An asymptotic expansion of Selberg's central limit theorem near the critical line
aa r X i v : . [ m a t h . N T ] J a n AN ASYMPTOTIC EXPANSION OF SELBERG’S CENTRAL LIMITTHEOREM NEAR THE CRITICAL LINE
YOONBOK LEE
Abstract.
We find an asymptotic expansion of Selberg’s central limit theorem for theRiemann zeta function on σ = + (log T ) − θ and t ∈ [ T, T ], where 0 < θ < is a constant. Introduction
Let θ > σ T := σ T ( θ ) = + (log T ) − θ throughout the paper. Selberg’s central limittheorem (Theorem 2 in [7]) says that for ≤ σ ≤ σ T , the functionlog ζ ( σ + it ) q π P p
Let < θ < , a < b and c < d be real numbers. There exist constants ǫ, η > and a sequence { d k,ℓ } k,ℓ ≥ of real numbers such that T meas { t ∈ [ T, T ] : log ζ ( σ T + it ) √ πψ T ∈ [ a, b ] × [ c, d ] } Date : January 5, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Selberg’s central limit theorem, Riemann zeta function.This work was supported by Incheon National University RIBS Grant in 2020. = X k + ℓ ≤ ǫψ T d k,ℓ √ ψ T k + ℓ Z dc Z ba e − π ( x + y ) H k ( √ πx ) H ℓ ( √ πy ) dxdy + O (cid:18) T ) η (cid:19) as T → ∞ , where the functions H n ( x ) are the Hermite polynomials defined by H n ( x ) := ( − n e x d n dx n ( e − x ) . (1.3) Moreover, d , = 1 and d k,ℓ = 0 for k + ℓ = 1 , { d k,ℓ } is defined by its generating function in (2.11). Since d k,ℓ = 0 for k + ℓ = 1 , T meas { t ∈ [ T, T ] : log ζ ( σ T + it ) √ πψ T ∈ [ a, b ] × [ c, d ] } = Z dc Z ba e − π ( x + y ) dxdy + O (cid:18) T ) / (cid:19) . Remark that for each k, ℓ ≥
1, the double integral in Theorem 1.2 is integrable by the identity Z x x e − x H n ( x ) dx = e − x H n − ( x ) − e − x H n − ( x )for n ≥ Estimates on the random model
The random Riemann zeta function is defined by the product ζ ( σ : X ) := Y p (cid:18) − X ( p ) p σ (cid:19) − , where X ( p ) is independent and identically distributed random variables on the unit circle | z | = 1 assigned for each prime p . The product converges almost surely for σ > . If σ > is not too close to , then the distribution of the random model log ζ ( σ : X ) approximatesthat of log ζ ( σ + it ). More precisely, the discrepancy defined by D σ ( T ) := sup R (cid:12)(cid:12)(cid:12)(cid:12) T meas { t ∈ [ T, T ] : log ζ ( σ + it ) ∈ R} − P [log ζ ( σ : X ) ∈ R ] (cid:12)(cid:12)(cid:12)(cid:12) is small for σ ≥ σ T , where the supremum is taken over rectangles R with sides parallel tothe coordinate axes. Lamzouri, Lester and Radziwi l l in [3] showed that D σ ( T ) = O (cid:18) T ) σ (cid:19) holds for