of eigenvalues Hyperbolic lattice-point problems Spectral bounds for cusp forms Classical analysis Special functions References Subject index Notation index.

On the Diophantine Equation n(n + d) · · · (n + (k − 1)d) = byl

2004-09-01

Kálmán Győry , Lajos Hajdu , N. Saradha

Abstract We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the … Abstract We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k ≥ 3, l ≥ 2 are fixed and k + l > 6.

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Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors

1913-10-01

L. E. Dickson

There are Infinitely Many Carmichael Numbers

1994-05-01

W. R. Alford , Andrew Granville , Carl Pomerance

A New Proof of Szemer�di's Theorem for Arithmetic Progressions of Length Four

1998-07-01

W. T. Gowers

Some extremal functions in Fourier analysis

1985-01-01

Jeffrey D. Vaaler

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Topics in Multiplicative Number Theory

1971-01-01

Hugh L. Montgomery

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An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II

2000-12-31

E M Matveev

On the estimation of Fourier coefficients of modular forms

1965-01-01

Atle Selberg

The Arithmetic of Elliptic Curves

2009-01-01

Joseph H. Silverman

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Modular Functions and Dirichlet Series in Number Theory

1976-01-01

Tom M. Apostol

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Multi-parameter families of exponential sums

2025-06-24

| Princeton University Press eBooks

Unipotent mixing for general moduli

2025-06-24

Edgar Assing | Forum Mathematicum

Abstract In this note we prove a joint equidistribution result for discrete low lying horocycles. This generalizes previous work of Blomer and Michel, where it was crucially assumed that the … Abstract In this note we prove a joint equidistribution result for discrete low lying horocycles. This generalizes previous work of Blomer and Michel, where it was crucially assumed that the number of discrete points is prime.

One-parameter families of exponential sums

2025-06-24

| Princeton University Press eBooks

None

2025-06-23

Yuchen Ding , Wenguang Zhai , Lilu Zhao | Journal de Théorie des Nombres de Bordeaux

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> be the number of primes not exceeding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math>. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>c</mml:mi><mml:mo><</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math> be two relatively prime integers and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>g</mml:mi> <mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow> </mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mi>d</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math>. We confirm, by combining the … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>π</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> be the number of primes not exceeding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>t</mml:mi></mml:math>. Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo><</mml:mo><mml:mi>c</mml:mi><mml:mo><</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math> be two relatively prime integers and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>g</mml:mi> <mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow> </mml:msub><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mi>d</mml:mi><mml:mo>-</mml:mo><mml:mi>c</mml:mi><mml:mo>-</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:math>. We confirm, by combining the Hardy–Littlewood method with the Siegel–Walfisz theorem, a 2020 conjecture of Ramírez Alfonsín and Skałba which states that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mo>#</mml:mo><mml:mfenced close="}" open="{" separators=""><mml:mi>p</mml:mi><mml:mo>≤</mml:mo><mml:msub><mml:mi>g</mml:mi> <mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow> </mml:msub><mml:mo>:</mml:mo><mml:mi>p</mml:mi><mml:mo>∈</mml:mo><mml:mi>𝒫</mml:mi><mml:mo>,</mml:mo><mml:mspace width="3.33333pt"/><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>c</mml:mi><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:mi>y</mml:mi><mml:mo>,</mml:mo><mml:mspace width="3.33333pt"/><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>ℤ</mml:mi> <mml:mrow><mml:mo>⩾</mml:mo><mml:mn>0</mml:mn></mml:mrow> </mml:msub></mml:mfenced><mml:mo>∼</mml:mo><mml:mfrac><mml:mn>1</mml:mn> <mml:mn>2</mml:mn></mml:mfrac><mml:mi>π</mml:mi><mml:mfenced close=")" open="(" separators=""><mml:msub><mml:mi>g</mml:mi> <mml:mrow><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow> </mml:msub></mml:mfenced></mml:mrow></mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>c</mml:mi><mml:mo>→</mml:mo><mml:mi>∞</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>𝒫</mml:mi></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℤ</mml:mi> <mml:mrow><mml:mo>⩾</mml:mo><mml:mn>0</mml:mn></mml:mrow> </mml:msub></mml:math> denote the sets of primes and nonnegative integers, respectively.

