Almost Primes of the Form p c

Transcription

1 Almost Primes of the Form p c University of Missouri zgbmf@mail.missouri.edu Pre-Conference Workshop of Elementary, Analytic, and Algorithmic Number Theory Conference in Honor of Carl Pomerance s 70th birthday June 8, 2015

2 Piatetski-Shapiro Primes We denote by t and {t} the greatest integer t and the fractional part of t, respectively.

3 Piatetski-Shapiro Primes We denote by t and {t} the greatest integer t and the fractional part of t, respectively. The Piatetski-Shapiro sequences are sequences of the form ( n c ) n N (c > 1, c N), named in honor of Piatetski-Shapiro, who showed that the set P (c) := { p prime : p = n c for some n N } is infinite for any c (1, ).

4 Piatetski-Shapiro Primes We denote by t and {t} the greatest integer t and the fractional part of t, respectively. The Piatetski-Shapiro sequences are sequences of the form ( n c ) n N (c > 1, c N), named in honor of Piatetski-Shapiro, who showed that the set P (c) := { p prime : p = n c for some n N } is infinite for any c (1, ). More precisely, Let π (c) (x) be the number of primes with n x, p P (c). The asymptotic formula is π (c) (x) x c log x.

5 Piatetski-Shapiro Primes We know there are infinitely many prime numbers in arithmetic progressions.

6 Piatetski-Shapiro Primes We know there are infinitely many prime numbers in arithmetic progressions. However, it is harder to count the primes represented by polynomials of degree greater than 1, for example, primes of the form n

7 Piatetski-Shapiro Primes We know there are infinitely many prime numbers in arithmetic progressions. However, it is harder to count the primes represented by polynomials of degree greater than 1, for example, primes of the form n One of the motivations for Piatetski-Shapiro primes is that we consider the exponent of integers between 1 and 2.

8 Piatetski-Shapiro Primes We know there are infinitely many prime numbers in arithmetic progressions. However, it is harder to count the primes represented by polynomials of degree greater than 1, for example, primes of the form n One of the motivations for Piatetski-Shapiro primes is that we consider the exponent of integers between 1 and 2. Moreover, the investigation for 1 < c < 2 is a test to measure progress in the theory of exponential sums.

9 Piatetski-Shapiro Primes The asymptotic formula is expected to be true in the range 1 < c < 2. The range of c has been extended by many number theorists.

10 Piatetski-Shapiro Primes The asymptotic formula is expected to be true in the range 1 < c < 2. The range of c has been extended by many number theorists. Kolesnik improved the result to 1 < c < 10 9 =

11 Piatetski-Shapiro Primes The asymptotic formula is expected to be true in the range 1 < c < 2. The range of c has been extended by many number theorists. Kolesnik improved the result to 1 < c < 10 9 = Graham and Leitmann, using the exponent pairs method, have, independently, give the range 1 < c < =

12 Piatetski-Shapiro Primes Heath-Brown, with a Weyl s shift technique as well as the exponent pair method, and his decomposition of the Van Mangoldt function, extended the range to 1 < c < =

13 Piatetski-Shapiro Primes Heath-Brown, with a Weyl s shift technique as well as the exponent pair method, and his decomposition of the Van Mangoldt function, extended the range to 1 < c < = Kolesnik improved the result to 1 < c < =

14 Piatetski-Shapiro Primes Rivat and Sargos extended the range to 1 < c < =

15 Piatetski-Shapiro Primes Rivat and Sargos extended the range to 1 < c < = In fact, Rivat has several papers with different collaborators about this question.

16 Piatetski-Shapiro Primes Rivat and Sargos extended the range to 1 < c < = In fact, Rivat has several papers with different collaborators about this question. Rivat and Wu prove that π (c) (x) if x x 0 (c) for c (1, ). x c log x

17 Question Recall the definition of the Piatetski-Shapiro primes. P (c) := { p prime : p = n c for some n N }.

18 Question Recall the definition of the Piatetski-Shapiro primes. P (c) := { p prime : p = n c for some n N }. There is no restriction for the integers n.