fixed σ > . By the same method, Ha and Lee in [1] showed that for each 0 < θ < ,there is a constant η > D σ T ( T ) = O η (cid:18) T ) η (cid:19) . (2.1)Note that there are several earlier results of Matsumoto [4], [5] and Harman and Matsumoto[2] in this direction, but their estimates depend on the size of R .Define Φ rand σ T ( B ) := P [log ζ ( σ T : X ) ∈ B ] N ASYMPTOTIC EXPANSION OF SELBERG’S CENTRAL LIMIT THEOREM 3 for a Borel set
B ⊂ C . By Proposition 3.1 of [1], this measure has a density function F σ T such that P [log ζ ( σ T : X ) ∈ B ] = Z Z B F σ T ( x, y ) dxdy (2.2)holds for any region B . Since we have1 T meas { t ∈ [ T, T ] : log ζ ( σ T + it ) √ πψ T ∈ [ a, b ] × [ c, d ] } = P (cid:20) log ζ ( σ T : X ) √ πψ T ∈ [ a, b ] × [ c, d ] (cid:21) + O (cid:18) T ) η (cid:19) = Z d √ πψ T c √ πψ T Z b √ πψ T a √ πψ T F σ T ( x, y ) dxdy + O (cid:18) T ) η (cid:19) (2.3)by (2.1) and (2.2), it is enough to find an asymptotic for F σ T ( x, y ) to prove Theorem 1.2.Since we have F σ T ( x, y ) = Z Z R b Φ rand σ T ( u, v ) e − πi ( ux + vy ) dudv (2.4)by the Fourier inversion, we estimate the Fourier transform b Φ rand σ T beforehand.By the definition we have b Φ rand σ T ( u, v ) = E (cid:2) e πi ( u Re(log ζ ( σ T : X ))+ v Im(log ζ ( σ T : X )) (cid:3) = Y p J ( πu, πv, p − σ T ) , where J ( u, v, w ) := E (cid:2) e − i ( u Re log(1 − wX )+ v Im log(1 − wX )) (cid:3) . Then we have the following lemma, which follows immediately from Lemma 3.3 in [1] andits proof.
Lemma 2.1.
Let < r < and C r = − r log(1 − r ) . Then we have series expansions J ( u, v, w ) = 1 + X k,ℓ ≥ i k + ℓ k ! ℓ ! a k,ℓ ( w )( u + iv ) k ( u − iv ) ℓ for any u, v ∈ R and < w < , and log J ( u, v, w ) = X k,ℓ ≥ i k + ℓ k ! ℓ ! b k,ℓ ( w )( u + iv ) k ( u − iv ) ℓ (2.5) for u + v ≤ (2 rC r ) − and | w | ≤ r . The coefficients a k,ℓ ( w ) and b k,ℓ ( w ) are defined by a k,ℓ ( w ) = X n ≥ max( k,ℓ ) (cid:18) X n + ··· + n k = nn i ≥ n · · · n k (cid:19)(cid:18) X m + ··· + m ℓ = nm i ≥ m · · · m ℓ (cid:19) w n , (2.6) b k,ℓ ( w ) = X n ≤ min( k,ℓ ) ( − n − n X k + ··· k n = kℓ + ··· + ℓ n = ℓk i ,ℓ i ≥ (cid:18) kk , . . . , k n (cid:19)(cid:18) ℓℓ , . . . , ℓ n (cid:19) a k ,ℓ ( w ) · · · a k n ,ℓ n ( w ) , (2.7) and satisfy (1) b k,ℓ ( w ) is real and b , ( w ) = P m ≥ m w m , (2) a k,ℓ ( w ) , b k,ℓ ( w ) ≪ k,ℓ w k,ℓ ) , Y. LEE (3) a k,ℓ ( w ) = a ℓ,k ( w ) and b k,ℓ ( w ) = b ℓ,k ( w ) , (4) 0 < a k,ℓ ( w ) ≤ C k + ℓr w k + ℓ and | b k,ℓ ( w ) | ≤ C k + ℓr min( k, ℓ ) k + ℓ w k + ℓ for < w ≤ r . Lemma 2.2.
