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A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$

S. I. Dimitrov

TL;DR

The paper proves the solvability of the ternary Piatetski-Shapiro inequality $|p_1^c+p_2^c+p_3^c-N|<ε$ with primes $p_1$ constrained to the Linnik form $p_1=x^2+y^2+1$ for $1<c<427/400$, by developing a new Bombieri–Vinogradov type bound for exponential sums over primes. The argument combines major/minor arc techniques with an intricate decomposition of the representation numbers $r(p_1-1)$ and a delicate analysis of the resulting exponential sums, including a new upper bound for the trivial arc and a sharp lower bound for the main term. The key components are the asymptotic formula for major-arc contributions, a Bombieri–Vinogradov type estimate for exponential sums over primes, and precise control of auxiliary sums $oldsymbol{Γ_i(X)}$ to show the dominant positive term drives the count to infinity as $X$ grows. This yields solvability for sufficiently large $N$ with the stated form constraint on $p_1$, and it highlights a pathway to further results with multiple special prime forms or broader ranges of $c$.

Abstract

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes.

A ternary diophantine inequality by primes with one of the form $\mathbf{p=x^2+y^2+1}$

TL;DR

The paper proves the solvability of the ternary Piatetski-Shapiro inequality with primes constrained to the Linnik form for , by developing a new Bombieri–Vinogradov type bound for exponential sums over primes. The argument combines major/minor arc techniques with an intricate decomposition of the representation numbers and a delicate analysis of the resulting exponential sums, including a new upper bound for the trivial arc and a sharp lower bound for the main term. The key components are the asymptotic formula for major-arc contributions, a Bombieri–Vinogradov type estimate for exponential sums over primes, and precise control of auxiliary sums to show the dominant positive term drives the count to infinity as grows. This yields solvability for sufficiently large with the stated form constraint on , and it highlights a pathway to further results with multiple special prime forms or broader ranges of .

Abstract

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed , every sufficiently large positive number and a small constant , the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} has a solution in prime numbers , such that . For this purpose we establish a new Bombieri -- Vinogradov type result for exponential sums over primes.

Paper Structure

This paper contains 18 sections, 24 theorems, 231 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Notations
  3. Preliminary lemmas
  4. Outline of the proof
  5. Asymptotic formula for $\mathbf{I^{(1)}_{l,d;J}(X)}$
  6. Upper bound of $\mathbf{I^{(3)}_{l,d;J}(X)}$
  7. Upper bound of $\mathbf{\Gamma_3(X)}$
  8. Estimation of $\mathbf{\Gamma_3^{(1)}(X)}$
  9. Estimation of $\mathbf{\Gamma_3^{(2)}(X)}$
  10. Estimation of $\mathbf{\Gamma_3^{(3)}(X)}$
  11. Estimation of $\mathbf{\Gamma_3(X)}$
  12. Upper bound of $\mathbf{\Gamma_2(X)}$
  13. Lower bound for $\mathbf{\Gamma_1(X)}$
  14. Estimation of $\mathbf{\Gamma_1^{(1)}(X)}$
  15. Estimation of $\mathbf{\Gamma_1^{(2)}(X)}$
  16. ...and 3 more sections

Key Result

Theorem 1

Let $1<c<\frac{427}{400}$. For every sufficiently large positive number $N$, the diophantine inequality has a solution in prime numbers $p_1,\,p_2,\,p_3$, such that $p_1=x^2 + y^2 +1$. Here $\theta_0$ is defined by theta0.

Theorems & Definitions (47)

  • Theorem 1
  • Conjecture 1
  • Conjecture 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 37 more