Monatshefte für Mathematik, ISSN 1436-5081, 2016, Volume 183, Issue 2, pp. 303 - 310

We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function H(k) in the Waring-Goldbach problem. We obtain new results for...

Weyl sums | Vinogradov mean-value theorem | Waring–Goldbach problem | MATHEMATICS | 4TH | Waring-Goldbach problem | MEAN-VALUE THEOREM

Weyl sums | Vinogradov mean-value theorem | Waring–Goldbach problem | MATHEMATICS | 4TH | Waring-Goldbach problem | MEAN-VALUE THEOREM

Journal Article

Mathematika, ISSN 0025-5793, 2016, Volume 62, Issue 2, pp. 348 - 361

Let $E(N)$ denote the number of positive integers $n\leqslant N$ , with $n\equiv 4\;(\text{mod}\;24)$ , which cannot be represented as the sum of four squares...

11N36 | 11P55 (secondary) | 11L20 | 11P05 | 11P32 (primary) | Mathematics - Number Theory

11N36 | 11P55 (secondary) | 11L20 | 11P05 | 11P32 (primary) | Mathematics - Number Theory

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 06/2016, Volume 93, Issue 3, pp. 811 - 824

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain...

MATHEMATICS | ITERATIVE METHOD | IMPROVEMENT | 4TH | ADDITIVE NUMBER-THEORY | MEAN-VALUE THEOREM | Functions (mathematics) | Theorems | Exponents | Mathematical analysis | Estimates | Formulas (mathematics) | Sums | Mathematics - Number Theory

MATHEMATICS | ITERATIVE METHOD | IMPROVEMENT | 4TH | ADDITIVE NUMBER-THEORY | MEAN-VALUE THEOREM | Functions (mathematics) | Theorems | Exponents | Mathematical analysis | Estimates | Formulas (mathematics) | Sums | Mathematics - Number Theory

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 1/2013, Volume 30, Issue 1, pp. 101 - 116

...Ramanujan J (2013) 30:101–116 DOI 10.1007/s11139-012-9381-y On the convergence of some alternating series Angel V . Kumchev Received: 22 June 2011 / Accepted...

Functions of a Complex Variable | Field Theory and Polynomials | Alternating series | Diophantine approximation | Mathematics | 11J70 | Fourier series | 11J82 | Convergence of series | Fourier Analysis | 40A05 | Number Theory | Combinatorics | MATHEMATICS | APPROXIMATIONS

Functions of a Complex Variable | Field Theory and Polynomials | Alternating series | Diophantine approximation | Mathematics | 11J70 | Fourier series | 11J82 | Convergence of series | Fourier Analysis | 40A05 | Number Theory | Combinatorics | MATHEMATICS | APPROXIMATIONS

Journal Article

Michigan Mathematical Journal, ISSN 0026-2285, 2006, Volume 54, Issue 2, pp. 243 - 268

MATHEMATICS | WARING-GOLDBACH PROBLEM | EXCEPTIONAL SETS | CUBES | 3 SQUARES | POWERS | MODULO ONE | 11N36 | 11L15 | 11P55 | 11L20

Journal Article

Proceedings of the American Mathematical Society, ISSN 0002-9939, 10/2005, Volume 133, Issue 10, pp. 2927 - 2937

We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of 46 seventh powers...

Integers | Prime numbers | Mathematical theorems | Real numbers | Generating function | Iterative methods | Number theory | MATHEMATICS | ITERATIVE METHOD | IMPROVEMENT | MATHEMATICS, APPLIED | ADDITIVE NUMBER-THEORY

Integers | Prime numbers | Mathematical theorems | Real numbers | Generating function | Iterative methods | Number theory | MATHEMATICS | ITERATIVE METHOD | IMPROVEMENT | MATHEMATICS, APPLIED | ADDITIVE NUMBER-THEORY

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 03/2009, Volume 5, Issue 2, pp. 281 - 292

Let c be a real number with 1 < c < 2. We study the representations of a large integer n in the form $$ [m^c] + [p^c] = n, $$ where m is an integer and p is a...

Prime numbers | Binary additive problem | Fractional powers | MATHEMATICS | prime numbers | fractional powers

Prime numbers | Binary additive problem | Fractional powers | MATHEMATICS | prime numbers | fractional powers

Journal Article

Canadian journal of mathematics, ISSN 0008-414X, 04/2005, Volume 57, Issue 2, pp. 298 - 327

We investigate exceptional sets in the Waring–Goldbach problem. For example, in the cubic case, we show that all but $O\left( {{N}^{79/84+\in }} \right)$...

MATHEMATICS | INTEGERS | PRIME SQUARES | P MODULO ONE

MATHEMATICS | INTEGERS | PRIME SQUARES | P MODULO ONE

Journal Article

Acta Arithmetica, ISSN 0065-1036, 2006, Volume 123, Issue 2, pp. 125 - 142

Acta Arith. 123 (2006), 125-142 We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use...

MATHEMATICS | Mathematics - Number Theory

MATHEMATICS | Mathematics - Number Theory

Journal Article

The Rocky Mountain Journal of Mathematics, ISSN 0035-7596, 1/2007, Volume 37, Issue 2, pp. 455 - 464

Journal Article

Monatshefte fur Mathematik, ISSN 0026-9255, 2009, Volume 157, Issue 4, pp. 335 - 363

Let H-2 denote the set of even integers n not equivalent to 1 (mod 3). We prove that when H >= X-0.33, almost all integers n is an element of H-2 boolean AND (...