19 Question Recall the definition of the Piatetski-Shapiro primes. P (c) := { p prime : p = n c for some n N }. There is no restriction for the integers n. Can we investigate a thinner set than integers?

20 Question Recall the definition of the Piatetski-Shapiro primes. P (c) := { p prime : p = n c for some n N }. There is no restriction for the integers n. Can we investigate a thinner set than integers? A reasonable thinner set is the set of almost primes.

21 Piatetski-Shapiro primes from almost primes We study sets of Piatetski-Shapiro primes of the form P (c) r := { p prime : p = n c for some n N r }.

22 Piatetski-Shapiro primes from almost primes We study sets of Piatetski-Shapiro primes of the form P (c) r := { p prime : p = n c for some n N r }. Theorem (Baker, Banks, G., Yeager, 2014) For any fixed c (1, ) the set P(c) 8 is infinite. More precisely, { n x : n N 8 and n c is prime } x (log x) 2, where the implied constant in the symbol depends only on c.

23 Question How about an even thinner set than the integers?

24 Question How about an even thinner set than the integers? The following subsequences are more important. P c = ( p c ) p P (c > 0, c / N).

25 Question How about an even thinner set than the integers? The following subsequences are more important. P c = ( p c ) p P (c > 0, c / N). Up to now very little has been established about the arithmetic structure of P c for fixed c.

26 Question How about an even thinner set than the integers? The following subsequences are more important. P c = ( p c ) p P (c > 0, c / N). Up to now very little has been established about the arithmetic structure of P c for fixed c. Balog asserts that the counting function Π c (x) = { prime p x : p c is prime }

27 Question How about an even thinner set than the integers? The following subsequences are more important. P c = ( p c ) p P (c > 0, c / N). Up to now very little has been established about the arithmetic structure of P c for fixed c. Balog asserts that the counting function Π c (x) = { prime p x : p c is prime } satisfies Π c (x) for any fixed c (0, 5 6 ). x c log 2 x (x )

28 Theorem We consider the question of whether or not P c contains infinitely many squarefree natural numbers.

29 Theorem We consider the question of whether or not P c contains infinitely many squarefree natural numbers. Theorem (Baker, Banks, G., Shperlinski) For every fixed c (1, ) the set Pc contains infinitely many squarefree natural numbers. Moreover, for some ε > 0 that depends only on c we have { prime p x : p c is squarefree } = 6 π 2 π(x) + O(x 1 ε ), where π(x) is the number of primes not exceeding x.

30 Discrepancy Bound We prove our theorem by studying the discrepancy D d (N) of the sequence (p c d 1 ) p N for any fixed d N; this is the quantity is given by D d (N) = sup {p N : {p c d 1 } λ} λ π(n). 0 λ 1 The case d = 1 has been previously studied by Baker and Kolesnik.

31 Discrepancy Bound Theorem (Baker, Banks, G., Shparlinski) Fix ε, ν > 0 and c (0, 3), c Z. If ε > 0 is small enough (depending on c and ν), then the discrepancy D d (N) satisfies the uniform bound D d (N) N 1 ε (1 d N νε ), where the implied constant depends only on c, ε, ν.

32 Another Theorem Eventually, we also consider the question of whether or not P c contains infinitely many almost primes.

33 Another Theorem Eventually, we also consider the question of whether or not P c contains infinitely many almost primes. Balog shows that there are a constant C > 1 and integers r 1, s 1 such that for any 0 < c < C, c is not an integer, we have P r x P r c is P s 1 x log 2 x.

34 Another Theorem Eventually, we also consider the question of whether or not P c contains infinitely many almost primes. Balog shows that there are a constant C > 1 and integers r 1, s 1 such that for any 0 < c < C, c is not an integer, we have P r x P r c is P s One can take, for example 1 x log 2 x. r s

35 Another Theorem However, Balog does not give the exact range of c.

36 Another Theorem However, Balog does not give the exact range of c. In the same paper, we prove Theorem (Baker, Banks, G., Shparlinski) For every fixed c 2.2 there is a positive integer R not exceeding 16c c 2 such that { prime p x : p c is an R-almost prime } x log 2 x, where the implied constant depends only on c. If c 3, then the same result holds with some R 16c c 2.