Let ψ T be as in (1.2) . Define ˜ b k,ℓ := ( πi ) k + ℓ k ! ℓ ! X p b k,ℓ ( p − / ) , where b k,ℓ ( w ) is defined in (2.7) . Then there is a constant δ > X p log J ( πu, πv, p − σ T ) = − π ( u + v ) ψ T + X k,ℓ ≥ k + ℓ ≥ ˜ b k,ℓ ( u + iv ) k ( u − iv ) ℓ + O (cid:18) T ) θ (cid:19) for u + v ≤ δ .Proof. For any ǫ >
0, there is a constant C ( ǫ ) > x ≤ C ( ǫ ) x ǫ for all x ≥ < a k,ℓ ( p − / ) − a k,ℓ ( p − σ T ) ≤ C (2 ǫ )(log T ) θ X n ≥ max( k,ℓ ) (cid:18) X n + ··· + n k = nn i ≥ n · · · n k (cid:19)(cid:18) X m + ··· + m ℓ = nm i ≥ m · · · m ℓ (cid:19) p (1 − ǫ ) n = 2 C (2 ǫ )(log T ) θ a k,ℓ ( p − / ǫ ) ≤ C (2 ǫ )(log T ) θ C k + ℓr p − (1 / − ǫ )( k + ℓ ) for k, ℓ ≥
1, any prime p and any ǫ > r = 2 − / ǫ . Since0 < n Y j =1 a k j ,ℓ j ( p − / ) − n Y j =1 a k j ,ℓ j ( p − σ T ) ≤ n X j =1 ( a k j ,ℓ j ( p − / ) − a k j ,ℓ j ( p − σ T )) Y i = j a k i ,ℓ i ( p − / ) ≤ n X j =1 C (2 ǫ )(log T ) θ C k j + ℓ j r p − (1 / − ǫ )( k j + ℓ j ) Y i = j C k i + ℓ i r p − / k i + ℓ i ) ≤ n C (2 ǫ )(log T ) θ C P j ( k j + ℓ j ) r p − (1 / − ǫ ) P j ( k j + ℓ j ) , by (2.7) we have | b k,ℓ ( p − / ) − b k,ℓ ( p − σ T ) |≤ X n ≤ min( k,ℓ ) n X k + ··· k n = kℓ + ··· + ℓ n = ℓk i ,ℓ i ≥ (cid:18) kk , . . . , k n (cid:19)(cid:18) ℓℓ , . . . , ℓ n (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 a k j ,ℓ j ( p − / ) − n Y j =1 a k j ,ℓ j ( p − σ T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ≤ min( k,ℓ ) n k + ℓ C (2 ǫ )(log T ) θ C k + ℓr p − (1 / − ǫ )( k + ℓ ) ≤ C (2 ǫ )(log T ) θ (min( k, ℓ )) k + ℓ C k + ℓr p − (1 / − ǫ )( k + ℓ ) . By Stirling’s formula and choosing 0 < ǫ < , we have N ASYMPTOTIC EXPANSION OF SELBERG’S CENTRAL LIMIT THEOREM 5 X p X k + ℓ ≥ π k + ℓ ( u + v ) ( k + ℓ ) / k ! ℓ ! | b k,ℓ ( p − / ) − b k,ℓ ( p − σ T ) |≪ T ) θ X p X k + ℓ ≥ (cid:18) π √ δ eC r p / − ǫ (cid:19) k + ℓ ≪ T ) θ (2.8)for u + v ≤ δ , where δ is a constant satisfying π √ δ eC r / − ǫ <
1. By (2.5), we have X p log J ( πu, πv, p − σ T ) = X k,ℓ ≥ ( πi ) k + ℓ k ! ℓ ! ( u + iv ) k ( u − iv ) ℓ X p b k,ℓ ( p − σ T ) (2.9)for u + v ≤ δ if δ ≤ ( π √ C / √ ) − . By (2.8), (2.9) and the identity ψ T = P p b , ( p − σ T ),the lemma follows. (cid:3) Lemma 2.3.