None

2025-06-23

Kristian Holm | Journal de Théorie des Nombres de Bordeaux

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝕂</mml:mi><mml:mo>=</mml:mo><mml:mi>ℚ</mml:mi><mml:mo>(</mml:mo><mml:msqrt><mml:mrow><mml:mo>-</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msqrt><mml:mo>)</mml:mo></mml:mrow></mml:math> be an imaginary quadratic number field of class number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒪</mml:mi> <mml:mi>𝕂</mml:mi> </mml:msub></mml:math> its ring of integers. We study a family of Hecke <mml:math … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝕂</mml:mi><mml:mo>=</mml:mo><mml:mi>ℚ</mml:mi><mml:mo>(</mml:mo><mml:msqrt><mml:mrow><mml:mo>-</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msqrt><mml:mo>)</mml:mo></mml:mrow></mml:math> be an imaginary quadratic number field of class number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒪</mml:mi> <mml:mi>𝕂</mml:mi> </mml:msub></mml:math> its ring of integers. We study a family of Hecke <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-functions associated to angular characters on the non-zero ideals of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>𝒪</mml:mi> <mml:mi>𝕂</mml:mi> </mml:msub></mml:math>. Using the powerful Ratios Conjecture (RC) due to Conrey, Farmer, and Zirnbauer, we compute a conditional asymptotic for the average <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>-level density of the zeros of this family, including terms of lower order than the main term in the Katz–Sarnak Density Conjecture coming from random matrix theory. We also prove an unconditional result about the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math>-level density, which agrees with the RC prediction when our test functions have Fourier transforms with support in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math>.

None

2025-06-23

Claire Frechette , Mathilde Gerbelli-Gauthier , Alia Hamieh +1 more | Journal de Théorie des Nombres de Bordeaux

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> be a fixed positive integer, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>S</mml:mi> <mml:mi>k</mml:mi> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> be a primitive cusp form given by the Fourier expansion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo> <mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup><mml:msub><mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> … Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math> be a fixed positive integer, and let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>S</mml:mi> <mml:mi>k</mml:mi> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> be a primitive cusp form given by the Fourier expansion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo> <mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow> <mml:mi>∞</mml:mi> </mml:msubsup><mml:msub><mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>n</mml:mi> <mml:mfrac><mml:mrow><mml:mi>k</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow> <mml:mn>2</mml:mn></mml:mfrac> </mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>. We consider the partial sum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mo>∑</mml:mo> <mml:mrow><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mi>x</mml:mi></mml:mrow> </mml:msub><mml:msub><mml:mi>λ</mml:mi> <mml:mi>f</mml:mi> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>. It is conjectured that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mi>o</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo form="prefix">log</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>≥</mml:mo><mml:msup><mml:mi>k</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msup></mml:mrow></mml:math>. Lamzouri proved in [8] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>ϵ</mml:mi><mml:mo>></mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo form="prefix">log</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mn>1</mml:mn> <mml:mn>8</mml:mn></mml:mfrac></mml:mrow> </mml:msup></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn><mml:mo>≤</mml:mo><mml:mi>T</mml:mi><mml:mo>≤</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mo form="prefix">log</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow> <mml:mfrac><mml:mn>1</mml:mn> <mml:mn>200</mml:mn></mml:mfrac> </mml:msup></mml:mrow></mml:math>, we have <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≪</mml:mo><mml:mfrac><mml:mrow><mml:mi>x</mml:mi><mml:mo form="prefix">log</mml:mo><mml:mi>x</mml:mi></mml:mrow> <mml:mi>T</mml:mi></mml:mfrac></mml:mrow></mml:math> in the range <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>≥</mml:mo><mml:msup><mml:mi>k</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msup></mml:mrow></mml:math> provided that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> has no more than <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>ϵ</mml:mi> <mml:mn>2</mml:mn> </mml:msup><mml:mo form="prefix">log</mml:mo><mml:mi>k</mml:mi><mml:mo>/</mml:mo><mml:mn>5000</mml:mn></mml:mrow></mml:math> zeros in the region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfenced close="}" open="{" separators=""><mml:mi>s</mml:mi><mml:mspace width="0.166667em"/><mml:mo>:</mml:mo><mml:mspace width="0.166667em"/><mml:mi>ℜ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>≥</mml:mo><mml:mfrac><mml:mn>3</mml:mn> <mml:mn>4</mml:mn></mml:mfrac><mml:mo>,</mml:mo><mml:mspace width="0.166667em"/><mml:mrow><mml:mo>|</mml:mo><mml:mi>ℑ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mo>≤</mml:mo><mml:mfrac><mml:mn>1</mml:mn> <mml:mn>4</mml:mn></mml:mfrac></mml:mfenced></mml:math> for every real number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ϕ</mml:mi></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>|</mml:mo><mml:mo>≤</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:math>.

