Talk:Riemann zeta function

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Gamma Function definition[edit]

In the Riemann zeta function page it is written that:

and that:

.

However looking at the Gamma function definition in Wikipedia one can see that:

.

Correspondingly .

More looking at Riemann's paper he has the s-1 also.

However Riemann instead of Gamma used factorial .

I tried to fix the page but it was rejected as vandalism.

Please correct it or explain to me what am I wrong about.

Adikatz (talk) 06:52, 26 July 2017 (UTC)

In the definition of the zeta function the definition of the Gamma function is wrong. It should be s-1 instead of s. The s-1 should also be in the integral representing the gamma function. I have tried to fix but it was revoked as vandalism.

Vandalism is "malicious or ignorant destruction" (Webster's). In your case the second definition occurs. The gamma function is defined as

. Or . But not .Sapphorain (talk) 07:57, 26 July 2017 (UTC)

Sorry. I was mistaken since I did not pay attention to the fact that the dx was divided by x. Adikatz (talk) 14:21, 26 July 2017 (UTC)

Simple Approximation[edit]


Also, for we have:



where, in both expressions, refers to the Euler-Mascheroni constant. — Craciun Lucian.

Anon edit needs vetting[edit]

The following anonymous edit comes from an IP with a very checkered history. It needs vetting:

Thanks. --Wetman 13:02, 18 Apr 2005 (UTC)

This is surely wrong, as it implies that the \zeta function is zero. Oleg Alexandrov 18:15, 18 Apr 2005 (UTC)
I don't see this exact thing in the article though. Oleg Alexandrov 18:18, 18 Apr 2005 (UTC)
Wetman posted a diff; he is talking about the edit from 9:32, 8 April 2005, which Oleg reverted a few hours later anyway. No matter, the edit was fine, the product is over primes. Case closed. linas 22:09, 18 Apr 2005 (UTC)

Perfect powers[edit]

Is there a formula out there that relates the zeta function to the perfect powers (1, 4, 8, 9, 16, 25, 27, 32, etc) in a similar way that Euler's product formula does for primes? The reason I ask is because I discovered a very simple one a while back, but I can't find information about any other such formulas. Thanks. --Vagodin 14:47, August 21, 2005 (UTC)

Globally convergent series[edit]

The 'globally convergent series' found by Hasse appears to be essentially just the Euler transform applied to the Dirichlet eta function.

I programmed the second of the two Hasse series, and the second one seems to be wrong. It returns large incorrect values. If there's a bug, I must be blind:

The first series (which works correct) uses the following code:

double hasse0(double s, uint64_t limit) {

 double u = 0;
 for (uint64_t n=0; n<limit; ++n) {
   double r = 0;
   for (uint64_t k=0; k<=n; ++k) {
     double t = choose(n,k) / pow(k+1,s);
     if (k%2) r -= t; else r += t;
   }
   u += r/pow(2,n+1);
 }
 return u/(1-pow(2,1-s));

}

The second series (which does not work) uses the exact same template:

double hasse1(double s, uint64_t limit) {

 double u = 0;
 for (uint64_t n=0; n<limit; ++n) {
   double r = 0;
   for (uint64_t k=0; k<=n; ++k) {
     double t = choose(n,k) / pow(k+1,s-1);
     if (k%2) r -= t; else r += t;
   }
   u += r/(n+1);
 }
 return u/(s-1);

}

Some more representations[edit]

Here are some more representations, but I don't know what subsection to put them in, or what commentary to give on them:

for positive integer n

for 0 < Re(s) < 1, where frac is the fractional part

for Re(s) > 1

(Reference: http://mathworld.wolfram.com/RiemannZetaFunction.html)

--AndreRD (talk) 16:29, 27 July 2017 (UTC)

Graph[edit]

Under "Specific values", the graph seems to be of three functions, only one of which is the Zeta function. The other two seem to be based on a finite number of terms of the infinite series in the definition of the Zeta function.

If s=1/2 in Riemann functional equation then the calculated result seems to disprove Riemann's hypothesis[edit]

If the complex number s = 1/2 + 0j is put into the Riemann functional equation after the equation is rearranged so the left hand side reads E{s}/E{1-s} then the left hand side with a value of 1 does not equal the right hand side and this means that the complex part is not zero and so there can be no solution on the 1/2,0 coordinate of the complex plane as Riemann says there is. Have I got this right?

Soopdish (talk) 12:48, 8 August 2017 (UTC)Soopdish

The conjecture does not say "every point on the critical line is a zero", it says "every zero is on the critical line". So the fact that the point you looked at is nonzero is uninteresting and irrelevant. —David Eppstein (talk) 16:29, 8 August 2017 (UTC)