There are constant δ , δ > and a sequence { d k,ℓ } k,ℓ ≥ of real numbers suchthat b Φ rand σ T ( u, v ) = e − π ( u + v ) ψ T (cid:18) X k,ℓ ≥ (2 πi ) k + ℓ d k,ℓ u k v ℓ + O (cid:18) T ) θ (cid:19)(cid:19) for u + v ≤ δ , where d , = 1 , d k,ℓ = 0 for k + ℓ = 1 , and d k,ℓ = O ( δ − ( k + ℓ )3 ) for k + ℓ ≥ .Proof. By Lemma 2.2, we have b Φ rand σ T ( u, v ) = Y p J ( πu, πv, p − σ T )= e − π ( u + v ) ψ T exp (cid:18) X k,ℓ ≥ k + ℓ ≥ ˜ b k,ℓ ( u + iv ) k ( u − iv ) ℓ (cid:19)(cid:18) O (cid:18) T ) θ (cid:19)(cid:19) for u + v ≤ δ . By Lemma 2.1 and Stirling’s formula, the sum X k,ℓ ≥ k + ℓ ≥ | ˜ b k,ℓ || ( u + iv ) k ( u − iv ) ℓ | ≤ X k,ℓ ≥ k + ℓ ≥ ( π √ u + v ) k + ℓ k ! ℓ ! X p C k + ℓ / √ min( k, ℓ ) k + ℓ p − ( k + ℓ ) / ≪ X p X k,ℓ ≥ k + ℓ ≥ (cid:18) C / √ πe √ u + v √ p (cid:19) k + ℓ (2.10)is convergent and bounded for u + v ≤ δ if C / √ πe √ δ < √
2. Thus we can find a powerseries expansion of g ( u, v ) := exp (cid:18) X k,ℓ ≥ k + ℓ ≥ ˜ b k,ℓ ( u + iv ) k ( u − iv ) ℓ (cid:19) for u + v ≤ δ . Let b ′ k,ℓ = ˜ b k,ℓ (2 πi ) − k − ℓ , x = 2 πiu and y = 2 πiv , then we see that g (cid:18) x πi , y πi (cid:19) = exp (cid:18) X k,ℓ ≥ k + ℓ ≥ b ′ k,ℓ ( x + iy ) k ( x − iy ) ℓ (cid:19) . Y. LEE
Since b ′ k,ℓ = b ′ ℓ,k and b ′ k,ℓ is real for every k, ℓ by Lemmas 2.1 and 2.2, the sum P k,ℓ ≥ k + ℓ ≥ b ′ k,ℓ ( x + iy ) k ( x − iy ) ℓ is a power series in x and y with real coefficients. Thus, we can find a sequence { d k,ℓ } k,ℓ ≥ of real numbers such that X k,ℓ ≥ d k,ℓ x k y ℓ := exp (cid:18) X k,ℓ ≥ k + ℓ ≥ b ′ k,ℓ ( x + iy ) k ( x − iy ) ℓ (cid:19) . (2.11)This is equivalent to g ( u, v ) = X k,ℓ ≥ (2 πi ) k + ℓ d k,ℓ u k v ℓ . Finding first few terms is easy. We see that d , = 1 and d k,ℓ = 0 for k + ℓ = 1 ,
2. Let δ be a constant such that 0 < δ < √ eC / √ . Since g ( u, v ) is bounded for | u | , | v | ≤ δ π by similarestimations to (2.10), we have d k,ℓ = 1(2 πi ) k + ℓ +2 I | u | = δ π I | v | = δ π g ( u, v ) u k +1 v ℓ +1 dvdu = O ( δ − ( k + ℓ )3 ) . (cid:3) Lemma 2.4.
