Minimizing GCD sums and applications joint work with Marc Munsch - PowerPoint PPT Presentation

Minimizing GCD sums and applications joint work with Marc Munsch and G´ erald Tenenbaum Symposium in Analytic Number Theory July 2019 R´ egis de la Bret` eche Universit´ e Paris Diderot France regis.delabreteche@imj-prg.fr

– 1 – 1. Previously in G´ al sums : Large values One traditionally defines the G´ al sum ( m, n ) X p mn , S ( M ) := m,n 2 M where ( m, n ) denotes the greatest common divisor of m and n .

– 1 – 1. Previously in G´ al sums : Large values One traditionally defines the G´ al sum ( m, n ) X p mn , S ( M ) := m,n 2 M where ( m, n ) denotes the greatest common divisor of m and n . Key point : no bound on the size of m 2 M , only bound on the size of | M |

– 1 – 1. Previously in G´ al sums : Large values One traditionally defines the G´ al sum ( m, n ) X p mn , S ( M ) := m,n 2 M where ( m, n ) denotes the greatest common divisor of m and n . Key point : no bound on the size of m 2 M , only bound on the size of | M | Improving Bondarenko and Seip (’15, ’17), Tenenbaum and dlB proved when N tends to infinity, p S ( M ) = L ( N ) 2 2+ o (1) , max | M | | M | = N with (s ) log N log 3 N L ( N ) := exp , log 2 N p where we denote by log k the k -th iterated logarithm. Gain : 2 2.

– 2 – The same estimate holds also for � � p ( m, n ) � � X � = L ( N ) 2 2+ o (1) . p mn Q ( M ) := sup c m c n � � � � c 2 C N � m,n 2 M k c k 2 =1

– 2 – The same estimate holds also for � � p ( m, n ) � � X � = L ( N ) 2 2+ o (1) . p mn Q ( M ) := sup c m c n � � � � c 2 C N � m,n 2 M k c k 2 =1 First application Let be � ζ ( 1 � � Z β ( T ) := max 2 + i τ ) (0 6 β < 1 , T > 1) � T β 6 τ 6 T Tenenbaum and dlB proved p 2(1 � β )+ o (1) . Z β ( T ) > L ( T ) p Improvement of Bondarenko and Seip by a 2 extra factor

– 3 – Second application χ ( n ) X L ( s, χ ) := ( χ 6 = χ 0 , < e ( s ) > 0) . n s n > 1 When q is prime and tends to 1 , Tenenbaum and dlB obtained s ( ) log q log 3 q � > L ( q ) 1+ o (1) = exp � L ( 1 � � � � max 2 , χ ) 1 + o (1) . log 2 q χ mod q χ 6 = χ 0 χ ( � 1)=1 p To compare with Hough’s theorems (’16), a log 3 q extra factor

– 4 – Third application Let be X S ( x, χ ) := χ ( n ) , ∆ ( x, q ) := max | S ( x, χ ) | , χ 6 = χ 0 n 6 x χ mod q When e (log q ) 1 / 2+ ε 6 x 6 q/ e (1+ ε ) ω ( q ) , Tenenbaum and dlB had ∆ ( x, q ) � p x L (3 q/x ) p 2+ o (1) ( q ! 1 ) . p Improvement of Hough by an extra factor log 3 (3 q/x ). Valid not only for q prime.

– 5 – 2. Small G´ al sums We define ( m, n ) X p mn c m c n , T ( c ; N ) := T N := N inf T ( c ; N ) , c 2 ( R + ) N m,n 6 N k c k 1 =1 ( m, n ) X V ( c ; N ) := m + nc m c n , V N := N inf V ( c ; N ) , c 2 ( R + ) N m,n 6 N k c k 1 =1

– 5 – 2. Small G´ al sums We define ( m, n ) X p mn c m c n , T ( c ; N ) := T N := N inf T ( c ; N ) , c 2 ( R + ) N m,n 6 N k c k 1 =1 ( m, n ) X V ( c ; N ) := m + nc m c n , V N := N inf V ( c ; N ) , c 2 ( R + ) N m,n 6 N k c k 1 =1 Trivial bounds : V N 6 1 2 T N ⌧ (log N )

