aa r X i v : . [ m a t h . G M ] M a y Primitive Roots In Short Intervals
N. A. Carella
Abstract : Let p ≥ N ≫ (log p ) ε . This note proves the existence ofprimitive roots in the short interval [ M, M + N ], where M ≥ ε > g ( p ) = O (cid:0) (log p ) ε (cid:1) , and the least primeprimitive root g ∗ ( p ) = O (cid:0) (log p ) ε (cid:1) unconditionally. Contents
May 27, 2020
Mathematics Subject Classifications : Primary 11A07, Secondary 11N37.
Keywords : Least primitive root; Least prime primitive root; Primitive root in short interval. rimitive Roots In Short Intervals Given a large prime p ≥
2, and a number N ≤ p . The standard analytic methods demonstrate theexistence of primitive roots in any short interval[ M, M + N ] (1)for any number N ≫ p / ε , where M ≥ ε > N ≫ p / ε , see [2]. Further, the explicit upper bound claims that the least primitive root g ( p ) ≥ g ( p ) < √ p − p > g ( p ) = O (cid:0) log p (cid:1) ,and the average value is g ( p ) = O (cid:0) (log log p ) (cid:1) , see [43] and [3] respectively.Almost all these results are based on the standard indicator function in Lemma 3.1. This noteintroduces a new technique based on the indicator function in Lemma 3.2 to improve the resultsfor primitive roots in short intervals. Theorem 1.1.
Given a small number ε > , and a sufficiently large prime p ≥ , let N ≫ (log p ) ε . Then, the short interval [ M, M + N ] (3) contains a primitive root for any fixed M ≥ . In particular, the least primitive root g ( p ) = O (cid:0) (log p ) ε (cid:1) unconditionally. As the probability of a primitive root modulo p is O (1 / log log p ), this result is nearly optimal, seeSection 5 for a discussion.The existence of prime primitive roots in short interval [ M, M + N ] requires information aboutprimes in short intervals such that N < p / , and M ≥ , x ], it is feasible. Recently, it was proved that theleast prime primitive root g ∗ ( p ) = O ( p ε ), unconditionally, see [10]. Moreover, assuming standardconjectures, the least prime primitive root is expected to be g ∗ ( p ) = O (cid:0) (log p )(log log p ) (cid:1) , see [4].A very close upper bound is provided here. Theorem 1.2. If p ≥ is a sufficiently large prime, then, the least prime primitive root satisfies g ∗ ( p ) = O (cid:0) (log p ) ε (cid:1) (4) for any small number ε > , unconditionally. Theorem 1.3.
Let p ≥ be a sufficiently large prime, and let N ≫ p . . Then, the short interval [ M, M + N ] (5) contains a prime primitive root for any fixed M ≥ unconditionally. The fundamental background materials are discussed in the earlier sections. Section 9 presentsa proof of Theorem 1.1, the penultimate section presents a proofs of Theorem 1.2, and the lastsection presents a proof of Theorem 1.3.
For a prime p ≥
2, the multiplicative group of the finite fields F p is a cyclic group for all primes. Definition 2.1.
The order min { k ∈ N : u k ≡ p } of an element u ∈ F p is denoted byord p ( u ). An element is a primitive root if and only if ord p ( u ) = p − rimitive Roots In Short Intervals ϕ ( n ) = { k : gcd( k, n ) =1 } . This counting function is compactly expressed by the analytic formula ϕ ( n ) = n Q p | n (1 − /p ) , n ∈ N . Lemma 2.1. (Fermat-Euler) If a ∈ Z is an integer such that gcd( a, n ) = 1 , then a ϕ ( n ) ≡ n . Lemma 2.2. (Primitive root test)
An integer u ∈ Z is a primitive root modulo an integer n ∈ N if and only if u ϕ ( n ) /p − n for all prime divisors p | ϕ ( n ) . The primitive root test is a special case of the Lucas primality test, introduced in [27, p. 302]. Amore recent version appears in [11, Theorem 4.1.1], and similar sources.
Lemma 2.3. (Complexity of primitive root test)
Given a prime p ≥ , and the squarefree part p p · · · p v | p − , a primitive root modulo p can be determined in deterministic polynomial time O (log c p ) , some constant c > .Proof. The mechanics of the deterministic polynomial time algorithm are specified in [44, Chap-ter 11]. By Theorem 1.2, the algorithm is repeated at most O (cid:0) (log p ) ε (cid:1) times for each u = O (cid:0) (log p ) ε (cid:1) . These prove the claim. (cid:4) The characteristic function Ψ : G −→ { , } of primitive elements is one of the standard an-alytic tools employed to investigate the various properties of primitive roots in cyclic groups G .Many equivalent representations of the characteristic function Ψ of primitive elements are possible.Several of these representations are studied in this section. A representation of the characteristic function dependent on the orders of the cyclic groups isgiven below. This representation is sensitive to the primes decompositions q = p e p e · · · p e t t , with p i prime and e i ≥
1, of the orders of the cyclic groups q = G . Lemma 3.1.