Оценки тригонометрических сумм по решениям квадратичных сравнений

2025-06-22

Ирина Юрьевна Реброва | Чебышевский сборник

В 1963 году, опираясь на оценки специальных тригонометрических сумм, Хули впервые доказал асимптотическую формулу для среднего числа делителей квадратичного полинома со степенным понижением в остаточном члене по сравнению с главным. … В 1963 году, опираясь на оценки специальных тригонометрических сумм, Хули впервые доказал асимптотическую формулу для среднего числа делителей квадратичного полинома со степенным понижением в остаточном члене по сравнению с главным. Позднее эти оценки были улучшены. В работе доказываются новые, более сильные результаты в этом направлении исследований аналитической теории чисел.

New insights into Glasser–Manna integrals and the harmonic zeta function

2025-06-22

Marc-Antoine Coppo , Bernard Candelpergher | The Journal of Analysis

Non-linear additive twists of GL(2) ×GL(2) cusp-form coefficients over primes

2025-06-21

Shu Luo , Guangshi Lü | International Journal of Number Theory

Let [Formula: see text] and [Formula: see text] be holomorphic or Maass cusp forms for [Formula: see text] with normalized Fourier coefficients [Formula: see text] and [Formula: see text], respectively. … Let [Formula: see text] and [Formula: see text] be holomorphic or Maass cusp forms for [Formula: see text] with normalized Fourier coefficients [Formula: see text] and [Formula: see text], respectively. In this paper, we study the non-linear additive twisted sum of [Formula: see text] forms [Formula: see text] where [Formula: see text]. Previously, [Formula: see text] and [Formula: see text] cases have been investigated extensively.

ON ESWARATHASAN–LEVINE AND BOYD’S CONJECTURES FOR HARMONIC NUMBERS

2025-06-20

Leonardo Carofiglio , Giacomo Cherubini , Alessandro Gambini | Bulletin of the Australian Mathematical Society

Abstract We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number … Abstract We provide numerical evidence towards three conjectures on harmonic numbers by Eswarathasan, Levine and Boyd. Let $J_p$ denote the set of integers $n\geq 1$ such that the harmonic number $H_n$ is divisible by a prime p . The conjectures state that: (i) $J_p$ is always finite and of the order $O(p^2(\log \log p)^{2+\epsilon })$ ; (ii) the set of primes for which $J_p$ is minimal (called harmonic primes) has density $e^{-1}$ among all primes; (iii) no harmonic number is divisible by $p^4$ . We prove parts (i) and (iii) for all $p\leq 16843$ with at most one exception, and enumerate harmonic primes up to $50\times 10^5$ , finding a proportion close to the expected density. Our work extends previous computations by Boyd by a factor of approximately $30$ and $50$ , respectively.

