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distinct primes?
is one of the most important functions in mathematics, deeply connected to number theory, complex analysis, and mathematical physics. Its study has led to profound insights into the distribution of prime numbers and the structure of the complex plane.
with real part greater than 
, the Riemann zeta function is defined by the absolutely convergent series:
. can be extended to a meromorphic function on the entire complex plane, except for a simple pole at 
. The extension is achieved using techniques like:
is the Gamma function. at and 
, and it is crucial for the study of its zeros. has an infinite product representation over prime numbers when :
and the distribution of prime numbers. It expresses the fundamental theorem of arithmetic (unique prime factorization) analytically.
are the Bernoulli numbers.
. are of two types:
.
.
.
as an infinite product:
, where both 
and 
are integers between 
and 
inclusive, such that 
.
can be factored as 
. Thus, we consider three cases: 
, 
, or both.
pairs. The second case yields 
pairs. The third case yields pair. This yields 
pairs.
is a solution then divide by 
, sub 
to get 
, so 
. Casework gives 
.
at the end
and 
can equal 
can equal 
so we have to subtract for overcount.
,assume some constant 
allows us to factor by grouping, thus the expression becomes 
or 
. In order to factor we want 
,we find 
and we can factor and solve as above
We get 
or 
so 
,assume some constant allows us to factor by grouping, thus the expression becomes or . In order to factor we want ,we find and we can factor and solve as above
, where both and are integers between and inclusive, such that .) this equation is equivalent to:
We now have two mutually exclusive cases:
. Since 
, we can constraint 
which rearranges to 
. Likewise, since 
, 
. All such 
with will produce a valid and unique 
satisfying the original equation, so it just remains to count the number of solutions for which is 
.
and 
and we proceed to bound . Since 
, we have 
, and similarly 
. All such 
with will produce a valid and unique solution for except for 
, which violates the constraint. Thus our answer for this case is (subtracting at the end so as not to count ) 
.
solutions.
, and solve over , we have![\[
x=\frac{-y\pm17y}{24}
\]](http://latex.artofproblemsolving.com/a/6/c/a6c330741d1e09ecbd859c435f9b3dbae664ebbc.png)
so either 
, or 
.
if , then we have 
, hence there are 
solutions. Similarly for , there are 
solutions.
as a solution, so the answer is 
.
,
or 
.
,
,
possible ordered pairs. If it is divisible by 
but not 
,
non-zero values of 
ordered pairs.
non-zero values of 
ordered pairs.
non-zero values of 
ordered pairs.
.
.
to 
.
.
,
The pairs of integers that satisfy this equation are 
and 
for 
For the first type of solution, we have 
so the number of solutions of that type are 
Then, for the second type, we get 
so the number of solutions of the second type is 
However, note that the solution 
So when we get that our first case is 3x+2y and 
that then we get 
but we over count (0,0) so we have 
then just find x and y and make sure not to double counted the zero 