Let G be a finite cyclic group of order p − G , and let = u ∈ G be an invertibleelement of the group. Then Ψ( u ) = ϕ ( p − p − X d | p − µ ( d ) ϕ ( d ) X ord( χ )= d χ ( u ) = (cid:26) if ord p ( u ) = p − , if ord p ( u ) = p − . (6) Proof.
Assume that u = τ qm is a q th power residue modulo p , where q | p − m, p −
1) = 1.Then, the inner sum X ord( χ )= q χ ( u ) = X ord( χ )= q χ ( τ qm ) = X ord( χ )= q χ ( τ m ) q = ϕ ( q ) = q − , (7)where χ ( v ) q = 1. Replacing this information into the product φ ( p − p − X d | p − µ ( d ) ϕ ( d ) X ord( χ )= d χ ( u ) = φ ( p − p − Y q | p − − P ord( χ )= q χ ( u ) q − ! = φ ( p − p − Y q | p − (cid:18) − q − q − (cid:19) = 0 . (8) rimitive Roots In Short Intervals u ∈ G has order ord p ( u ) = q | p − q < p −
1. Now, assume that u = τ m is not q th power residue modulo p for any q | p − m, p −
1) = 1. Then, the inner sum X ord( ψ )= q χ ( u ) = X ord( ψ )= q χ ( τ m ) = − . (9)Replacing this information into the product φ ( p − p − X d | p − µ ( d ) ϕ ( d ) X ord( χ )= d χ ( u ) = φ ( p − p − Y q | p − − P ord( χ )= q χ ( u ) q − ! = φ ( p − p − Y q | p − (cid:18) − − q − (cid:19) = 1 . (10)These verify that both sides of the equation vanishes if and only if the element u ∈ G has orderord p ( u ) = q | p − q < p − (cid:4) The precise source of formula (6) is not clear. The authors in [14], and [45] attributed this formulato Vinogradov, and other authors have attributed it to Landau. The proof and other details onthe characteristic function are given in [18, p. 863], [29, p. 258], [31, p. 18]. The characteristicfunction for multiple primitive roots is used in [13, p. 146] to study consecutive primitive roots. In[16] it is used to study the gap between primitive roots with respect to the Hamming metric. Andin [45] it is used to prove the existence of primitive roots in certain small subsets A ⊂ F p . In [14]it is used to prove that some finite fields do not have primitive roots of the form aτ + b , with τ primitive and a, b ∈ F p constants. In addition, the Artin primitive root conjecture for polynomialsover finite fields was proved in [37] using this formula. It often difficult to derive any meaningful result using the usual divisors dependent characteristicfunction of primitive elements given in Lemma 3.1. This difficulty is due to the large number ofterms that can be generated by the divisors, for example, d | p −
1, involved in the calculations,see [18], [16] for typical applications and [30, p. 19] for a discussion.A new divisors-free representation of the characteristic function of primitive element is developedhere. This representation can overcomes some of the limitations of its counterpart in certainapplications. The divisors dependent representation of the characteristic function of primitiveroots, Lemma 3.1, detects the order ord p ( u ) of the element u ∈ F p by means of the divisors of thetotient p −
1. In contrast, the divisors-free representation of the characteristic function, Lemma3.2, detects the order ord p ( u ) ≥ u ∈ F p by means of the solutions of the equation τ n − u = 0 in F p , where u, τ are constants, and 1 ≤ n < p − , gcd( n, p −
1) = 1 , is a variable. Lemma 3.2.
Let p ≥ be a prime, and let τ be a primitive root mod p . If u ∈ F p is a nonzeroelement, and ψ = 1 is a nonprincipal additive character of order ord ψ = p , then Ψ( u ) = X gcd( n,p − p X ≤ m ≤ p − ψ (( τ n − u ) m ) = (cid:26) if ord p ( u ) = p − , if ord p ( u ) = p − . (11) Proof.
As the index n ≥ p −
1, the element τ n ∈ F p ranges over the primitive roots mod p . Ergo, the equation τ n − u = 0 (12)has a solution if and only if the fixed element u ∈ F p is a primitive root. Next, replace ψ ( z ) = e i πz/p to obtain Ψ( u ) = X gcd( n,p − p X ≤ m ≤ p − e i π ( τ n − u ) m/p = (cid:26) p ( u ) = p − , p ( u ) = p − . (13)This follows from the geometric series identity P ≤ m ≤ N − w m = ( w N − / ( w −
1) with w = 1,applied to the inner sum. (cid:4) rimitive Roots In Short Intervals Some prime numbers results focusing on the local minima of the ratio ϕ ( n ) n = Y p | n (cid:18) − p (cid:19) > e γ log log n + 5 / (2 log log n ) (14)are recorded in this section. The conditional results are studied in [35], and the unconditionalresults are proved by various authors as [40, Theorem 7 and Theorem 15], and [34, Theorem 2.9]. Lemma 4.1.
Let n ≥ be a large integer, and let ω ( n ) be the number of prime divisors p | n .Then (i) ω ( n ) ≪ log log n, the average number of prime divisors. (ii) ω ( n ) ≪ log n/ log log n, the maximal number of prime divisors.Proof. These are standard results in analytic number theory, see [34, Theorem 2.6]. (cid:4)
Lemma 4.2.