Irrationality and transcendence questions in the ‘poor man’s adèle ring’

2025-06-18

Florian Luca , Wadim Zudilin | The Ramanujan Journal

On the average number of cycles in conjugacy class products

2025-06-18

Jesse Campion Loth , Amarpreet Rattan | Advances in Applied Mathematics

The Schinzel-Szekeres function

2025-06-17

Andreas Weingartner | Research in Number Theory

Stability under scaling in the local phases of multiplicative functions

2025-06-16

Miguel N. Walsh | Inventiones mathematicae

On the mean values of the error terms in Mertens’ theorems

2025-06-16

Tiebin Zhao | Research in Number Theory

Upper bounds for L-functions in positive characteristic

2025-06-13

S.P. Liu , Jia-Yan Yao | Finite Fields and Their Applications

Arithmetic constants for symplectic variances of the divisor function

2025-06-13

Vivian Kuperberg , Matilde Laĺın | Mathematika

Abstract Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function over number fields, which were inspired by analogous results over function fields proven … Abstract Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function 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.

Extending the unconditional support in an Iwaniec–Luo–Sarnak family

2025-06-12

Lucile Devin , Daniel Fiorilli , Anders Södergren | Algebra & Number Theory

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On the Euler-Stieltjes constants for functions from the generalized Selberg class

2025-06-10

Almasa Odžak , Medina Zubača | Glasnik Matematicki

The class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ is a very broad class of $L$ functions that contains the Selberg class, the class of all automorphic $L$ functions and the Rankin–Selberg $L$ … The class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ is a very broad class of $L$ functions that contains the Selberg class, the class of all automorphic $L$ functions and the Rankin–Selberg $L$ functions, as well as products of suitable shifts of those functions. In this paper, we consider generalized Euler-Stieltjes constants $\gamma_n(F)$ attached to functions $F(s)$ from the class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$. These are coefficients in Laurent series expansion of function $F(s)$ at its pole. We derive an integral representation and an upper bound for these constants. The application of the obtained results in the case of product of suitable shifts of the Riemann zeta function is presented.

A note on zero-density approaches for the difference between consecutive primes

2025-06-04

Valeriia V. Starichkova | Journal of Number Theory

The asymptotics for the shifted convolutions of a class of multiplicative functions

2025-06-04

C. P. Huang , Guangshi Lü | Journal of Number Theory

Mean values of long Dirichlet polynomials with divisor coefficients

2025-06-03

Fatma Çi̇çek , Alia Hamieh , Nathan Ng | Algebra & Number Theory

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On the moments of averages of quadratic twists of the Möbius function

2025-06-02

Yuichiro Toma

On Gaussian primes in sparse sets

2025-06-01

Jori Merikoski | Compositio Mathematica

Abstract We show that there exists some $\delta > 0$ such that, for any set of integers B with $|B\cap[1,Y]|\gg Y^{1-\delta}$ for all $Y \gg 1$ , there are infinitely … Abstract We show that there exists some $\delta > 0$ such that, for any set of integers B with $|B\cap[1,Y]|\gg Y^{1-\delta}$ for all $Y \gg 1$ , there are infinitely many primes of the form $a^2+b^2$ with $b\in B$ . We prove a quasi-explicit formula for the number of primes of the form $a^2+b^2 \leq X$ with $b \in B$ for any $|B|=X^{1/2-\delta}$ with $\delta < 1/10$ and $B \subseteq [\eta X^{1/2},(1-\eta)X^{1/2}] \cap {\mathbb{Z}}$ , in terms of zeros of Hecke L -functions on ${\mathbb{Q}}(i)$ . We obtain the expected asymptotic formula for the number of such primes provided that the set B does not have a large subset which consists of multiples of a fixed large integer. In particular, we get an asymptotic formula if B is a sparse subset of primes. For an arbitrary B we obtain a lower bound for the number of primes with a weaker range for $\delta$ , by bounding the contribution from potential exceptional characters.

