r:=2; n := 12; p:=2; PRIME 2

NiM+SSJyRzYiIiIjNiM+SSJuRzYiIiM3NiM+SSJwRzYiIiIj

f := x-> (1-x)*cos(Pi*x) + (1/Pi)*sin(Pi*x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmIiIiRi85JCEiIkYvLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjVJKF9zeXNsaWJHRiU2IyomSSNQaUdGNUYvRjBGL0YvRi8qJkY5RjEtSSRzaW5HRjRGN0YvRi9GJUYlRiU=

bnd := b->r*Pi/2+ n*(gamma+log(8*Pi))-(n/2)*int((1-f(x/b))/sinh(x/2),x=0..b) -(n/2)*int(csch(x/2),x=b..infinity)-(r/2)*int((1-f(x/b))/cosh(x/2),x=0..b) -(r/2)*int(sech(x/2),x=b..infinity)-4*int(f(x/b)*cosh(x/2),x=0..b);

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

lcl := b->2*sum(log(p)*f(m*log(p)/b)*p^(-m/2),m=1..6);

NiM+SSRsY2xHNiJmKjYjSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJHN1bUc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiQqKC1JJGxvZ0dGLzYjSSJwR0YlIiIiLUkiZkdGJTYjKihJIm1HRiVGOEY0Rjg5JCEiIkY4KUY3LCRGPSNGPyIiI0Y4L0Y9O0Y4IiInRkNGJUYlRiU=

num := 1; NUMBER OF PRIMES IN S

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLy85eVMqM1AlISM4

evalf(bb/log(p)); CHECK UPPER LIMIT ON SUM IN LCL

NiMkIi9WeTVzJ2VJJyEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

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num := 2;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiL201Ij4kZWlYISM4

evalf(bb/log(p));

NiMkIi9JWFZoVCNlJyEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

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num :=3;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiL0A7KikqUjF1JSEjOA==

evalf(bb/log(p));

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evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi8iSDY3RiQ+PCIiIQ==

n := 12; p:=3; PRIME 3

NiM+SSJuRzYiIiM3NiM+SSJwRzYiIiIk

f := x-> (1-x)*cos(Pi*x) + (1/Pi)*sin(Pi*x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmIiIiRi85JCEiIkYvLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjVJKF9zeXNsaWJHRiU2IyomSSNQaUdGNUYvRjBGL0YvRi8qJkY5RjEtSSRzaW5HRjRGN0YvRi9GJUYlRiU=

lcl := b->2*sum(log(p)*f(m*log(p)/b)*p^(-m/2),m=1..3);

NiM+SSRsY2xHNiJmKjYjSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJHN1bUc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiQqKC1JJGxvZ0dGLzYjSSJwR0YlIiIiLUkiZkdGJTYjKihJIm1HRiVGOEY0Rjg5JCEiIkY4KUY3LCRGPSNGPyIiI0Y4L0Y9O0Y4IiIkRkNGJUYlRiU=

num := 1; NUMBER OF PRIMES IN S

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiL043JEdFKnBWISM4

evalf(bb/log(p));

NiMkIi9uTz8keXcoUiEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi8mWytwKTRvXCEiIw==

num := 2;

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lcl := b->2*sum(log(p)*f(m*log(p)/b)*p^(-m/2),m=1..4);

NiM+SSRsY2xHNiJmKjYjSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJHN1bUc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiQqKC1JJGxvZ0dGLzYjSSJwR0YlIiIiLUkiZkdGJTYjKihJIm1HRiVGOEY0Rjg5JCEiIkY4KUY3LCRGPSNGPyIiI0Y4L0Y9O0Y4IiIlRkNGJUYlRiU=

bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLzJsKWY5K2MlISM4

evalf(bb/log(p));

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evalf(exp(bnd(bb)+num*lcl(bb)));

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num :=3;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLyNvNGFpc3QlISM4

evalf(bb/log(p));

NiMkIi8jPjs+VT9KJSEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi8jZW5ENEo6JyEiIg==

n := 12; p:=5; PRIME 5

NiM+SSJuRzYiIiM3NiM+SSJwRzYiIiIm

f := x-> (1-x)*cos(Pi*x) + (1/Pi)*sin(Pi*x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmIiIiRi85JCEiIkYvLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjVJKF9zeXNsaWJHRiU2IyomSSNQaUdGNUYvRjBGL0YvRi8qJkY5RjEtSSRzaW5HRjRGN0YvRi9GJUYlRiU=

lcl := b->2*sum(log(p)*f(m*log(p)/b)*p^(-m/2),m=1..2);

NiM+SSRsY2xHNiJmKjYjSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJHN1bUc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiQqKC1JJGxvZ0dGLzYjSSJwR0YlIiIiLUkiZkdGJTYjKihJIm1HRiVGOEY0Rjg5JCEiIkY4KUY3LCRGPSNGPyIiI0Y4L0Y9O0Y4RkNGQ0YlRiVGJQ==

num := 1; NUMBER OF PRIMES IN S

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLyNSVT9oY08lISM4

evalf(bb/log(p));

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evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi81TlIwJ2ZRJCEiIw==

num := 2;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLyxiIUheUmIlISM4

evalf(bb/log(p));

NiMkIi87NUYhSCZIRyEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi9cYSd6MicpKnkhIiM=

num :=3;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLykqSCJRTydHWiEjOA==

evalf(bb/log(p));

NiMkIi82YHdwMVFIISM4

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi9CeDg1WFA+ISIi

n := 12; p:=7; PRIME 7

NiM+SSJuRzYiIiM3NiM+SSJwRzYiIiIo

f := x-> (1-x)*cos(Pi*x) + (1/Pi)*sin(Pi*x);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmIiIiRi85JCEiIkYvLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjVJKF9zeXNsaWJHRiU2IyomSSNQaUdGNUYvRjBGL0YvRi8qJkY5RjEtSSRzaW5HRjRGN0YvRi9GJUYlRiU=

lcl := b->2*sum(log(p)*f(m*log(p)/b)*p^(-m/2),m=1..2);

NiM+SSRsY2xHNiJmKjYjSSJiR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJC1JJHN1bUc2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlNiQqKC1JJGxvZ0dGLzYjSSJwR0YlIiIiLUkiZkdGJTYjKihJIm1HRiVGOEY0Rjg5JCEiIkY4KUY3LCRGPSNGPyIiI0Y4L0Y9O0Y4RkNGQ0YlRiVGJQ==

num := 1; NUMBER OF PRIMES IN S

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLzZsYyVIWU4lISM4

evalf(bb/log(p));

NiMkIi92U2ZvJHlCIyEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi9iaU02MUJGISIj

num := 2;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiL15qJCoza05YISM4

evalf(bb/log(p));

NiMkIi9OI29MZTNMIyEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi8vYjlTQCEzJiEiIw==

num :=3;

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLy8uVXB0MlohIzg=

evalf(bb/log(p));

NiMkIi9CIzNAKUg+QyEjOA==

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi90RjlfZkgqKiEiIw==

num :=0; BOUND WITHOUT LOCAL CORRECTIONS

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bb := fsolve(diff(bnd(b)+num*lcl(b),b)=0,b,0..8);

NiM+SSNiYkc2IiQiLyYpejVdXGtUISM4

evalf(exp(bnd(bb)+num*lcl(bb)));

NiMkIi8qR0wjPSVcYCIhIiM=