Let x ≥ be a large number, then (i) Y p ≤ x (cid:18) − p (cid:19) = 1 e γ log x + O (cid:16) e − c √ log x (cid:17) , unconditionally. (ii) Y p ≤ x (cid:18) − p (cid:19) = 1 e γ log x + Ω ± (cid:18) log log log xx / (cid:19) , unconditional oscillation. (iii) Y p ≤ x (cid:18) − p (cid:19) = 1 e γ log x + O (cid:18) log xx / (cid:19) , conditional on the RH.The symbol γ is the Euler constant, and c > is an absolute constant. The explicit estimates are given in [40, Theorem 7], and the results for products over arithmeticprogression are proved in [28], et alii. The nonquantitative unconditional oscillations of the errorof the product of primes is implied by the work of Phragmen, refer to equation ( ?? ), and [36, p.182]. Since then, various authors have developed quantitative versions, see [40], [15], et alii. The probability of primitive roots in a finite field F p has the closed form ϕ ( p − / ( p − ≤ / ϕ ( p − / ( p −
1) = 1 / F = { p = 2 n + 1 : n ≥ } = { , , , , , . . . } . (15)This is followed by the subset of Germain primes S = { p = 2 a q + 1 : q ≥ a ≥ } = { , , , , , , . . . } , (16)which has ϕ ( p − /p = (1 / − /q ), et cetera. Some basic questions such as the sizes of thesesubsets of primes are open problems. In contrast, the minimal probabilities occur on the varioussubsets of primes with highly composite totients p −
1. For example, the subset R = { p ≥ p − v · v · v · · · q v q , and v i ≥ } = { , , , , . . . } . (17)In these cases, the probability function can have a complicated expression such as ϕ ( p − p − ≍ Y q ≪ log p (cid:18) − q (cid:19) = 1 e γ log log p + Ω ± (cid:18) log log log log p (log p ) / (cid:19) . (18) rimitive Roots In Short Intervals p ≤ x is a well known constant a = 1 π ( x ) X p ≤ x ϕ ( p − p − Y p> (cid:18) − p ( p − (cid:19) + o (1) = 0 . . . . . (19)The analysis of the average appears in [22], [41], and an early numerical calculations is given in[46]. The distribution of primitive root for highly composite totients p − λ >
0. For k ≥
0, and 1 ≤ t ≤ δ log log p , with δ >
0, theprobability function has the asymptotic formula P k ( t ) ∼ e − λ λ k k ! , (20)confer [13, Theorem 2] for the finer details. Let p ≥ g , g , . . . , g t be the sequence of primitive roots in increasing order,with t = ϕ ( p − p ≥
2, the average gap between a pair of consecutiveprimitive roots is defined by d n = g n +1 − g n = p − ϕ ( p − ≪ log log p. (21) Lemma 5.1.
Let x ≥ be a large number, then the average gap between consecutive primitiveroots over all the primes p ≤ x is bounded by a constant. In particular, for any constant c > , d n = Y p ≥ (cid:18) − p − (cid:19) li( x ) + O (cid:18) x log c − x (cid:19) . (22) Proof.
The identity n/ϕ ( n ) = P d | n µ ( d ) /ϕ ( d ) is used here to compute the average over all theprimes p ≤ x : X p ≤ x p − ϕ ( p −
1) = X p ≤ x X d | p − µ ( d ) ϕ ( d ) (23)= X d ≤ x µ ( d ) ϕ ( d ) X p ≤ xp ≡ d . To apply the prime number theorem to the inner sum, use a dyadic partition X d ≤ x µ ( d ) ϕ ( d ) X p ≤ xp ≡ d X d ≤ log c x µ ( d ) ϕ ( d ) X p ≤ xp ≡ d X d ≥ log c x µ ( d ) ϕ ( d ) X p ≤ xp ≡ d , (24)where c > X d ≤ log C x µ ( d ) ϕ ( d ) X p ≤ xp ≡ d X d ≤ log C x µ ( d ) ϕ ( d ) (cid:18) li( x ) ϕ ( d ) + O (cid:18) x log b x (cid:19)(cid:19) (25)= li( x ) X d ≥ µ ( d ) ϕ ( d ) + O (cid:18) x log b x (cid:19) , where b > c + 1. The second sum has the asymptotic expression X d ≥ log C x µ ( d ) ϕ ( d ) X p ≤ xp ≡ d ≪ x log c x X d ≥ log c x ϕ ( d ) = O (cid:18) x log c − x (cid:19) , (26)Combining the last two expressions (25) and (26) completes the proof. (cid:4) rimitive Roots In Short Intervals Y p ≥ (cid:18) − p − (cid:19) = 2 . . . .. (27) Lemma 5.2.