– 5 – 2. Small G´ al sums We define ( m, n ) X p mn c m c n , T ( c ; N ) := T N := N inf T ( c ; N ) , c 2 ( R + ) N m,n 6 N k c k 1 =1 ( m, n ) X V ( c ; N ) := m + nc m c n , V N := N inf V ( c ; N ) , c 2 ( R + ) N m,n 6 N k c k 1 =1 Trivial bounds : V N 6 1 2 T N ⌧ (log N ) Theorem 1 (BMT ’19). Let be η := 0 . 16656 . . . < 1 / 6 . There exists c > 0 such that (log N ) η ⌧ V N 6 1 2 T N ⌧ (log N ) η L ( N ) c p log 2 N . with L ( N ) := e

– 6 – Application : Improvement of Burgess’ bound Let X S ( M, N ; χ ) := χ ( n ) , M<n 6 M + N where χ is a Dirichlet character to the modulus p .

– 6 – Application : Improvement of Burgess’ bound Let X S ( M, N ; χ ) := χ ( n ) , M<n 6 M + N where χ is a Dirichlet character to the modulus p . olya and Vinogradov’s bound in O ( p p log p ) is non trivial for N > p 1 / 2+ ε . P´

– 6 – Application : Improvement of Burgess’ bound Let X S ( M, N ; χ ) := χ ( n ) , M<n 6 M + N where χ is a Dirichlet character to the modulus p . olya and Vinogradov’s bound in O ( p p log p ) is non trivial for N > p 1 / 2+ ε . P´ Burgess proved the following inequality S ( M, N ; χ ) ⌧ N 1 � 1 /r p ( r +1) / 4 r 2 (log p ) b ( ⇤ ) ( r > 1) with b = 1. It is non trivial for N > p 1 / 4+ ε .

– 6 – Application : Improvement of Burgess’ bound Let X S ( M, N ; χ ) := χ ( n ) , M<n 6 M + N where χ is a Dirichlet character to the modulus p . olya and Vinogradov’s bound in O ( p p log p ) is non trivial for N > p 1 / 2+ ε . P´ Burgess proved the following inequality S ( M, N ; χ ) ⌧ N 1 � 1 /r p ( r +1) / 4 r 2 (log p ) b ( ⇤ ) ( r > 1) with b = 1. It is non trivial for N > p 1 / 4+ ε . 1 Recently, Kerr, Shparlinski and Yau proved (*) for b = 4 r + o (1). Theorem 2 (BMT’19). For r > 1 , p 6 1 2 N S ( M, N ; χ ) ⌧ N 1 � 1 /r p ( r +1) / 4 r 2 max 1 6 x 6 p T 1 / 2 r . x η Hence we have (*) for b = 2 r + o (1).

– 7 – Let us consider the weighted version of the multiplicative energy ! 2 X X X E ( c ; N ) := c d c t = c d 1 c t 1 c d 2 c t 2 1 6 n 6 N 2 dt = n 1 6 d 1 ,t 1 ,d 2 ,t 2 6 N d,t 6 N d 1 t 1 = d 2 t 2 and define N 2 E ( c ; N ) . E N := inf c 2 ( R + ) N k c k 1 =1

– 7 – Let us consider the weighted version of the multiplicative energy ! 2 X X X E ( c ; N ) := c d c t = c d 1 c t 1 c d 2 c t 2 1 6 n 6 N 2 dt = n 1 6 d 1 ,t 1 ,d 2 ,t 2 6 N d,t 6 N d 1 t 1 = d 2 t 2 and define N 2 E ( c ; N ) . E N := inf c 2 ( R + ) N k c k 1 =1 Let δ := 1 � (1 + log 2 2) / log 2 ⇡ 0 . 08607. Appears in table multiplication problem (Hall, Tenenbaum ’88 and Ford ’06) N 2 � ⇣ n 6 N 2 � � �� H ( N ) := 9 a, b 6 N n = ab (log N ) δ (log 2 N ) 3 / 2 ·

