Формулы для мультипликативных функций, представимых степенями функций разбиений на слагаемые

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    04-Apr-2017

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  • 4

    511.3

    ,

    . . . . .

    , .

    ( )p n n , ( ) ( )1

    n

    np q p n q

    =

    = - . , -

    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )24 2 12 4 6 8 3, , ,q q q qa q b q c q d q

    p q p q p q p q= = = = .

    p , . ( )a n .

    ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

    2 2 216 1 6 5 6 124

    1 1 1 .k k kk k kk k

    qa q q q qp q

    + ++

    = =

    = = = +

    :

    ( ) ( )( ) ( )( ) ( ) ( )

    2

    2

    2 2

    1, 5,7 mod12 ,

    1, 1,11 mod12 ,

    0, 6 5 , 6 1 .

    a m m

    a m m

    a n n k k

    = =

    =

    : ( ) 0a p = ,

    ( ) ( )( )2 1, 5,7 mod12 ,

    1, 1,11 mod12

    pa p

    p

    =

    ( )2,3p ( )b n .

    ( ) ( ) ( )( ) ( )2 26 1 6 1

    22 12

    ,1

    k lk l

    k l

    qb q qp q

    + + ++

    = = Z

    12 1n m= + ( ) ( )1 k lb n += ( ) ( )2 22 6 1 6 1n k l= + + + .

    12 1p m= + ( )224 2 1 ,p x y= + ( )

    ( )( )

    2, 0 mod3 ,

    2, 0 mod3

    xb p

    x

    =

    12 1p m + ( ) 0b p = . ( )c n .

    , ( )

    ( ) ( ) ( )( )2

    6

    2 3 2 123k+1

    k

    qp q3k +1 q

    p q p q =

    Z, ( )

    ( ) ( )( )

    22 2 4

    3 5 2l

    l

    p q p qq q

    p q = =

    Z.

  • 0

    - 8 (12) 2008 .

    ( ) ( )4 6qc q

    p q= =

    ( )( ) ( )

    ( ) ( )( )

    6 2 3 2 12

    2 3 2 12 5 6

    qp q p q p q

    p q p q p q = ( ) ( ) ( ) ( )

    2 2 22 33

    ,

    3k+1 3k+1 ll

    k l k l3k +1 q q 3k +1 q +

    = = Z Z Z

    .

    6 1n m= + ( ) ( )3 1c n k= +( )2 23 1 3n k l= + + .

    6 1p m= + ( )22 2 23 3 1 3p x y k l= + = + + , ( ) ( )2 3 1 2c p k x= + =

    ( )( )

    ( )2 , 1 mod3 ,

    2 , 1 mod3

    x xc p

    x x

    = .

    ( )d n .

    ( ) ( )( ) ( ) ( )( ) ( ) ( )2 22 2 3 1 3 3

    48 3

    ,

    3 1 9 3 3 1

    8

    l k

    k l

    l k lqd q qp q

    + +

    =

    + += =

    Z

    3 1n m= +

    ( )( ) ( ) ( )( )2 2

    ,

    3 1 9 3 3 1

    8k l

    l k ld q

    =

    + +=

    Z,

    4 ( ) ( )2 23 1 3 3n k k= + + . 3 1p m= + 2 23p x y= + ,

    2 2

    2 2

    2 9 , 1 mod3

    2 9 , 1 mod

    x x y xd p

    x x y x

    ,

    3 .

    , ( ) ( ) ( )3 3d p c p pc p= .

    511.3

    . .

    . . .

    - .

    = + . .

    ( ) ( ) ( )1 1 1 1 = + = + = , 1 = ., ( ) ( ) ( ) ( )1n n n = + . ( ), 1m n = . ( ) ( ) ( )mn m n = ,

    ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )1 1 1m m n n mn mn + + = + ( ) ( ) ( )( ) ( ) ( )( )1 0m m n n = .

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