Let p ≥ be a large prime, and let t ≤ ϕ ( p − be a large number. Then, the g , g , . . . , g t be the sequence of primitive roots in increasing order are uniformly distributed overthe interval [2 , p − .Proof. Apply Theorem 6.2 to the Bohl-Weil criterion1 p X ≤ n ≤ t e i πg n /p = o (1) , (28)where p / < t ≤ ϕ ( p − (cid:4) This section provides simple estimates for the exponential sums of interest in this analysis. Thereare two objectives: To determine an upper bound, proved in Theorem 6.2, and to show that X gcd( n,p − e i πbτ n /p = X gcd( n,p − e i πτ n /p + E ( p ) , (29)where E ( p ) is an error term, this is proved in Lemma 6.1. The proofs of these results are entirelybased on established results and elementary techniques. Let f : C −→ C be a function, and let q ∈ N be a large integer. The finite Fourier transformˆ f ( t ) = 1 q X ≤ s ≤ q − e iπst/q (30)and its inverse are used here to derive a summation kernel function, which is almost identical tothe Dirichlet kernel. Definition 6.1.
Let p and q be primes, and let ω = e i π/q , and ζ = e i π/p be roots of unity. The finite summation kernel is defined by the finite Fourier transform identity K ( f ( n )) = 1 q X ≤ t ≤ q − , X ≤ s ≤ p − ω t ( n − s ) f ( s ) = f ( n ) . (31)This simple identity is very effective in computing upper bounds of some exponential sums X n ≤ x f ( n ) = X n ≤ x K ( f ( n )) , (32)where x ≤ p < q . Two applications are illustrated here. Theorem 6.1. ([42], [32])
Let p ≥ be a large prime, and let τ ∈ F p be an element of largemultiplicative order ord p ( τ ) | p − . Then, for any b ∈ [1 , p − , and x ≤ p − , X n ≤ x e i πbτ n /p ≪ p / log p. (33) rimitive Roots In Short Intervals Proof.
Let q = p + o ( p ) > p be a large prime, and let f ( n ) = e i πbτ n /p , where τ is a primitive rootmodulo p . Applying the finite summation kernel in Definition 6.1, yields X n ≤ x e i πbτ n /p = X n ≤ x q X ≤ t ≤ q − , X ≤ s ≤ p − ω t ( n − s ) e i πbτ s /p . (34)The term t = 0 contributes − x/q , and rearranging it yield X n ≤ x e i πbτ n /p = 1 q X n ≤ x, X ≤ t ≤ q − , X ≤ s ≤ p − ω t ( n − s ) e i πbτ s /p − xq (35)= 1 q X ≤ t ≤ q − X ≤ s ≤ p − ω − ts e i πbτ s /p X n ≤ x ω tn − xq . Taking absolute value, and applying Lemma 6.2, and Lemma 6.4, yield (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x e i πbτ n /p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q X ≤ t ≤ q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ s ≤ p − ω − ts e i πbτ s /p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n ≤ x ω tn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + xq ≪ q X ≤ t ≤ q − (cid:16) q / log q (cid:17) · (cid:18) qπt (cid:19) + xq (36) ≪ p / log p. The last summation in (36) uses the estimate X ≤ t ≤ q − t ≪ log q ≪ log p (37)since q = p + o ( p ) > p , and x/q ≤ (cid:4) This appears to be the best possible upper bound. The above proof generalizes the sum of resolventsmethod used in [32]. Here, it is reformulated as a finite Fourier transform method, which isapplicable to a wide range of functions. A similar upper bound for composite moduli p = m is alsoproved, [op. cit., equation (2.29)]. Theorem 6.2.
Let p ≥ be a large prime, and let τ be a primitive root modulo p . Then, X gcd( n,p − e i πbτ n /p ≪ p − ε (38) for any b ∈ [1 , p − , and any arbitrary small number ε ∈ (0 , / .Proof. Let q = p + o ( p ) > p be a large prime, and let f ( n ) = e i πbτ n /p , where τ is a primitive rootmodulo p . Start with the representation X gcd( n,p − e i πbτnp = X gcd( n,p − q X ≤ t ≤ q − , X ≤ s ≤ p − ω t ( n − s ) e i πbτsp , (39)see Definition 6.1. Use the inclusion exclusion principle to rewrite the exponential sum as X gcd( n,p − e i πbτnp = X n ≤ p − q X ≤ t ≤ q − , X ≤ s ≤ p − ω t ( n − s ) e i πbτsp X d | p − d | n µ ( d ) . (40) rimitive Roots In Short Intervals t = 0 contributes − ϕ ( p ) /q , and rearranging it yield X gcd( n,p − e i πbτnp (41)= X n ≤ p − q X ≤ t ≤ q − , X ≤ s ≤ p − ω t ( n − s ) e i πbτsp X d | p − d | n µ ( d ) − ϕ ( p ) q = 1 q X ≤ t ≤ q − X ≤ s ≤ p − ω − ts e i πbτsp X d | p − µ ( d ) X n ≤ p − ,d | n ω tn − ϕ ( p ) q . Taking absolute value, and applying Lemma 6.3, and Lemma 6.4, yield (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X gcd( n,p − e i πbτnp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (42) ≤ q X ≤ t ≤ q − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ s ≤ p − ω − ts e i πbτ s /p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d | p − µ ( d ) X n ≤ p − ,d | n ω tn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ϕ ( p ) q ≪ q X ≤ t ≤ q − (cid:16) q / log q (cid:17) · (cid:18) q log log pπt (cid:19) + ϕ ( p ) q ≪ p / log p. The last summation in (42) uses the estimate X ≤ t ≤ q − t ≪ log q ≪ log p (43)since q = p + o ( p ) > p , and ϕ ( p ) /q ≤
1. This is restated in the simpler notation p / log p ≤ p − ε for any arbitrary small number ε ∈ (0 , / (cid:4) The upper bound given in Theorem 6.2 seems to be optimum. A different proof, which has aweaker upper bound, appears in [21, Theorem 6], and related results are given in [7], [20], [23],and [24, Theorem 1].