– 7 – Let us consider the weighted version of the multiplicative energy ! 2 X X X E ( c ; N ) := c d c t = c d 1 c t 1 c d 2 c t 2 1 6 n 6 N 2 dt = n 1 6 d 1 ,t 1 ,d 2 ,t 2 6 N d,t 6 N d 1 t 1 = d 2 t 2 and define N 2 E ( c ; N ) . E N := inf c 2 ( R + ) N k c k 1 =1 Let δ := 1 � (1 + log 2 2) / log 2 ⇡ 0 . 08607. Appears in table multiplication problem (Hall, Tenenbaum ’88 and Ford ’06) N 2 � ⇣ n 6 N 2 � � �� H ( N ) := 9 a, b 6 N n = ab (log N ) δ (log 2 N ) 3 / 2 · Theorem 3 (BMT’19). For N > 3 and suitable constant c , we have (log N ) δ (log 2 N ) 3 / 2 ⌧ E N ⌧ (log N ) δ L ( N ) c .

– 8 – First application : Non vanishing of theta functions Balasubramanian and Murty ’92 proved that a positive proportion of characters verify L ( 1 2 , χ ) 6 = 0. We consider χ ( n )e � π n 2 x/p X ( χ 2 X + ϑ ( x ; χ ) = p = { χ mod p : χ 6 = χ 0 , χ ( � 1) = 1 } ) . n > 1

– 8 – First application : Non vanishing of theta functions Balasubramanian and Murty ’92 proved that a positive proportion of characters verify L ( 1 2 , χ ) 6 = 0. We consider χ ( n )e � π n 2 x/p X ( χ 2 X + ϑ ( x ; χ ) = p = { χ mod p : χ 6 = χ 0 , χ ( � 1) = 1 } ) . n > 1 The function ϑ satisfies for any even non-principal character τ ( χ ) ϑ ( x ; χ ) = ( q/x ) 1 / 2 ϑ (1 /x ; χ )

– 8 – First application : Non vanishing of theta functions Balasubramanian and Murty ’92 proved that a positive proportion of characters verify L ( 1 2 , χ ) 6 = 0. We consider χ ( n )e � π n 2 x/p X ( χ 2 X + ϑ ( x ; χ ) = p = { χ mod p : χ 6 = χ 0 , χ ( � 1) = 1 } ) . n > 1 The function ϑ satisfies for any even non-principal character τ ( χ ) ϑ ( x ; χ ) = ( q/x ) 1 / 2 ϑ (1 /x ; χ ) Let M 0 ( p ) = { χ mod p : χ 6 = χ 0 , χ ( � 1) = 1 , ϑ (1; χ ) 6 = 0 } .

– 8 – First application : Non vanishing of theta functions Balasubramanian and Murty ’92 proved that a positive proportion of characters verify L ( 1 2 , χ ) 6 = 0. We consider χ ( n )e � π n 2 x/p X ( χ 2 X + ϑ ( x ; χ ) = p = { χ mod p : χ 6 = χ 0 , χ ( � 1) = 1 } ) . n > 1 The function ϑ satisfies for any even non-principal character τ ( χ ) ϑ ( x ; χ ) = ( q/x ) 1 / 2 ϑ (1 /x ; χ ) Let M 0 ( p ) = { χ mod p : χ 6 = χ 0 , χ ( � 1) = 1 , ϑ (1; χ ) 6 = 0 } . 2 ( p � 1). Checked for 3 6 p 6 10 6 by Molin. Louboutin conjectured M 0 ( p ) = 1 Louboutin and Munsch ’13 showed that M 0 ( p ) � p/ log p . Theorem 4 (BMT’19). With δ := 1 � (1 + log 2 2) / log 2 ⇡ 0 . 08607 , we have p p M 0 ( p ) � q/ 3 ⇧ � (log p ) δ L ( p ) c . E ⌅ p

– 9 – Second application : Lower bounds for low moments of character sums Recently, Harper ’17 announced p � � 1 N X X � � χ ( n ) � ⌧ � � min (log 2 L, log 3 6 p ) 1 / 4 p � 2 � χ 6 = χ 0 n 6 N where L := min { N, p/N } . More than squareroot cancellation !