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Consider a primitive Dirichlet Character $\chi$ (non principal) and the partial Euler product attached to the L-function $L(\chi,s)$ ($p_i$ are the prime numbers) :

$$P(\chi,N)=\prod_{i=1}^{N} \frac{1}{1-\chi(p_i) p_i^{-s}} $$

According to my knowledge, this product (for $0<Re(s)<1$) is not known to converge. I expect it to have a quite erratic behavior (for reason explained here - concerning Zeta case : Is the Euler product formula always divergent for 0<Re(s)<1?)

But can we expect $|P(\chi,N)|$ to be lower than a fixed constant for an infinity of N values ? What is known on the behavior of this partial product ?

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  • $\begingroup$ $P(\chi,N)$ depends on $s$, of course. Do you want to fix $s$, and consider the various values of $P(\chi,N)$ for that $s$? Or vary $s$ and $N$? $\endgroup$
    – Greg Martin
    Commented Mar 6, 2016 at 20:29
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    $\begingroup$ if $\chi$ is a non-principal character and assuming RH then $\ln L(s,\chi)$ is holomorphic for $Re(s)> 1/2$ and the Euler product (your $\lim_{N \to \infty} P(s,N)$) converges to $L(s,\chi)$ there. on $0 < Re(s) < 1/2$, the product diverges everywhere, and on $Re(s) < 0$ it diverges to $|\infty|$ $\endgroup$
    – reuns
    Commented Mar 6, 2016 at 21:37
  • $\begingroup$ @Greg, $s$ is fixed and $N$ varies. $\endgroup$
    – Bertrand
    Commented Mar 7, 2016 at 14:09
  • $\begingroup$ @user1952009 On $0<Re(s)<\frac{1}{2}$ you write the product diverges but does it oscillate ? Is there an inifinty of value with $|P(\chi,N)|$ lower than a fixed constant ? Any reference about the divergence you mention ? $\endgroup$
    – Bertrand
    Commented Mar 7, 2016 at 14:12
  • $\begingroup$ @Bertrand : look at the Dirichlet series $\sum_{i} \chi(p_i^k) p_i^{-sk}$ and see yourself $\endgroup$
    – reuns
    Commented Mar 7, 2016 at 19:21

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