For any fixed 0 = b ∈ F p , the map τ n −→ bτ n is one-to-one in F p . Consequently, the subsets { τ n : gcd( n, p −
1) = 1 } and { bτ n : gcd( n, p −
1) = 1 } ⊂ F p (44)have the same cardinalities. As a direct consequence the exponential sums X gcd( n,p − e i πbτ n /p and X gcd( n,p − e i πτ n /p , (45)have the same upper bound up to an error term. An asymptotic relation for the exponential sums(45) is provided in Lemma 6.1. This result expresses the first exponential sum in (45) as a sum ofsimpler exponential sum and an error term. Lemma 6.1.
Let p ≥ be a large primes. If τ be a primitive root modulo p , then, X gcd( n,p − e i πbτ n /p = X gcd( n,p − e i πτ n /p + O ( p / log p ) , (46) for any b ∈ [1 , p − . rimitive Roots In Short Intervals Proof.
For b = 1, the exponential sum has the representation X gcd( n,p − e i πbτnp (47)= 1 q X ≤ t ≤ q − X ≤ s ≤ p − ω − ts e i πbτsp X d | p − µ ( d ) X n ≤ p − ,d | n ω tn − ϕ ( p ) q , confer equation (41) for details. And, for b = 1, X gcd( n,p − e i πτnp (48)= 1 q X ≤ t ≤ q − X ≤ s ≤ p − ω − ts e i πτsp X d | p − µ ( d ) X n ≤ p − ,d | n ω tn − ϕ ( p ) q , respectively, see (41). Differencing (47) and (48) produces X gcd( n,p − e i πbτ n /p − X gcd( n,p − e i πτ n /p (49)= 1 q X ≤ t ≤ q − X ≤ s ≤ p − ω − ts e i πbτsp − X ≤ s ≤ p − ω − ts e i πτsp × X d | p − µ ( d ) X n ≤ p − ,d | n ω tn . By Lemma 6.3, the relatively prime summation kernel is bounded by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d | p − µ ( d ) X n ≤ p − ,d | n ω tn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X gcd( n,p − ω tn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q log log pπt , (50)and by Lemma 6.4, the difference of two Gauss sums is bounded by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ s ≤ p − ω − ts e i πbτsp − X ≤ s ≤ p − ω − ts e i πτsp (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ s ≤ p − χ ( s ) ψ b ( s ) − X ≤ s ≤ p − χ ( s ) ψ ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ p / log p, (51)where χ ( s ) = e iπst/p , and ψ b ( s ) = e i πbτ s /p . Taking absolute value in (49) and replacing (50), and rimitive Roots In Short Intervals (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X gcd( n,p − e i πbτ n /p − X gcd( n,p − e i πτ n /p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q X ≤ t ≤ q − (cid:16) q / log q (cid:17) · (cid:18) q log log pt (cid:19) (52) ≤ q / (log q )(log q )(log log p ) ≤ p / log p, where q = p + o ( p ). (cid:4) The same proof works for many other subsets of elements
A ⊂ F p . For example, X n ∈A e i πbτ n /p = X n ∈A e i πτ n /p + O ( p / log c p ) , (53)for some constant c > Lemma 6.2.
Let p ≥ and q = p + o ( p ) > p be large primes. Let ω = e i π/q be a q th root ofunity, and let t ∈ [1 , p − . Then, (i) X n ≤ p − ω tn = ω t − ω tp − ω t , (ii) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n ≤ p − ω tn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ qπt . Proof. (i) Use the geometric series to compute this simple exponential sum as X n ≤ p − ω tn = ω t − ω tp − ω t . (ii) Observe that the parameters q = p + o ( p ) > p is prime, ω = e i π/q , the integers t ∈ [1 , p − d ≤ p − < q −
1. This data implies that πt/q = kπ with k ∈ Z , so the sine function sin( πt/q ) = 0is well defined. Using standard manipulations, and z/ ≤ sin( z ) < z for 0 < | z | < π/
2, the lastexpression becomes (cid:12)(cid:12)(cid:12)(cid:12) ω t − ω tp − ω t (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) πt/q ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ qπt . (54) (cid:4) Lemma 6.3.
