Fractional Parts of Noninteger Powers of Primes
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ractional Parts of Noninteger Powers of Primes* A. V. Shubin1** 1
Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia Received April 30, 2020; in final form, April 30, 2020; accepted May 12, 2020
Abstract—Let α > 0 be any fixed noninteger. In the paper, we prove an analogue of the Bombieri–Vinogradov theorem for the set of primes p satisfying the condition {pα } < 1/2. This generalizes the previous result of Gritsenko and Zinchenko. DOI: 10.1134/S0001434620090084 Keywords: prime numbers, arithmetic progressions, fractional parts, Bombieri–Vinogradov theorem, exponential sums.
1. INTRODUCTION The problem concerning the distribution of primes satisfying the condition {pα } < σ for fixed numbers α, σ such that 0 < α, σ < 1 was first considered by Vinogradov in [1]. Using his method of exponential sums, he proved the formula 1 = σπ(X) + R(X), (1) p≤X {pα } 1 and sharpened in [9]–[13]. Vinogradov gave an interesting interpretation of the subset of primes
p satisfying the condition {pα } < 1/2: all such primes lie in intervals of the form k1/α ; (k + 1/2)1/α . Clearly, the length of such interval grows as k → +∞ for α < 1 and tends to zero for α > 1. Thus, in the second case, most such intervals do not contain even a single integer. This is the reason why the case α > 1 seems to be more difficult. Many results in number theory are based on the Bombieri–Vinogradov theorem, which is an estimate of the following type: π(Y ) X max max 1− . Y ≤X (a,q)=1 ϕ(q) (log X)A q≤Q
p≡a
p≤Y (mod q)
Here A > 0 is an arbitrary fixed constant, Q = X θ−ε with any fixed θ ≤ 1/2. The exponent θ is usually called “level distribution.” In the case of some special set E of integers, an analogue of Bombieri–Vinogradov theorem has the form: 1 X max max 1− 1 . (2) Y ≤X (a,q)=1 ϕ(q) (log X)A q≤Q
p≤Y,p∈E p≡a (mod q)
p≤Y p∈E
√ For the set E = n ∈ N | { n} < 1/2 , such an estimate was first proved by Tolev [14] for any θ ≤ 1/4. Later, this result was improved by Gritsenko and Zinchenko [15], who showed that (2) holds for all θ ≤ 1/3 for any set
1/2 ≤ α < 1. E = n ∈ N | {nα } < 1/2 , In the present paper, we show that the similar statement holds for any fixed noninteger α > 0. The main result is the following theorem. Theorem 1. Suppose that α > 0 is fixed noninteger and let E be the set of integers n satisfying the condition {nα } < 1/2. Further, let θ, ε and A > 0 be some fixed numbers such that 0 < ε < θ < 1/3, ε < α/20 and let 2 < Q ≤ X θ−ε . Then the inequality 1 cX max 1− 1 ≤ ϕ(q) (log X)A (a,q)=1 q≤Q
X≤p 0 is fixed noninteger, E is the set of integers n satisfying the condition {nα } < 1/2. Further, let θ, ε, D be fixed constants satisfying the conditions 0 < ε < θ < 1/3, ε < α/20, D > 1, and suppose that 1 ≤ h ≤ (log X)D , 2 < Q ≤ X θ−ε . Then the sum α max e(hp ) T = (a,q)=1 q≤Q
MATHEMATICAL NOTES
Vol. 108 No. 3 2020
X≤p
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