Let { d k,ℓ } k,ℓ ≥ be the sequence of real numbers in Lemma 2.3. There existconstants ǫ, η > such that F σ T ( x, y ) = e − ( x + y ) /ψ T X k + ℓ ≤ ǫψ T d k,ℓ π √ ψ T k + ℓ +2 H k (cid:18) x √ ψ T (cid:19) H ℓ (cid:18) y √ ψ T (cid:19) + O (cid:18) T ) η (cid:19) for all x, y ∈ R , where the Hermite polynomials H n ( x ) are defined in (1.3) .Proof. Let δ be a constant satisfying 0 < δ < min( δ , δ (2 π ) − ). By applying Lemma 3.5of [1] to (2.4), there is a constant η > F σ T ( x, y ) = Z Z u + v ≤ δ b Φ rand σ T ( u, v ) e − πi ( ux + vy ) dudv + O (cid:18) T ) η (cid:19) . Let ǫ be a constant satisfying 0 < ǫ < e δ . By Lemma 2.3, we have F σ T ( x, y ) = X k,ℓ ≥ (2 πi ) k + ℓ d k,ℓ Z Z u + v ≤ δ e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv + O (cid:18) T ) η (cid:19) = X k + ℓ ≤ ǫψ T (2 πi ) k + ℓ d k,ℓ Z Z u + v ≤ δ e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv + O (cid:18) X k + ℓ>ǫψ T (2 π ) k + ℓ δ ( k + ℓ ) / δ k + ℓ ψ T + 1(log T ) η (cid:19) where η = min( η , θ ). Since δ < δ (2 π ) − , the O -term is O ((log T ) − η ) for some η > Z Z R e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv − Z Z u + v >δ e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv. (2.12) N ASYMPTOTIC EXPANSION OF SELBERG’S CENTRAL LIMIT THEOREM 7
The second integral in (2.12) is (cid:12)(cid:12)(cid:12)(cid:12) Z Z u + v >δ e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z u + v >δ e − π ( u + v ) ψ T | u | k | v | ℓ dudv ≤ (cid:18) Z Z u + v >δ e − π ( u + v ) ψ T dudv (cid:19) / (cid:18) Z Z R e − π ( u + v ) ψ T u k v ℓ dudv (cid:19) / = e − π δ ψ T √ π ( π √ ψ T ) k + ℓ +2 r Γ( k + 12 )Γ( ℓ + 12 )by the Cauchy-Schwartz inequality. By Stirling’s formula, the above is ≪ e − π δ ψ T π √ ψ T ) k + ℓ +2 ( k + 1 / k/ ( ℓ + 1 / ℓ/ e ( k + ℓ ) / ≤ e − π δ ψ T π ψ T (cid:18) √ ǫπ √ e (cid:19) k + ℓ for k + ℓ ≤ ǫψ T . Since d k,ℓ = O ( δ − ( k + ℓ )3 ) by Lemma 2.3, the contribution of the secondintegral in (2.12) to F σ T ( x, y ) is O (cid:18) X k + ℓ>ǫψ T e − π δ ψ T ψ T (cid:18) √ ǫδ √ e (cid:19) k + ℓ (cid:19) = O (cid:18) T ) η (cid:19) for some η > ǫ < e δ . Therefore, we have F σ T ( x, y ) = X k + ℓ ≤ ǫψ T (2 πi ) k + ℓ d k,ℓ Z Z R e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv + O (cid:18) T ) η (cid:19) = e − ( x + y ) /ψ T X k + ℓ ≤ ǫψ T d k,ℓ π √ ψ T k + ℓ +2 H k (cid:18) x √ ψ T (cid:19) H ℓ (cid:18) y √ ψ T (cid:19) + O (cid:18) T ) η (cid:19) with η = min( η , η ). The last equality holds by Z Z R e − π ( u + v ) ψ T u k v ℓ e − πi ( ux + vy ) dudv = 1( − πi ) k + ℓ ∂ k + ℓ ∂x k ∂y ℓ Z Z R e − π ( u + v ) ψ T e − πi ( ux + vy ) dudv = 1( − πi ) k + ℓ ∂ k + ℓ ∂x k ∂y ℓ (cid:18) πψ T e − ( x + y ) /ψ T (cid:19) = 1 πψ T πi √ ψ T ) k + ℓ e − ( x + y ) /ψ T H k (cid:18) x √ ψ T (cid:19) H ℓ (cid:18) y √ ψ T (cid:19) . (cid:3) Proof of Theorem 1.2.
The theorem holds by (2.3) and Lemma 2.4. (cid:3)
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Department of Mathematics, Research Institute of Basic Science, Incheon National Uni-versity, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Korea
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