Let p ≥ and q = p + o ( p ) > p be large primes, and let ω = e i π/q be a q th root ofunity. Then, (i) X gcd( n,p − ω tn = X d | p − µ ( d ) ω dt − ω dt (( p − /d +1) − ω dt , (ii) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X gcd( n,p − ω tn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q log log pπt , where µ ( k ) is the Mobius function, for any fixed pair d | p − and t ∈ [1 , p − . rimitive Roots In Short Intervals Proof. (i) Use the inclusion exclusion principle to rewrite the exponential sum as X gcd( n,p − ω tn = X n ≤ p − ω tn X d | p − d | n µ ( d )= X d | p − µ ( d ) X n ≤ p − d | n ω tn = X d | p − µ ( d ) X m ≤ ( p − /d ω dtm (55)= X d | p − µ ( d ) ω dt − ω dt (( p − /d +1) − ω dt . (ii) Observe that the parameters q = p + o ( p ) > p is prime, ω = e i π/q , the integers t ∈ [1 , p − d ≤ p − < q −
1. This data implies that πdt/q = kπ with k ∈ Z , so the sine function sin( πdt/q ) = 0is well defined. Using standard manipulations, and z/ ≤ sin( z ) < z for 0 < | z | < π/
2, the lastexpression becomes (cid:12)(cid:12)(cid:12)(cid:12) ω dt − ω dtp − ω dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) πdt/q ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ qπdt (56)for 1 ≤ d ≤ p −
1. Finally, the upper bound is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X d | p − µ ( d ) ω dt − ω dt (( p − /d +1) − ω dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ qπt X d | p − d (57) ≤ q log log pπt . The last inequality uses the elementary estimate P d | n d − ≤ n . (cid:4) Lemma 6.4. (Gauss sums)
Let p ≥ and q be large primes. Let χ ( t ) = e i πt/q and ψ ( t ) = e i πτ t /p be a pair of characters. Then, the Gaussian sum has the upper bound (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ≤ t ≤ q − χ ( t ) ψ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ q / log q. (58) The upper bounds for exponential sums over subsets of elements in finite fields F p studied inSection 6 are used to estimate the error terms E ( x, y ) and E ( x, Λ) in the proofs of Theorem 1.1and Theorem 1.2 respectively.
Lemma 7.1.
Let p ≥ be a large prime, let ψ = 1 be an additive character, and let τ be aprimitive root mod p . If the element u = 0 is not a primitive root, then, p X x ≤ u ≤ y, X gcd( n,p − , X 1. Since ϕ ( p − /p ≤ / 2, anontrivial error term | E ( x, y ) | < (cid:12)(cid:12)(cid:12)(cid:12) − ϕ ( p − p ( y − x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ y − x rimitive Roots In Short Intervals ψ ( z ) = e i πz/p , and rearrange the triple finite sum in theform E ( x, y ) = 1 p X x ≤ u ≤ y, X The results available in the literature for primes in small intervals of the forms [ x, x + y ] with y < x / are not uniform. In light of this fact, only the error term for the simpler intervals [2 , x ]can be computed effectively. Lemma 7.2. Let p ≥ be a large prime, let ψ = 1 be an additive character, and let τ be aprimitive root mod p . If the element u = 0 is not a primitive root, then, p X u ≤ x, X gcd( n,p − , X The notation f ( x ) ≍ g ( x ) is defined by af ( x ) < g ( x ) < bf ( x ) for some constants a, b > Lemma 8.1. Let p ≥ be a large prime, and let ≤ x < y < p be a pair of numbers. Then, X x ≤ u ≤ y p X gcd( n,p − ≫ y − x log log p (cid:16) O (cid:16) (log log p ) e − c √ log log p (cid:17)(cid:17) . (66) Proof. The maximal number ω ( p − 1) of prime divisors of highly composite totients p − ω ( p − ≫ log p/ log log p . This implies that z ≍ log p . An application of Lemma 4.2 to the ratioreturns ϕ ( p − p = p − p p − Y q | p − (cid:18) − q (cid:19) ≥ Y q ≤ z (cid:18) − q (cid:19) = 1 e γ log z + O (cid:16) e − c √ log z (cid:17) (67) ≫ e γ log log p + O (cid:16) e − c √ log log p (cid:17) . Substituting this, the main term reduces to M ( x, y ) = X x ≤ u ≤ y p X gcd( n,p − ϕ ( p − p ( y − x ) (68) ≫ (cid:18) e γ log log p + O (cid:16) e − c √ log log p (cid:17)(cid:19) ( y − x ) . The proves the claim. (cid:4) Lemma 8.2. Let p ≥ be a large prime, and let x < p be a number. Then, X u ≤ x p X gcd( n,p − Λ( u ) ≫ x log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) (69) for some constant c > .Proof. The maximal number ω ( p − 1) of prime divisors of