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On complete monotonicity of the Riemann zeta function
Journal of Inequalities and Applications volume 2014, Article number: 15 (2014)
Under the assumption of the Riemann hypothesis for the Riemann zeta function and some Dirichlet L-series we demonstrate that certain products of the corresponding zeta functions are completely monotonic. This may provide a method to disprove a certain Riemann hypothesis numerically.
The Riemann zeta function can be defined by
and on the rest of the complex plane by analytic continuation. It is known that the extended is meromorphic with infinitely many zeros at for (a.k.a trivial zeros) and with infinitely many zeros within the vertical strip (nontrivial zeros). The Riemann hypothesis for says that all nontrivial zeros are actually on the critical line .
is an even entire function of order 1. The celebrated Riemann hypothesis is equivalent to the statement that has only real zeros.
is an even entire function of order 1. The Riemann hypothesis for is equivalent to having only real zeros.
Given real numbers a, b with and an indefinite differentiable real valued function on , is called completely monotonic on if for all and . In this work, under the assumptions of the Riemann hypothesis for the Riemann zeta function and certain L-series, we apply the ideas from [8, 9] to prove that some products of these zeta functions are completely monotonic. This complete monotonicity may provide a method to disprove a certain Riemann hypothesis via numerical methods.
2 Main results
Lemma 1 Given a non-increasing sequence of positive numbers such that
then, the entire function
is completely monotonic on .
Proof It is a direct consequence of Theorem 1 of . □
Assuming the Riemann hypothesis is true, we list all positive zeros of as
and is approximately 14.1347. Then,
for , where . In fact, for any positive integer and assume that is a primitive ℓ th root of unity; then we have
Corollary 2 Under the Riemann hypothesis, let be the least positive zeros of ; then the function is completely monotonic for , is completely monotonic for , and is completely monotonic for . Let be a primitive ℓth root of unity for some positive integer ℓ; then is completely monotonic for .
Proof Notice that is a positive constant, and the claims are obtained by applying Corollary 1 to equations (2.5)-(2.8). □
Assuming the Riemann hypothesis for , we list all the positive zeros for as 
otherwise , which is clearly false. Thus,
for . Furthermore,
for , where . Let be a primitive ℓ th root of unity for some positive integer ℓ; then we have
Corollary 3 Assume that the Riemann hypothesis is true for and is the least positive zero of ; then the function is completely monotonic for , is completely monotonic for , and is completely monotonic for . Let be a primitive ℓth root of unity for some positive integer ℓ, then is completely monotonic for .
Proof These are consequences of Lemma 1 and equations (2.12)-(2.15). □
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This research is partially supported by National Natural Science Foundation of China, grant No. 11371294.
The author declares that they have no competing interests.
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Zhang, R. On complete monotonicity of the Riemann zeta function. J Inequal Appl 2014, 15 (2014). https://doi.org/10.1186/1029-242X-2014-15
- Riemann zeta function
- Dirichlet series
- Riemann hypothesis
- complete monotonic functions