Has any professional mathematician ever attempted to solve the Riemann hypothesis using only number theory? [closed]

Closed. This question is opinion-based. It is not currently accepting answers. Want to improve this question? Update the question so it can be answered with facts and citations by editing this post. Closed 2 years ago. Improve this question I have often heard people saying that ”all attempts at solving the Riemann hypothesis using number … Read more

Seek a reference for Theorem 1.2 on p. 6 of the Riemann Hypothesis sourcebook of Borwein et. al

The book “The Riemann Hypothesis, A Resource for the Afficionado and Virtuoso Alike”, Borwein, Choi, Rooney, Weirathmueller, Eds., states on its page six the following theorem (Theorem 1.2): The Riemann hypothesis is equivalent to the statement that for every fixed ϵ>0, lim (Here, \lambda is the Liouville function \lambda: n \mapsto (-1)^{\omega(n)} where \omega(n) is … Read more

Can the Lagarias inequality be written as a “kernel inequality”?

The Lagarias inequality, which is equivalent to the Riemann hypothesis, is: σ(n)≤Hn+exp(Hn)log(Hn)=:L(n) for all natural numbers n, where σ= sum of divisors, Hn=n-th harmonic number. Lagarias inequality is equivalent to: σ(abgcd for all natural numbers a,b. (We see this by plugging in n=\frac{ab}{\gcd(a,b)^2} in one direction, and by setting a=n,b=1 in the other direction.) We … Read more

A Hadamard product of the zeros of the Riemann integral. Does it put any constraints on where the ρ\rho’s can reside in the critical strip?

I have deleted a previous, now obsolete question on the same topic. Take the well-known Riemann integral: π−s2Γ(s2)ζ(s)=∫∞1(xs2−1+x−s2−12)∞∑n=1e−πn2xdx−1s(1−s) and move 1s(1−s) to the RHS to make both sides entire and call it ˆξ(s): ˆξ(s)=π−s2Γ(s2)ζ(s)+1s(1−s)=∫∞1(xs2−1+x−s2−12)∞∑n=1e−πn2xdx The function ˆξ(s)=ˆξ(1−s) induces an infinite number of symmetrical zero (μ) foursomes. After calculating the first 100.000 μ‘s (i.e. 400.000 zeros), … Read more

Estimate on the prime-counting function ψ(x)\psi(x).

There is an elementary statement that I believe I have read somewhere, but I can’t remember where. I’d like to know if the statement is correct (in which case it is surely standard) and if so, where I can find a proof of it. The statement is about the prime-counting function ψ(x)=∑pn<xlogp. It is well-known … Read more

Exact formula for partial sums of Liouville function L(n)L(n) (OEIS sequence A002819)

I am wondering if it is possible to get a useful exact formula, or at least some useful asymptotics, for the partial sums of the Liouville function (OEIS sequence A002819) L(n)=n∑k=1λ(k), by exploiting the identity n∑k=1λ(k)⌊nk⌋=⌊√n⌋. The behavior of L(n) is strongly related to the Riemann hypothesis; see here for a great related MO question … Read more

Riemann Hypothesis and Euler product

It is conjecture that under certain conditions a L-function satisfies RH. Among these conditions there is the necessity for the L-function to have an Euler product. (Some L-functions with a functional equation but without Euler product are known to have non trivial zeros with real part between 1/2 and 1). So the Euler product seems … Read more

Question on coefficient of $\exp(H_n).\log(H_n)$ in Lagarias equivalence of RH

In page 197, Equivalents of the Riemann Hypothesis Vol 1, the following statement caught my eye There is an editorial comment in [102] that includes an observation by the GCHQ Problem Solving Group. They contest that one could replace the number 2 in inequality (7.94) by any constant $K > 1$ at the cost of … Read more

What happens to ζ(s)\zeta(s) when all its ℑ(ρn)\Im(\rho_n) are “scaled” linearly?

I found that the following infinite product with μ=a+nbi and a,b real, s∈C: ∞∏n=1(1−sμ)(1−s1−μ) can be expressed in a closed form (with poles at a,b=0 or a=1 and when a=s): (a2−a)(a2−a+s−s2)Γ(−iab)Γ(−i(a−1)b)Γ(−i(a−s)b)Γ(−i(a+s−1)b) When a=12 this could be further reduced to (poles at s=12 and b=0): 1(2s−1)sinh((2s−1)π2b)sinh(π2b) Encouraged by this result, my wish was to use it … Read more

Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?

The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the critical strip off the critical line? This question came to my mind considering the sequence of trivial zeros … Read more