highly composite totients p − ω ( p − ≫ log p/ log log p . This implies that z ≍ log p . An application of Lemma 4.2 to the ratio rimitive Roots In Short Intervals ϕ ( p − p = p − p p − Y q | p − (cid:18) − q (cid:19) ≥ Y q ≤ z (cid:18) − q (cid:19) = 1 e γ log z + O (cid:16) e − c √ log z (cid:17) (70) ≫ e γ log log p + O (cid:16) e − c √ log log p (cid:17) . In addition, using the prime number theorem in the form P n ≤ x Λ( n ) = x + O (cid:16) xe − c √ log x (cid:17) , themain term reduces to M ( x, Λ) = X u ≤ x p X gcd( n,p − Λ( u )= ϕ ( p − p X u ≤ x Λ( u )= ϕ ( p − p (cid:16) x + O (cid:16) xe − c √ log x (cid:17)(cid:17) (71) ≫ (cid:18) e γ log log p + O (cid:16) e − c √ log log p (cid:17)(cid:19) (cid:16) x + O (cid:16) xe − c √ log x (cid:17)(cid:17) ≫ x log log p (cid:16) O (cid:16) (log log p ) e − c √ log log p (cid:17)(cid:17) (cid:16) O (cid:16) e − c √ log x (cid:17)(cid:17) ≫ x log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) . This proves the claim. (cid:4) Lemma 8.3. Let p ≥ be a large prime, and let ≤ p . < N < p be a pair of numbers. Then,forany number M < p , X M ≤ u ≤ M + N p X gcd( n,p − Λ( u ) ≫ Ne γ log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) . (72) Proof. The maximal number ω ( p − 1) of prime divisors of highly composite totients p − ω ( p − ≫ log p/ log log p . This implies that z ≍ log p . An application of Lemma 4.2 to the ratioreturns ϕ ( p − p = p − p p − Y q | p − (cid:18) − q (cid:19) ≥ Y q ≤ z (cid:18) − q (cid:19) = 1 e γ log z + O (cid:16) e − c √ log z (cid:17) (73) ≫ e γ log log p + O (cid:16) e − c √ log log p (cid:17) . Let x = M , and y = M + N . Substituting this, the main term reduces to rimitive Roots In Short Intervals M ( x, y, Λ) = X x ≤ u ≤ y p X gcd( n,p − Λ( u )= ϕ ( p − p X x ≤ u ≤ y Λ( u ) (74) ≫ (cid:18) e γ log log p + O (cid:16) e − c √ log log p (cid:17)(cid:19) X x ≤ u ≤ y Λ( u ) . Applying the prime number theorem in short intervals P x ≤ n ≤ y Λ( n ) ≫ y − x = N , see [6], to thelast inequality yields M ( x, y, Λ) ≫ (cid:18) e γ log log p + O (cid:16) e − c √ log log p (cid:17)(cid:19) ( y − x ) (75) ≫ Ne γ log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) . The proves the claim. (cid:4) The previous sections provide sufficient background materials to assemble the proof of the existenceof primitive roots in a short interval [ M, M + N ] for any sufficiently large prime p ≥ 2, a number N ≫ (log p ) ε , and the fixed parameters M ≥ ε > ϕ ( p − /p at the highly compositetotients p − Proof. (Theorem 1.1) Suppose that the short interval [ M, M + N ] = [ x, y ], with 1 ≤ x < y < p ,does not contain a primitive root modulo a large primes p ≥ 2, and consider the sum of thecharacteristic function over the short interval, that is,0 = X x ≤ u ≤ y Ψ( u ) . (76)Replacing the characteristic function, Lemma 3.2, and expanding the nonexistence equation (76)yield 0 = X x ≤ u ≤ y Ψ( u )= X x ≤ u ≤ y p X gcd( n,p − , X ≤ m ≤ p − ψ (( τ n − u ) m ) (77)= c p p X x ≤ u ≤ y, X gcd( n,p − p X x ≤ u ≤ y, X gcd( n,p − , X 2. The main term M ( x, y ) is determined by a finite sum over the trivial additive character ψ = 1, and the error term E ( x, y ) is determined by a finite sum over the nontrivial additive characters ψ ( t ) = e i πt/p = 1. rimitive Roots In Short Intervals X x ≤ u ≤ y Ψ( u ) = M ( x, y ) + E ( x, y ) ≫ (cid:18) e γ log log p + O (cid:16) e − c √ log log p (cid:17)(cid:19) ( y − x ) + O (cid:18) y − xp ε (cid:19) ≫ y − x log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) > , where the implied constant d p = e − γ a p c p ≥ p ≥ 2. However, a short interval [ x, y ] of length y − x = N ≫ (log p ) ε > p ≥ 2. Ergo, the short interval [ M, M + N ] containsa primitive root for any sufficiently large prime p ≥ M ≥ ε > (cid:4) 10 Least Prime Primitive Roots A modified version of the previous result demonstrate the existence of prime primitive roots in aninterval [2 , x ] for any sufficiently large prime p ≥ 2. The analysis below indicates that the localminima of the ratio ϕ ( p − /p at the highly composite totients p − 1, and the number of primes P p ≤ x Λ( n ) are the primary factors determining the size of the interval [2 , x ]. Proof. (Theorem 1.2) Suppose that the interval [2 , x ], with 1 ≤ x < p , does not contain a primeprimitive root modulo a large primes p ≥ 2, and consider the sum of the weighted characteristicfunction over the integers u ≤ x , that is,0 = X u ≤ x Ψ( u )Λ( u ) . (78)Replacing the characteristic function, Lemma 3.2, and expanding the nonexistence equation (76)yield 0 = X u ≤ x Ψ( u )Λ( u )= X u ≤ x p X gcd( n,p − , X ≤ m ≤ p − ψ (( τ n − u ) m ) Λ( u ) (79)= c p p X u ≤ x Λ( u ) X gcd( n,p − p X u ≤ x Λ( u ) X gcd( n,p − , X 2. The main term M ( x, Λ) is determined by a finite sum over the trivial additive character ψ = 1, and the error term E ( x, Λ) is determined by a finite sum over the nontrivial additive characters ψ ( t ) = e i πt/p = 1.An application of Lemma 8.2 to the main term, and an application of Lemma 7.2 to the error term rimitive Roots In Short Intervals X u ≤ y Ψ( u )Λ( u ) = M ( x, Λ) + E ( x, Λ) ≫ x log log p (cid:16) O (cid:16) (log log p ) e − c √ log log p (cid:17)(cid:17) + O (cid:18) xp ε (cid:19) ≫ x log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) > , where the implied constant d p = e − γ a p c p ≥ p ≥ 2. But, an interval [2 , x ] of length x − ≫ (log p ) ε > p ≥ 2. Ergo, the short interval [2 , x ] contains a prime primitive rootfor any sufficiently large prime p ≥ ε > (cid:4) 11 Prime Primitive Roots In Short Intervals The prime number theorem in short intervals P M ≤ n ≤ M + N Λ( n ) ≫ N , see [6]. A modified versionof the previous result will prove the existence of prime primitive roots in short interval [ M, M + N ]for any sufficiently large prime p ≥ N ≫ p . and any M < p . The analysis below indicatesthat the number of primes P M ≤ p ≤ M + N Λ( n ) in a short interval [ M, M + N ] is the primary factordetermining the size of the interval N . The local minima of the ratio ϕ ( p − /p at the highlycomposite totients p − Proof. (Theorem 1.3) Suppose that the interval [2 , x ], with 1 ≤ x < p , does not contain a primeprimitive root modulo a large primes p ≥ 2, and consider the sum of the weighted characteristicfunction over the integers u ≤ x , that is,0 = X M ≤ u ≤ M + N Ψ( u )Λ( u ) . (80)Replacing the characteristic function, Lemma 3.2, and expanding the nonexistence equation (76)yield0 = X M ≤ u ≤ M + N Ψ( u )Λ( u )= X M ≤ u ≤ M + N p X gcd( n,p − , X ≤ m ≤ p − ψ (( τ n − u ) m ) Λ( u ) (81)= c p p X M ≤ u ≤ M + N Λ( u ) X gcd( n,p − p X M ≤ u ≤ M + N Λ( u ) X gcd( n,p − , X 2. The main term M ( N, Λ) is determined by a finite sum over the trivial additive character ψ = 1, and the error term E ( N, Λ) is determined by a finite sum over the nontrivial additive characters ψ ( t ) = e i πt/p = 1.An application of Lemma 8.3 to the main term, and an application of Lemma 7.2 to the error term rimitive Roots In Short Intervals X M ≤ u ≤ M + N Ψ( u )Λ( u ) = M ( N, Λ) + E ( N, Λ) ≫ N log log p (cid:16) O (cid:16) (log log p ) e − c √ log log p (cid:17)(cid:17) + O (cid:18) xp ε (cid:19) ≫ N log log p (cid:18) O (cid:18) e γ log log pe c √ log log p (cid:19)(cid:19) > , where the implied constant d p = e − γ a p c p ≥ p ≥ 2. But, an interval [ M, M + N ] of length N ≫ p . > p ≥ 2. Ergo, the short interval [ M, M + N ] contains a prime primitiveroot for any sufficiently large prime p ≥ M ≥ (cid:4) 12 Problems Exercise 12.1. Determine an explicit interval [ M, M + N ], where N ≥ c (log log p ) ε , c > ε ≤ 2, such the the interval contains a primitive root for any prime p ≥ p , and M ≥ Exercise 12.2. Let a = Q p> (1 − /p ( p − . . . . be the average probabilityof a primitive root modulo a prime p ≥ 2. Determine the length N ≥ M, M + N ] that contains N · (0 . . . . ) k (1 − . . . . ) N − k ≥ k primitive roots, where N ≥ (log log p ) ε ≥ k , k ≥ 1, and ε = 1. Exercise 12.3. Show that the distribution of primitive root modulo a large Germain prime p =2 a q + 1 with q ≥ a ≥ 1, has a normal approximation with mean µ ≈ a − q (1 − /q )and standard deviation σ ≈ p a − q (1 − /q ). Exercise 12.4. Estimate the number of highly composite totients p − X x ≤ p ≤ x + yω ( p − ≫ log p/ log log p , where x ≥ < y < x . 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