37 P-S Primes in Arithmetic Progressions Another way of investigating a subset of Piatetski-Shapiro primes: we find the intersection with other sequences.

38 P-S Primes in Arithmetic Progressions Another way of investigating a subset of Piatetski-Shapiro primes: we find the intersection with other sequences. Leitmann and Wolke show that for any coprime integers a, d with 1 a d and any real number c (1, ) the counting function satisfies π (c) (x; d, a) = # { p x : p P (c) and p a mod d }, π c (x; d, a) x 1/c φ(d) log(x) as x, where φ is the Euler function.

39 P-S Primes in Arithmetic Progressions Denote γ = 1 c. Baker, Banks, Brüdern, Shparlinski and Weingartner show that for fixed c (1, ) π c (x; d, a) = γx γ 1 π(x; d, a) + γ(1 γ) + O(x 17/39+7γ/13+ε ). 2 x u γ 2 π(u; d, a)du

40 P-S Primes in Arithmetic Progressions Denote γ = 1 c. Baker, Banks, Brüdern, Shparlinski and Weingartner show that for fixed c (1, ) π c (x; d, a) = γx γ 1 π(x; d, a) + γ(1 γ) + O(x 17/39+7γ/13+ε ). 2 x u γ 2 π(u; d, a)du The range of c is improved from (1, ) to (1, 13 ) and the error term is improved from O(x 17/39+7γ/13+ε ) to O(x 3/7+7γ/13+ε ).

41 Carmichael Numbers Composed of P-S Primes A Carmichael number is a composite number n which satisfies the modular arithmetic congruence relation: b n 1 1 (mod n) for all integers 1 < b < n which are relatively prime to n.

42 Carmichael Numbers Composed of P-S Primes A Carmichael number is a composite number n which satisfies the modular arithmetic congruence relation: b n 1 1 (mod n) for all integers 1 < b < n which are relatively prime to n. In the same paper, Baker et al show that for every c (1, ) there are infinitely many Carmichael numbers composed entirely of primes from the set P (c).

43 Carmichael Numbers Composed of P-S Primes A Carmichael number is a composite number n which satisfies the modular arithmetic congruence relation: b n 1 1 (mod n) for all integers 1 < b < n which are relatively prime to n. In the same paper, Baker et al show that for every c (1, ) there are infinitely many Carmichael numbers composed entirely of primes from the set P (c). The range of c can be extended from (1, ) to (1, 561 ), with a small improvement of

44 P-S Primes from a Beatty Sequence For two fixed real numbers α and β, the corresponding non-homogeneous Beatty sequence is the sequence of integers defined by B α,β = ( αn + β ) n=1.

45 P-S Primes from a Beatty Sequence For two fixed real numbers α and β, the corresponding non-homogeneous Beatty sequence is the sequence of integers defined by B α,β = ( αn + β ) n=1. The type τ = τ(α) of the irrational number α is defined by τ = sup { t R : lim inf n nt αn = 0 }, where t denotes the distance from a real number t to the nearest integer.

46 P-S Primes from a Beatty Sequence For two fixed real numbers α and β, the corresponding non-homogeneous Beatty sequence is the sequence of integers defined by B α,β = ( αn + β ) n=1. The type τ = τ(α) of the irrational number α is defined by τ = sup { t R : lim inf n nt αn = 0 }, where t denotes the distance from a real number t to the nearest integer. A Liouville number is not of finite type.

47 P-S Primes from a Beatty Sequence Theorem (G.) Let α, β R, and suppose that α > 1 is irrational and of finite type. Let c (1, ). There are infinitely many primes in both the Beatty sequence B α,β and the Piatetski-Shapiro sequence N (c). Moreover, the counting function satisfies π (c) α,β (x) = { prime p x : p B α,β N (c)} ( π (c) x 1/c α,β (x) = α log x + O x 1/c log 2 x where the implied constant depends only on α and c. ),

48 End Thank You