# 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 = .