The Exotic Inverted Kloosterman Sum

2025-06-01

Lei Fu , Daqing Wan | International Mathematics Research Notices

Abstract Let $B$ be a product of finitely many finite fields containing $\mathbb{F}_{q}$, $\psi :\mathbb{F}_{q}\to \overline{\mathbb{Q}}_\ell ^{*}$ a nontrivial additive character, and $\chi : B^{*}\to \overline{\mathbb{Q}}_\ell ^{*}$ a multiplicative character. … Abstract Let $B$ be a product of finitely many finite fields containing $\mathbb{F}_{q}$, $\psi :\mathbb{F}_{q}\to \overline{\mathbb{Q}}_\ell ^{*}$ a nontrivial additive character, and $\chi : B^{*}\to \overline{\mathbb{Q}}_\ell ^{*}$ a multiplicative character. Katz introduced the so-called exotic inverted Kloosterman sum $$ \begin{eqnarray*} \mathrm{EIK}(\mathbb{F}_{q}, a):=\sum_{\substack{x\in B^{*} \\ \mathrm{Tr}_{B/\mathbb{F}_{q}}(x)\not =0\\ \mathrm{N}_{B/\mathbb{F}_{q}}(x)=a}} \chi(x)\psi\Big(\frac{1}{\mathrm{Tr}_{B/\mathbb{F}_{q}}(x)}\Big), \ \ a\in \mathbb{F}_{q}^{*}. \end{eqnarray*} $$ We estimate this sum using $\ell $-adic cohomology theory. Our main result is that, up to a trivial term, the associated exotic inverted Kloosterman sheaf is lisse of rank at most $2(n+1)$ and mixed of weight at most $n$, where $n+1 = \dim _{\mathbb{F}_{q}}B$. Up to a trivial main term, this gives the expected square root cancellation.

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Lp$L^p$‐norm bounds for automorphic forms via spectral reciprocity

2025-06-01

Peter Humphries , Rizwanur Khan | Proceedings of the London Mathematical Society

Abstract Let be a Hecke–Maaß cusp form on the modular surface , namely an ‐normalised non‐constant Laplacian eigenfunction on that is additionally a joint eigenfunction of every Hecke operator. We … Abstract Let be a Hecke–Maaß cusp form on the modular surface , namely an ‐normalised non‐constant Laplacian eigenfunction on that is additionally a joint eigenfunction of every Hecke operator. We prove the ‐norm bound , where denotes the Laplacian eigenvalue of , which improves upon Sogge's ‐norm bound for Laplacian eigenfunctions on a compact Riemann surface by more than a six‐fold power‐saving. Interpolating with the sup‐norm bound due to Iwaniec and Sarnak, this yields ‐norm bounds for Hecke–Maaß cusp forms that are power‐saving improvements on Sogge's bounds for all . Our paper marks the first improvement of Sogge's result on the modular surface. Furthermore, these methods yield for compact arithmetic surfaces the best ‐norm bound to date. Via the Watson–Ichino triple product formula, bounds for the ‐norm of are reduced to bounds for certain mixed moments of ‐functions. We bound these using two forms of spectral reciprocity: identities between two different moments of central values of ‐functions. The first is a form of spectral reciprocity, which relates a moment of Rankin–Selberg ‐functions to a moment of Rankin–Selberg ‐functions; this can be seen as a cuspidal analogue of Motohashi's formula relating the fourth moment of the Riemann zeta function to the third moment of central values of Hecke ‐functions. The second is a form of spectral reciprocity, which is a cuspidal analogue of a formula of Kuznetsov for the fourth moment of central values of Hecke ‐functions.