Exceptional sets | Waring-Goldbach problem | Distribution of primes | Exponential sums | Sieve methods | MATHEMATICS | EXCEPTIONAL SET | THEOREM | P MODULO ONE | VARIABLES

Exceptional sets | Waring-Goldbach problem | Distribution of primes | Exponential sums | Sieve methods | MATHEMATICS | EXCEPTIONAL SET | THEOREM | P MODULO ONE | VARIABLES

Journal Article

Archiv der Mathematik, ISSN 0003-889X, 4/2017, Volume 108, Issue 4, pp. 341 - 350

... Archiv der Mathematik The strong symmetric genus spectrum of abelian groups Angel V. Kumchev, Coy L. May, and Jay J. Zimmerman Abstract. Let S denote the set...

Primary 57M60 | Mathematics | 30F99 | Unions of arithmetic progressions | Secondary 11B05 | Riemann surface | 11N25 | Abelian groups | Mathematics, general | 20F38 | 11N37 | Strong symmetric genus | Genus spectrum | Asymptotic density | MATHEMATICS

Primary 57M60 | Mathematics | 30F99 | Unions of arithmetic progressions | Secondary 11B05 | Riemann surface | 11N25 | Abelian groups | Mathematics, general | 20F38 | 11N37 | Strong symmetric genus | Genus spectrum | Asymptotic density | MATHEMATICS

Journal Article

Monatshefte für Mathematik, ISSN 0026-9255, 6/2019, Volume 189, Issue 2, pp. 355 - 376

...Monatshefte für Mathematik (2019) 189:355–376 https://doi.org/10.1007/s00605-018-1217-4 A hybrid of two theorems of Piatetski-Shapiro Angel Kumchev 1 · Zhivko...

11L20 | 11J25 | 11P32 | Exponential sums over primes | Mathematics, general | Diophantine inequalities | 11P55 | Mathematics | Additive prime number theory | Piatetski-Shapiro primes | MATHEMATICS

11L20 | 11J25 | 11P32 | Exponential sums over primes | Mathematics, general | Diophantine inequalities | 11P55 | Mathematics | Additive prime number theory | Piatetski-Shapiro primes | MATHEMATICS

Journal Article

06/2020

Integration over curved manifolds with higher codimension and, separately, discrete variants of continuous operators, have been two important, yet separate...

Journal Article

Journal of Number Theory, ISSN 0022-314X, 2010, Volume 130, Issue 9, pp. 1969 - 2002

In this paper we continue our study, begun in G. Harman and A.V. Kumchev (2006) [10], of the exceptional set of integers, not restricted by elementary...

Exponential sums | Squares of primes | Sieve methods | MATHEMATICS | THEOREM | P MODULO ONE | LINEAR-EQUATIONS | VARIABLES | VALUES | QUADRATIC EQUATIONS

Exponential sums | Squares of primes | Sieve methods | MATHEMATICS | THEOREM | P MODULO ONE | LINEAR-EQUATIONS | VARIABLES | VALUES | QUADRATIC EQUATIONS

Journal Article

Journal of Number Theory, ISSN 0022-314X, 04/2012, Volume 132, Issue 4, pp. 608 - 636

We study the representations of large integers n as sums p12+⋯+ps2, where p1,…,ps are primes with |pi−(n/s)1/2|⩽nθ/2, for some fixed θ<1. When s=5 we use a...

Exceptional sets | Almost equal squares | Distribution of primes | Waring–Goldbach problem | Exponential sums | Sieve methods | Secondary | Primary | Waring-Goldbach problem | HUAS THEOREM | MATHEMATICS | VARIABLES | SHORT INTERVALS

Exceptional sets | Almost equal squares | Distribution of primes | Waring–Goldbach problem | Exponential sums | Sieve methods | Secondary | Primary | Waring-Goldbach problem | HUAS THEOREM | MATHEMATICS | VARIABLES | SHORT INTERVALS

Journal Article

02/2016

Monatsh. Math. 183, 303-310 (2017) We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function $H(k)$ in the Waring-Goldbach...

Mathematics - Number Theory

Mathematics - Number Theory

Journal Article

Journal of Number Theory, ISSN 0022-314X, 2004, Volume 107, Issue 2, pp. 357 - 367

Let b 1,⋯, b 5 be non-zero integers satisfying gcd( b i , b j , b k )=1, for 1⩽ i< j< k⩽5 and | b j |⩽| b 5| for 1⩽ j⩽5 and n an integer satisfying b 1+⋯+b 5≡n...

Small prime solutions | Quadratic equations | Hardy–Littlewood method | Hardy - Littlewood method | MATHEMATICS | quadratic equations | small prime solutions | Hardy-Littlewood method

Small prime solutions | Quadratic equations | Hardy–Littlewood method | Hardy - Littlewood method | MATHEMATICS | quadratic equations | small prime solutions | Hardy-Littlewood method

Journal Article

12/2011

The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.

Mathematics - Number Theory

Mathematics - Number Theory

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 06/2005, Volume 1, Issue 2, pp. 161 - 173

Let r3(n) be the number of representations of a positive integer n as a sum of three squares of integers. We give two alternative proofs of a conjecture of...

MATHEMATICS | modular forms | DAS EIGENWERTPROBLEM | FOURIER COEFFICIENTS | circle method | Sums of squares | AUTOMORPHEN FORMEN

MATHEMATICS | modular forms | DAS EIGENWERTPROBLEM | FOURIER COEFFICIENTS | circle method | Sums of squares | AUTOMORPHEN FORMEN

Journal Article

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