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An MSE user and myself were discussing the Dirichlet L-function from one of their posts when we thought of a generalisation to the Riemann Hypothesis.

The Dirichlet L-series is defined by $$L(s,\chi)=\sum_{n=1}^\infty\frac{\chi(n)}{n^s}=\sum_{n=1}^\infty\frac{\Re\chi(n)+i\Im\chi(n)}{n^s}$$ and $\chi$ is the Dirichlet character, which can be extended through analytic continuation.

The Generalised Riemann Hypothesis is equivalent to proving that $L(s,\chi)=0\implies \Re s=1/2$ for every $s,\chi$. Now what if we further extend this hypothesis? Clearly $\chi$ is not real for all values of $n$ modulo $k$ for some $k\in\Bbb Z^+$, so what if we define the function $$L_k(s,\chi)=\sum_{n=1}^\infty\frac{(\Re\chi(n))^k+(\Im\chi(n))^k}{n^{s/k}}$$ so that the numerator is always real? It can be shown that it converges for $\Re s>1$, and we also have the fact that $\exists k\in\Bbb Z^+: \chi(n)=\chi(n+k)\,\forall n$. I have added the $s/k$ power in the denominator as otherwise, the numerator quickly tends to zero.

Questions:

  1. What do the zeros of $L_k$ look like? A complex plot would be preferable.

  2. Is there a link/expression between this and the standard Dirichlet L-function?

Thanks.

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    $\begingroup$ related math.stackexchange.com/questions/15350/… $\endgroup$ – Ultradark Apr 27 at 20:50
  • $\begingroup$ Your statement of GRH is incorrect: only for primitive $\chi$ do we expect GRH, as other Dirichlet $L$-functions have extra zeros on the imaginary axis. And you are leaving out the trivial zeros. For $k=2$ your numerator is $|\chi(n)|^2$, which will be 1 except periodically, so that $L$-function is $\zeta(s)$ with missing Euler factors. For other $k$, is there a real purpose to what you are doing or are you just making up conjectures? I see no reason these series for $k\not= 2$ should have any reasonable properties. With no Euler product or functional equation, GRH is totally unrealistic. $\endgroup$ – KCd Apr 27 at 21:34
  • $\begingroup$ It is desired, to introduce a real integer sequence that exists in the numerator of a zeta function that is non-periodic, and to examine such roots of said function in terms of a complex plot. In order to generate said non-periodic sequence, it has been proposed that using a dirichlet character in the numerator could suffice. The goal is simple: to find a generator that generates a pseudorandom sequence in the numerator of a zeta function, examine the zeros of the function, and explore how this function relates to the zeta function and dirichlet L-functions. $\endgroup$ – Ultradark Apr 27 at 22:25
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    $\begingroup$ It is a linear combination of L-functions, there is a whole theory about that (extended selberg class). @Kcd Its functional equation expresses $F(1-s)$ as the sum of two linear combination of L-functions - gamma factors. No Euler product implies a bunch of off- line zeros. There is still a GRH for those linear combinations : instead of looking at the zeros (the singularities of the logarithm of the Euler product) look at the $s$ such that $\lim_{N\to\infty}\sum_{n=1,gcd(n,N)=1}^\infty a_nn^{-s} $ converges, to $1$. If there is no pole at $s=1$ then GRH is that it converges for $\Re(s) >1/2$. $\endgroup$ – reuns Apr 27 at 22:58
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    $\begingroup$ Also the OP replaced $\frac{1}{n^s}$ by $\frac{1}{n^{s/k}}$ indicating some confusion : the Dirichlet characters are periodic (and completely multiplicative), so $n \mapsto (\Re\chi(n))^k+(\Im\chi(n))^k$ is periodic, thus it doesn't make sense to scale $s$ by $1/k$. And $\frac{-C}{s-1}+\sum_{n=1}^\infty ((\Re\chi(n))^k+(\Im\chi(n))^k) n^{-s}$ is entire with $C$ the mean value of the sequence, if the mean value is $0$ it converges for $ \Re(s) > 0$, otherwise for $\Re(s) > 1$. $\endgroup$ – reuns Apr 27 at 23:12

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